Trditev o singularnosti v končnem času za 3D Navier–Stokesove enačbe

Vir, odlomek za odlomkom

1. u1 FINITE TIME BLOWUP FOR NAVIER–STOKES
2. u2 Abstract. For every positive viscosity, we construct a solution of the three-dimensional
3. u3 incompressible Navier–Stokes equations that starts from rest and develops unbounded velocity
4. u4 in finite time while maintaining uniformly bounded kinetic energy.
5. u5 1.
6. u6 1
7. u7 2.
8. u8 3
9. u9 3.
10. u10 6
11. u11 4.
12. u12 24
13. u13 5.
14. u14 Correcting the base flow to every order
15. u15 45
16. u16 6.
17. u17 Auxiliary torus and separation of oscillatory supports
18. u18 62
19. u19 7.
20. u20 Oscillatory realization and correction of the residual stress
21. u21 73
22. u22 8.
23. u23 Compactly supported mean corrections
24. u24 88
25. u25 9.
26. u26 Residual improvement and the local field
27. u27 100
28. u28 10.
29. u29 Compact forcing and whole-space breakdown
30. u30 116
31. u31 Appendix A.
32. u32 Matching radial moments and constructing the heat exterior
33. u33 126
34. u34 Appendix B.
35. u35 Analytic profiles near the axis and their continuation
36. u36 144
37. u37 Appendix C.
38. u38 157
39. u39 References
40. u40 165
41. u41 We consider the three-dimensional incompressible Navier–Stokes equations. A fundamental question about the nature of these equations is whether a solution starting from smooth data can develop unbounded velocity in finite time while its kinetic energy remains bounded. We construct such a flow with zero initial velocity and a smooth force compactly supported in space and time.
42. u42 c (R3 × (0, ∞); R3), a compact set K ⊂R3, Theorem 1.1. For every ν > 0 there exist a force f ∈C∞ c and smooth velocity and pressure fields u, p on R3 × [0, c 1) satisfying
43. u43 (1.1) such that supp u(·, t) ∪supp p(·, t) ⊂K for every 0 ≤t < 1,
44. u44 Consequently, there is no smooth solution (u, P) on R3 × [0, ∞) with the same force and initial datum whose kinetic energy is uniformly bounded supt≥0
45. u45 This establishes alternative (C) in the Millennium problem statement for Navier–Stokes as stated by Fefferman in [13]. Compact support also yields the corresponding construction on T3 = R3/Z3, establishing alternative (D) in [13]; see Corollary 10.6.
46. u46 1.1. Historical context and previous work. Incompressible and inviscid flows are described by the classical Euler equations [12]:
47. u47 (1.2) The material acceleration Dtu describes the acceleration of a parcel of fluid along its trajectory, due both to the changing flow and the parcel’s changing position. Crucially, the latter contribution (u · ∇)u is nonlinear in the velocity u, creating most of the difficulty in the analysis. The first equation in (1.2) says that this acceleration is related by Newton’s second law to the sum of the negative pressure gradient and the external force. The second equation tells us that the fluid is incompressible: we cannot squeeze the fluid in one direction without simultaneously stretching it in another.
48. u48 The Navier–Stokes equations [18, 21], given in (1.1), add a viscous force that resists shear between neighbouring fluid parcels. The viscous force is linear in u. However, it significantly changes the character of the flows by smoothing velocity gradients. Whether this smoothing is always sufficient to prevent a finite-time singularity in three dimensions lies at the heart of the regularity problem.
49. u49 In 1934, Leray [16] constructed global finite-energy weak solutions of the three-dimensional equations satisfying an energy inequality. Whether solutions starting from smooth data remain smooth was left unresolved. Caffarelli, Kohn, and Nirenberg [5] proved that the singular set of a suitable weak solution has zero one-dimensional parabolic Hausdorff measure, a bound that allows isolated singularities. Escauriaza, Seregin, and Šverák [11] proved regularity for the unforced Cauchy problem under a bounded scale-invariant L∞ t
50. u50 t L3 x
51. u51 x norm.
52. u52 Tao [22] constructed finite-time blowup for an averaged Navier–Stokes equation that retains the energy cancellation of the original nonlinearity. For the original equations, Buckmaster and Vicol [4] used convex integration to establish nonuniqueness among finiteenergy weak solutions. Albritton, Brué, and Colombo [1] subsequently constructed distinct suitable Leray–Hopf solutions with zero initial velocity and the same force, using an unstable vortex in similarity variables. Their force lies in L1 t
53. u53 The methods of Lifschitz and Hameiri and of Friedlander and Vishik [17, 14] describe the evolution of wavevectors and velocity polarizations along a background flow. Craik and Criminale [9] constructed exact finite-amplitude waves on affine background flows, exploiting the cancellation of the wave’s quadratic self-interaction. Further precedents for the wave dynamics include the centrifugal-instability criteria of Leibovich and Stewartson [15] and of Billant and Gallaire [2, 3] and the exact viscous shearing waves of Singh and Sridhar [19]. The use of oscillations to realize a prescribed stress is central to the Euler constructions of Daneri and Székelyhidi [10].
54. u54 Córdoba and Martínez-Zoroa [6] constructed finite-time singularities for the forced threedimensional Euler equations through successive amplification of increasingly concentrated vortex layers. Córdoba, Martínez-Zoroa, and Zheng [8] extended this approach to hypodissipative Navier–Stokes with small positive dissipation orders and forcing in a local well-posedness class. These works established a strategy for singularity formation based on amplification across scales while controlling the regularity of the external force. In their constructions, larger-scale strain amplifies smaller-scale vorticity, with leading self-interactions and feedback on the larger scales suppressed. For the incompressible porous media equation, Córdoba and Martínez-Zoroa [7] constructed singularities from smooth initial data by successive amplification of oscillatory layers, using approximations of increasing order to keep every spatial derivative of the source uniformly bounded. Our construction also exploits
55. u55 dynamical amplification, with a different role for the amplified disturbances: oscillatory pulses generate a mean momentum flux that supplies the missing force on a collapsing background vortex.
56. u56 2. Physical description of the blowup
57. u57 We first describe the underlying physics of the flow realizing Theorem 1.1; we then provide a full sketch of the proof in Section 3.
58. u58 For any incompressible flow u and pressure p, we can always define the external force f to be the residual in (1.1). The Navier–Stokes equations then hold by construction. The challenge is to choose a flow that blows up while this residual remains smooth.
59. u59 The individual terms in the momentum residual can diverge, but we must arrange sufficient cancellation that their sum and all its derivatives extend smoothly through the singular time.
60. u60 Specifically, we construct a vortex whose leading profile is self-similar, with radial width decreasing faster than axial length. Its central flow combines inward spiralling with axial outflow. As the core contracts, the azimuthal and axial velocity scales increase, strengthening the shear in the surrounding annulus.
61. u61 In the inner core, the profile equations impose the leading momentum balance. Joining this flow to a smooth exterior whose speed decreases with radius leaves a nonzero residual in the intervening annulus. For the background alone, the momentum residual becomes unbounded as t ↑1, so it cannot supply the smooth force required by the theorem. We add spatially oscillatory pulses whose nonlinear momentum fluxes cancel the singular part of this residual. Further corrections remove the remaining singular errors, leaving a force that extends smoothly through t = 1.
62. u62 2.1. Inner core. We construct our flow so that the singularity forms at the spatial origin at time t = 1. We use cylindrical coordinates about the vertical axis:
63. u63 where r is distance from the axis, θ is angle around it, and z is height.
64. u64 The velocity components ur, uθ, and uz describe motion toward or away from the axis, around the axis, and along it, respectively. The leading flow u(0) is axisymmetric: its components depend on r, z, t, but not on θ.
65. u65 In the central part of the inner region, fluid spirals inward toward the axis and flows upward and downward on opposite sides of a dividing layer close to z = 0 (Figure 1). Farther above and below this central region, the inner profile also contains radial outflow. Pressure decreases toward the axis, supplying the leading centripetal force. Its axial gradient enters the axial momentum balance together with transport and viscosity. Near the axis, the axial pressure force points toward the plane z = 0.
66. u66 Radial inflow contributes to spin-up by carrying angular momentum toward smaller radii. For a fluid particle experiencing no torque, the angular momentum per unit mass ruθ is conserved, so moving inward increases uθ. In the actual core, viscosity transports angular momentum outwards; the increasing characteristic speed reflects the balance between inward transport and viscous loss. Incompressibility prevents the incoming fluid from accumulating near the axis: the axial outflow carries it away, allowing the inward motion and spin-up to continue.
67. u67 To leading order, the core evolves self-similarly: its velocity profile has a fixed shape when distances and velocities are measured in their respective time-dependent scales. The radial
68. u68 Figure 1. The central part of the inner core at successive times t1 < t2 < t3 <
69. u69 1. In this region, fluid spirals inward and flows axially on opposite sides of a
70. u70 dividing layer close to z = 0. As the singular time approaches, the region of
71. u71 intense flow shrinks and the velocity increases. Its radius shrinks faster than
72. u72 its height; this difference is exaggerated in the schematic.
73. u73 and axial scales shrink at different rates. Writing τ = 1 −t for the time remaining before the singularity, we have
74. u74 where h is fixed and ≍denotes bounds above and below by positive factors independent of τ. Both lengths tend to zero, while ℓr/ℓz ≍τh →0: the core becomes an increasingly slender column concentrated at the origin, with volume of order τ3/2−h.
75. u75 The characteristic velocity magnitudes of the leading flow satisfy
76. u76 These are typical scales across the core; in particular, the azimuthal velocity vanishes on the axis. The flow is smooth for every t < 1, but its largest speeds become unbounded in a shrinking neighbourhood of the origin as t ↑1. The total kinetic energy of the core is of order τ1/2−3h, which tends to zero despite the increasing speeds.
77. u77 The different velocity scales give different Reynolds numbers. For a speed U, length ℓ, and viscosity ν, the Reynolds number Uℓ/ν compares the viscous diffusion time ℓ2/ν with the time ℓ/U for fluid motion over that distance. Using the core radius as the length scale, the characteristic angular and radial Reynolds numbers are
78. u78 for fixed ν > 0.
79. u79 The angular Reynolds number grows without bound: fluid makes increasingly many turns during a radial diffusion time. The radial Reynolds number remains bounded, so viscosity continues to compete with radial inflow. The corresponding transport and diffusion
80. u80 Figure 2. Pulse geometry and perturbation velocities. Left: At fixed height,
81. u81 pulse envelopes occupy complete rings; the close-ups show radial and az-
82. u82 imuthal perturbation velocities. Right: A radial–axial section, where each
83. u83 localized patch is a cross-section of a complete ring; the close-ups show radial
84. u84 and axial perturbation velocities. Colors distinguish the two families. The
85. u85 specific pulse arrangement shown is schematic; active locations, spacing, and
86. u86 rates satisfy
87. u87 Thus radial diffusion and transport by the radial and axial flow remain in the leading balance as the core contracts. Axial diffusion is weaker: the ratio of axial to radial diffusion rates is
88. u88 The core must also provide enough shear at its outer edge to amplify the pulses placed there. Away from the middle plane, axial flow carries angular momentum from more rapidly rotating layers, supporting a steep radial decrease in angular velocity. With exact reflection symmetry, this transport would vanish at z = 0. Symmetry would also impose uz(r, 0, t) = 0, leaving no radial shear of the axial velocity to compensate. We therefore choose a slightly asymmetric axial profile, with a small upward bias and nonzero velocity at z = 0. This separates the layer where rotational amplification is weak from the layer where axial shear vanishes. The radial shear of axial velocity then supplies the additional amplification needed near the middle plane; farther away, the rotational mechanism already suffices. 2.2. Annulus and pulses. At the edge of the core, there is an annular region where the inner vortex joins onto an exterior flow. The background satisfies its leading momentum equations inside the core, but this transition leaves an imbalance in the annulus. The external force needed to sustain the background becomes unbounded as t ↑1, so it cannot serve as the smooth external force required by the construction.
89. u89 We therefore perturb the velocity so that the fluid’s own motion supplies the missing momentum transport. Through the nonlinear advection term, the perturbation produces additional momentum fluxes whose spatial variation acts as an internal force on the background. We require this contribution to cancel the singular part of the background residual, leaving only a remainder that extends smoothly through t = 1. These internal forces redistribute momentum within the fluid.
90. u90 Our perturbation consists of a sequence of spatially oscillatory pulses that are localized in radius, height, and time, but extend around a complete ring. An exponentially small external force seeds each pulse; the background shear supplies its subsequent growth. Their cylindrical velocity components have zero angular average, but their momentum fluxes need not. For example, outward motion carrying an azimuthal velocity surplus and inward motion carrying a deficit both increase outward angular momentum flux. Reversing both velocity components leaves their product unchanged. Axial momentum is transported in the same way. We arrange the net transport into or out of each region to supply the contribution missing from the background’s momentum balance.
91. u91 The pulses grow by extracting energy from the background shear. A simple example illustrates how this amplification occurs. In a rotating flow, an azimuthal velocity surplus produces outward acceleration. If angular velocity decreases sufficiently rapidly with radius, this outward motion increases the surplus relative to the surrounding fluid. The two effects reinforce one another, giving exponential growth when the amplification exceeds viscous damping. In the full flow, axial shear also contributes to the amplification.
92. u92 The same shear also changes the oscillation’s spatial pattern: fluid at different radii carries it along at different speeds. We choose the pulse orientations so that this progressively shortens the radial wavelength. The changing orientation weakens the amplification, while shortening the radial wavelength increases the total squared wavenumber and hence viscous damping. We choose the initial wavelength so that amplification dominates at first, but damping overtakes it later. The tails are sufficiently small that the cutoff errors and all their derivatives vanish at the singularity.
93. u93 We combine two families of pulses with different ratios of radial angular-momentum flux to radial axial-momentum flux. The background profile and oscillation directions are chosen together so that the averaged quadratic velocity products of the two families supply both components of the required leading stress. We arrange the pulses on successively finer spatial and temporal scales as the singularity approaches. Further corrections cancel the remaining singular terms in the momentum residual, giving a flow whose velocity becomes unbounded while its residual force extends smoothly through the singular time. 2.3. Exterior flow and localization. Beyond the annulus, the flow is purely azimuthal and independent of height: fluid circles the axis, with speed decreasing as the radius increases. Pressure supplies the centripetal acceleration, while the azimuthal velocity evolves by viscous diffusion. We choose this outer flow to satisfy the radial heat equation for the azimuthal velocity exactly, and call it the heat exterior. Its momentum residual vanishes, so no external force is needed to sustain this part of the local flow.
94. u94 At every fixed positive radius, the exterior velocity and pressure have smooth limits, together with all their derivatives, as the singular time approaches. Together with the regularity of the inner construction away from the origin, this allows us to cut off the flow smoothly in space. The additional force produced by this cutoff remains smooth through the singular time, while the cutoff leaves the concentrating core unchanged.
95. u95 The physical description in Section 2 leads to a precise cancellation problem. We must construct a velocity and pressure whose momentum residual extends smoothly through the singular time, while the velocity becomes unbounded. At viscosity one, the residual is
96. u96 R(u, p) = ∂tu + (u · ∇)u −∆u + ∇p.
97. u97 (3.1)
98. u98 The estimates must control every Cartesian space-time derivative of this residual.
99. u99 First we construct the axisymmetric background (uB, pB) of Proposition 5.5, whose residual in (5.41) is the negative divergence of an annular stress, up to a remainder vanishing to every order at the singular point. Oscillatory velocities supply the leading stress through their averaged quadratic products (Proposition 7.5). Successive corrections improve the decay of the remaining residual (Proposition 9.6). We then sum and localize the fields, preserving incompressibility and the leading velocity growth, and extend the residual to the smooth compactly supported force in Theorem 1.1 (Propositions 10.1 and 9.9 and Lemma 10.3). All estimates used below are proved for the viscous equations. The construction is summarized in Figures 5 and 6.
100. u100 For any ν > 0, a solution at viscosity one gives a solution at viscosity ν by
101. u101 The singular time is unchanged. The equation and energy scaling are verified in (10.22)– (10.23). We use ν = 1 for the rest of this outline.
102. u102 3.1. The concentrating leading field. We encode the geometry of Section 2.1 by the fixed velocity profiles E, U and pressure profile Π in (4.3). The similarity coordinates (X, η) in (3.2) resolve the different radial and axial contraction rates and express the velocity growth as powers of a concentration scale. We first specify this representation; the next subsection constructs profiles whose remaining momentum residual can be supplied by the pulses.
103. u103 We write
104. u104 with h fixed in the profile construction. In cylindrical coordinates,
105. u105 The leading field u(0) = u(0) r
106. u106 z ez is axisymmetric: its cylindrical components are independent of θ. The tangential components are the azimuthal and axial components, tangent to cylinders of constant r.
107. u107 We introduce the similarity coordinates
108. u108 (3.2) For τ > 0, eliminating η gives
109. u109 ∂q(q −z2q2h) = 1 −2hη2 ≥1 −2h > 0 Thus q = q(z, τ) is uniquely determined, and q ≍τ + |z|1/D, where ≍denotes bounds above and below by fixed positive factors. On a fixed compact set of similarity coordinates away from η = ±1, we have q ≍τ. For bounded X, letting q ↓0 approaches the singular point. The endpoints η = ±1, with q > 0, describe t = 1 away from that point; see Lemma 4.1.
110. u110 The azimuthal and axial profiles E(X, η), U(X, η) determine the leading fields by
111. u111 Incompressibility and regularity at the axis determine V0:
112. u112 The leading radial pressure balance and the normalization at radial infinity give
113. u113 2X, U, and V0/X are smooth at X = 0, giving a smooth The construction ensures that E/ Cartesian field at the axis for every t < 1. In particular, u(0) θ
114. u114 = 0 on the axis, while E(X, η) > 0 for X > 0.
115. u115 We choose Xc > 0 and 0 < ηc < 1 so that [0, Xc] × [−ηc, ηc] lies in the inner region of Theorem 4.6, and define the core at time t = 1 −τ by
116. u116 Cτ = {(r cos θ, r sin θ, z) : 0 ≤X(r, z, τ) ≤Xc, |η(z, τ)| ≤ηc}. Since q ≍τ there, its radial and axial extents satisfy
117. u117 Both extents vanish; their ratio diverges. The constructed profiles give
118. u118 The axial lower bound follows from the nonzero axis datum in Proposition B.2. For any The fixed axial 0 < X∗< lower Xc, bound the circle follows z = from 0, r = the √2X∗τ nonzero has axis q = datum τ, η = 0, in and
119. u119 The core is a time-dependent region in space; its boundary is determined by fixed similarity coordinates. Incompressibility preserves the volume of material regions transported by the velocity. The radial regions are shown in Figure 3(a).
120. u120 The leading field extends beyond the core. Outside a fixed radial interval in X, U = V0 = 0, while
121. u121 uniformly in η. The exact exterior formula (4.29) gives a purely azimuthal solution of the radial heat equation for the azimuthal velocity. The final spatial localization in Proposition 10.1 gives the completed velocity a fixed compact support.
122. u122 At viscosity one, the local length scales ℓr = q1/2, ℓz = qD give
123. u123 The rotation rate exceeds the radial diffusion rate. In the axisymmetric tangential equations, radial diffusion and radial and axial transport enter the leading balance at the rates
124. u124 Axial diffusion is smaller than radial diffusion by the factor ℓ2
125. u125 z = q2h, which supplies the expansion parameter for the background corrections.
126. u126 Figure 3. The concentrating geometry and the reference pulse.
127. u127 schematic radial–axial section at fixed time: the pulse annulus lies between
128. u128 the level surfaces X = Xa and X = Xb. Horizontal and vertical axes use their
129. u129 respective similarity scales; both physical length scales shrink. (b) A reference
130. u130 amplitude P(v), normalized to one at the midpoint of a pulse interval of
131. u131 length Ls in v, grows and then decays. The shaded regions contain the small
132. u132 cutoff tails. The spatial radii and pulse parameters shown are representative;
133. u133 the proof chooses them by the estimates in the profile and pulse sections.
134. u134 3.2. Construction of the background stress. We now choose the profiles E, U so that the pulses can supply their remaining momentum transport. The resulting stress T, defined by the radial integrals below, must vanish outside the pulse annulus and be a positive combination of the fluxes carried by the two wave families. The profile construction in Theorem 4.6(ii) gives this support; the admissible stress cone condition characterized in Lemma 4.5 ensures the positive representation of Proposition 7.5.
135. u135 We denote by R(0) θ
136. u136 the tangential momentum residuals of (u(0), p(0)), retaining radial viscosity and omitting axial viscosity. We construct physical stress components T = (Trθ, Trz) satisfying
137. u137 Regularity at the axis fixes their integration constants:
138. u138 We require T to be nonzero precisely in the annulus Xa < X < Xb, or
139. u139 2qXb, with 0 < Xa < Xb. We make the residual zero in the inner region and in the heat exterior, and impose
140. u140 These identities make the stress zero beyond the exterior radius as well as near the axis. The prescribed exterior moment identities give these zero integrals (Lemma A.8). Matching the profiles and the five radial integrals defined in (4.15) preserves the exterior fields (Lemma 4.4).
141. u141 The admissible stress cone condition permits the pulse construction to select two families with linearly independent covariance vectors v1, v2 ∈R2 at each point of the annulus. These vectors depend on the local background and give the azimuthal and axial momentum fluxes per unit squared amplitude. They represent the required stress as
142. u142 Thus T lies in the interior of the cone generated by v1, v2. Proposition 7.5 constructs this representation from the admissible stress cone condition on the profiles. Figure 4 illustrates this positive representation.
143. u143 Figure 4. Positive representation of the target stress. The shaded cone con-
144. u144 sists of nonnegative linear combinations of the covariance directions v1, v2.
145. u145 An interior target T = c1v1 + c2v2 has c1, c2 > 0; the dashed lines show vector
146. u146 addition. The coefficients remain positive under sufficiently small perturba-
147. u147 These conditions couple the inner and exterior profiles through the pressure and radial velocity, both of which depend on cumulative radial integrals. We choose the pressure datum, solve near the axis, and match these integrals before imposing the cone condition: 1. Exterior and axis pressure. Proposition A.4 constructs reference profiles with a purely
148. u148 azimuthal tail. Their radial pressure integral (4.31) fixes Π(0, η). Replacing the tail by the
149. u149 exact heat flow of Lemma A.6 changes this integral. The corrections in Proposition A.7
150. u150 restore it at smaller radii, preserving the axis pressure.
151. u151 2. Inner solution. √ With this pressure datum and the specified axis data, Proposition B.2
152. u152 2X, U, Π analytic in X, η near the axis and satisfying (4.13). Their leading
153. u153 tangential residual and stress vanish on X ≤Xa.
154. u154 3. Joining and moment matching. Corollary B.10 connects the inner profiles to the prescribed
155. u155 outer profiles. The correction in Proposition B.8 matches the five cumulative radial
156. u156 integrals governing the pressure, radial velocity, and stress. Equality of the profiles and
157. u157 these integrals at the joining radius restores the prescribed exterior fields by Lemma 4.4.
158. u158 The inner identity (4.13) and the exterior conclusion of Lemma A.8 give T = 0 for X ≤Xa 4. Enforcing the cone condition. On a compact subinterval of the annulus, Proposition C.2
159. u159 introduces a radial oscillation with phase N log X, where N is a large fixed integer.
160. u160 Its amplitude is O(N−1); applying X∂X produces an order-one change in the radial
161. u161 derivatives. This changes the leading tangential shear while changing profile values
162. u162 and radial moments by only O(N−1). The shear follows a periodic curve satisfying the
163. u163 admissible stress cone condition. A separate localized correction restores all five radial
164. u164 integrals exactly. The inner solution and heat exterior are preserved, and the resulting
165. u165 stress admits positive covariance weights throughout the annulus (Proposition C.3).
166. u166 The leading profiles are now fixed. Before adding pulses, Section 5 constructs corrections in successive powers q2nh, n = 1, 2, . . .. At each order, the previously determined coefficients supply the source terms in the coefficient equations. This recursion accounts for the radial momentum equation, axial viscosity, and the remaining lower-order terms. The coefficients are summed with smooth cutoffs on their vector potentials and pressures, equal to one near the singularity and supported in successively smaller neighborhoods. Taking curls after multiplication preserves incompressibility. The resulting smooth axisymmetric background (uB, pB) satisfies
167. u167 The stress Tphys is supported in the annulus and has leading term T. A residual is flat as q ↓0 if every Cartesian space-time derivative is O(qN) for every N ≥0, uniformly on each bounded interval in X. The remainder EB has this property by Proposition 5.5. 3.3. Oscillations and momentum transport. We’ve now arrived at a background (uB, pB) with the required growth and leading annular stress T that the two pulse families can realize. We next construct divergence-free pulses (Lemma 7.7) whose averaged quadratic flux supplies T to leading order (Propositions 7.5 and 9.5). The exact increment identity separates this cancellation from the linear pulse evolution and the interactions left for later corrections.
168. u168 For a divergence-free velocity increment w with pressure increment π, the exact residual identity is
169. u169 R(uB + w, pB + π) = R(uB, pB) + LuB(w, π) + ∇· (w ⊗w), where
170. u170 LuB(w, π) := ∂tw + (uB · ∇)w + (w · ∇)uB −∆w + ∇π.
171. u171 The linear term describes the evolution of the pulses in the background; the quadratic term provides the stress that cancels the leading part of R(uB, pB).
172. u172 We write Awave for the characteristic pulse oscillation amplitude and ℓwave for its wavelength. At fixed nonzero profile values in the pulse annulus, suppressing logarithmic factors and weights that vanish at the support boundary, the construction gives
173. u173 The square of the amplitude has the required stress scale:
174. u174 The last expression is the scale of the averaged radial stress divergence, matching the leading time derivative, transport, and radial viscosity in the tangential momentum equations.
175. u175 Freezing the amplitude and phase gradient at a point gives the schematic local form
176. u176 Here a ∈R3 is a constant amplitude, ξ ∈R3 is the wavevector, and φ ∈R is a phase. The velocity oscillates orthogonally to its wavevector. To separate distinct localized pulses while
177. u177 independent retaining periodic auxiliary averaging, variable we Y construct ∈T2. Its an physical extended value field is ˜︁w(r, θ, z, t, Y) depending on an independent auxiliary variable Y ∈T2. Its physical value is
178. u178 where the fixed phase map Y(r, t) is defined in (6.3). The covariance is computed before this evaluation. We use angular and auxiliary averages
179. u179 Their composition is denoted by ⟨F⟩:= ⟨⟨F⟩θ⟩Y. The averaging conventions are collected in Equations (3.7) to (3.9). Components are taken in the cylindrical basis. Localized pulses with distinct labels whose slow space-time supports overlap receive disjoint supports in Y. Products between such pulses vanish pointwise, including after evaluation at Y(r, t) (Lemma 6.1). Harmonics and corrections with the same label retain their interactions.
180. u180 In the actual construction the amplitudes, phases, and cutoffs vary. We evaluate localized vector potentials at Y(r, t) to obtain Awave, then take the physical curl:
181. u181 All chain-rule terms from Y(r, t) are included. Differentiating the amplitudes and cutoffs produces smaller velocity terms, retained in Lemma 7.7.
182. u182 The chosen phases have nonzero integer angular frequencies, so each wave has zero angular mean and the cosine square has angular average 1/2. The two families produce the momentum-flux directions used to choose c1, c2 > 0 above. These coefficients become squared amplitude weights. The covariance construction, with its physical scaling and curl corrections, gives
183. u183 The higher-order terms include the curl corrections and are smaller by positive powers of q, with derivative bounds proved in Corollary 7.8. The radial divergence of the displayed leading covariance cancels the leading negative stress divergence of the background (Propositions 7.5 and 9.5). The extended residual still contains oscillatory terms and angular means that depend on Y; these are evaluated at Y(r, t) in the physical equation.
184. u184 The same quadratic products govern energy transfer from the background. We consider the homogeneous linearized model
185. u185 Under sufficient spatial decay, integration by parts gives
186. u186 The first term transfers energy between the background and the perturbation; the second dissipates it. The selected transverse velocity amplitudes extract energy from the shear and provide the two momentum-flux directions whose positive span contains T. During each pulse, shear increases the magnitude of the radial component of the wavevector. Viscous damping strengthens and eventually exceeds the amplification, so the pulse grows and then decays. The temporal cutoff acts only in its exponentially small tails. The phase and amplitude equations establish these properties in Lemmas 7.1 and 7.4; Figure 3(b) illustrates the resulting envelope.
187. u187 The correction cycle treats the remaining linear and quadratic interactions, including the curl and cutoff errors in R(uB + w, pB + π).
188. u188 Concentrating axisymmetric background
189. u189 Represent the remaining residual by an annular stress
190. u190 Separate a flat remainder and write the remaining momentum residual as a stress
191. u191 divergence, supported in a cylindrical annulus.
192. u192 Construct two wave families
193. u193 Background shear amplifies the waves; viscous dissipation makes them decay.
194. u194 Choose the squared amplitudes
195. u195 Match the averaged radial flux of tangential momentum
196. u196 Add exactly divergence-free waves
197. u197 Take curls. The divergence of the leading averaged flux cancels
198. u198 Figure 5. Construction of the background and the first oscillatory correction.
199. u199 The averaged pulse flux cancels the leading background stress divergence.
200. u200 The remaining residual is corrected by the cycle in Figure 6. A flat remainder
201. u201 and all curl and cutoff terms are retained.
202. u202 3.4. Correction and summation. The averaged wave flux is chosen to cancel the leading stress divergence associated with T (Propositions 7.5 and 9.5). The remaining angular harmonics, auxiliary means, and radial integral defects require separate corrections. We treat them in a fixed cycle and recompute the full residual after each operation, as in Proposition 9.6, so newly created terms enter the next stage.
203. u203 Starting from (uB + w, pB + π), we reconstruct the pressure and correct the angularly averaged velocity and its radial integral defects. Proposition 9.5 proves that the resulting pair (u[0], p[0]) satisfies the bounds required for the correction cycle. Square brackets count these cycles; the superscript (0) continues to denote the leading field (u(0), p(0)).
204. u204 At stage j, we add a divergence-free velocity increment δuj and a pressure increment δpj:
205. u205 The residual changes according to
206. u206 R(u[j+1], p[j+1]) = R(u[j], p[j]) + Lu[j](δuj, δpj)
207. u207 Here Lv is the linear residual operator defined above, with background v. Thus canceling a selected source also introduces linear remainders and quadratic interactions. The four operations in Proposition 9.6 control these terms as follows.
208. u208 1. We solve the inhomogeneous wave-amplitude equations for the supported nonzero
209. u209 angular Fourier modes. Their principal operator cancels the prescribed source. The
210. u210 remaining linear terms and the interactions of the new pulses enter the next residual 2. We construct signed amplitude increments using the fixed positive leading amplitudes.
211. u211 Their symmetrized cross covariance with the leading pulses supplies the prescribed
212. u212 correction to the averaged stress. Interactions with earlier corrections and the quadratic
213. u213 self-interaction remain as higher-order terms (Proposition 7.6, (9.13)).
214. u214 3. We correct the angularly averaged residual with zero auxiliary average by inverting the
215. u215 fast auxiliary-time derivative (8.20). The axial increment is realized through a vector
216. u216 potential in (8.14), which also supplies its radial velocity. The reconstruction remainder,
217. u217 whose Cartesian space-time derivatives vanish faster than every power of q, is retained in 4. We solve the five radial moment equations in (8.25). Two equations preserve the zero
218. u218 angular-momentum and axial-flux integrals in (9.10). Three cancel the linear contributions
219. u219 to (P, Jθ, Jz), the radial integral defects for pressure and tangential momentum defined
220. u220 in (8.12) and (8.15); their nonlinear remainders have improved decay (8.27). The radial
221. u221 integral constructions keep the velocity and pressure corrections compactly supported. Pressure is reconstructed from the updated radial equation after each operation, and its contribution to the axial equation is included in the new residual. All products use the updated fields, including the curl and cutoff corrections. The background, leading amplitudes, and inverse operators remain fixed throughout the induction.
222. u222 The parameter σj in (9.8) measures residual decay. For Cartesian derivatives of order m, the bound in (9.18) contains the power qhσj−Km, where Km is independent of j, together with logarithmic factors and the separately retained remainder. Constants and logarithmic powers may depend on j, m. Initialization and one correction cycle give
223. u223 The cumulative velocity bounds (9.9), support conditions, and two preserved integrals (9.10) persist, so the cycle can be repeated. All finite stages are defined on the common domain of Lemma 9.7.
224. u224 To sum the corrections, we apply cutoffs to their vector potentials and pressures, then take curls; the direct axisymmetric azimuthal increments receive the same cutoffs. Each cutoff equals one sufficiently close to q = 0, and its support shrinks with the stage. The bounds in (9.17) have a loss in the power of q that depends on derivative order but is independent of the stage. They allow the cutoffs to be chosen so that every differentiated tail satisfies (5.35). The sum is locally finite for q > 0 and defines smooth fields (uloc, ploc) for t < 1. Comparing their residual with that of a fixed finite stage proves (9.20), while preserving the leading velocity growth (Proposition 9.9).
225. u225 The summation gives the required local velocity growth. To pass to the whole space, we also need a vector-potential representation and regularity up to time one away from the origin. Proposition 9.9 proves these properties together with flatness of the residual. In the theorem, we write (u, p) for the local pair (uloc, ploc).
226. u226 Compute the full remaining residual
227. u227 Retain oscillatory interactions, curl and cutoff corrections, and all other error terms.
228. u228 Correct waves, mean flow, pressure, and radial moments
229. u229 Each cycle improves the decay of the residual.
230. u230 Sum the corrections with shrinking cutoffs
231. u231 The residual and all its derivatives vanish to every order at the singularity.⃓⃓
232. u232 Localize and extend the residual as a smooth force
233. u233 The force is smooth and compactly supported; the leading velocity blowup persists.
234. u234 Figure 6. Residual correction and completion, continuing Figure 5. The
235. u235 correction cycle improves the residual exponent σj while retaining all linear
236. u236 and nonlinear terms. The flatness estimate holds uniformly for 0 ≤X ≤X1
237. u237 as q ↓0, for each finite X1. Incompressibility is preserved throughout.
238. u238 We write ∂α
239. u239 t for Cartesian space-time derivatives, where α is a spatial multi-index and b ≥0 is an integer.
240. u240 Theorem 3.1. There are constants 0 < h < 1/100, q∗> 0, 0 < Xa < Xext < ∞, and smooth fields on
241. u241 with the following properties.
242. u242 (i) There is a vector potential A and an axisymmetric scalar B such that
243. u243 (3.3)
244. u244 Both A and Beθ are smooth in Cartesian coordinates at the axis. The field is exactly divergence-
245. u245 (ii) Every Cartesian space-time derivative of A, Beθ, p is uniformly bounded on compact spatial
246. u246 subsets with c ≤q ≤c′ < q∗, for each 0 < c < c′, up to the one-sided boundary τ = 0. The
247. u247 resulting limits of the derivatives are compatible under spatial and time differentiation.
248. u248 (iii) For every α, b, N and every finite X1,
249. u249 (3.4)
250. u250 For X ≥Xext the residual is identically zero, and
251. u251 A = 0,
252. u252 (3.5)
253. u253 Here Hext is smooth on [0, ∞), with each fixed derivative bounded. The radial swirl heat
254. u254 In this statement the second argument of K is τ; when written as K(r, t) below, the same field
255. u255 (iv) For some fixed Xin ∈(0, Xa) and e0 > 0,
256. u256 (3.6)
257. u257 The entries on the left specify cylindrical position and time.
258. u258 3.5. Localization and completion. Finally, using the decomposition in (3.3), we multiply the vector potential A, azimuthal coefficient B, and pressure ploc by c = χxχt. Here χx is smooth and axisymmetric with compact support K, and χt is smooth with χt = 0 on [0, 1 −τ0] for some 0 < τ0 < 1/2. Both cutoffs equal one near (0, 1). Since q ≍τ + |z|1/D, we can choose their supports so that q < q∗/2 wherever c̸ = 0 and t < 1. We take the curl after multiplication and extend by zero outside the cutoff support:
259. u259 The curl term and the axisymmetric azimuthal term are separately divergence-free. Cartesian smoothness of the chosen representatives and the margin from q = q∗give a smooth zero extension. Thus u, p have fixed spatial support in K, vanish on the stated initial time interval, and agree with the local fields near (0, 1) (Proposition 10.1).
260. u260 For t < 1, we set f = R(u, p). In f −cR(uloc, ploc), differentiating the cutoffs produces terms such as (∂tc −∆c)uloc and ploc∇c; the curl correction ∇c × A contributes its derivatives and advection products. The change in self-advection also contains (c2 −c)(uloc · ∇)uloc. These additional terms are supported in the cutoff transition regions, away from a neighborhood of (0, 1).
261. u261 Near (0, 1), the cutoffs equal one, so f is the local residual. Its flatness estimate (3.4) on X ≤Xext, together with the identically zero residual for larger X, gives the bound (10.9) for every Cartesian space-time derivative. On the cutoff transition regions near time one, Theorem 3.1(ii) bounds all derivatives of the actual potentials and pressure where q stays positive. In the remaining region, r stays positive while q →0; there A = 0, and the heat bounds (10.8) and the normalized pressure integral in (3.5) give the derivative limits. These estimates, the uniform flatness bound near the origin, and the fixed support K give compatible limits uniformly on R3 for every derivative of f (Lemma 10.2). Lemma 10.3 realizes these limits from t > 1 with a force supported in K × [0, 2]:
262. u262 The growth path in Theorem 3.1(iv) eventually lies where the cutoffs equal one. Hence the localized velocity retains the asymptotic (3.6) and becomes unbounded as t ↑1.
263. u263 The concentration scales are compatible with bounded energy. The core Cτ has volume of order τ3/2−h. Its leading kinetic energy Ecore and the integral of the squared radial derivatives of its leading velocity, Dcore, have scales
264. u264 The factors are the regional volume and the squared velocity or squared radial derivative. Since h < 1/6,
265. u265 Lemma 10.4 proves the global energy and integrated dissipation bounds for the localized fields directly from their equation.
266. u266 Finally, Lemma 10.5 applies to any smooth solution with the same force and zero initial datum whose kinetic energy is uniformly bounded. It must agree with u on every [0, T], T < 1. A global smooth solution would therefore agree with the constructed velocity for 0 ≤t < 1, contradicting its boundedness on a compact neighborhood of (0, 1) along the growth path. This proves Theorem 1.1.
267. u267 3.6. Notation. All profile choices are fixed before q ↓0. Constants may depend on those choices and on the stated derivative order, but are independent of the concentration scale. The notation O(qa) means a bound by Cqa on the specified domain; uniformity in other parameters is stated when needed. The background expansion order n, correction stage j, dyadic band index ℓ, and Fourier harmonic index are distinct.
268. u268 Averaging conventions. For a scalar field F(r, θ, z, t, Y) on the extended domain, with the auxiliary variable Y ∈T2 = R2/Z2 independent of the physical coordinates, we define
269. u269 (3.7)
270. u270 Thus angular averaging holds the independent auxiliary coordinate fixed, while auxiliary averaging holds all physical coordinates fixed. For vector and tensor fields, these averages act on cylindrical components. Their composition, also called the fast average for wave fields, is
271. u271 (3.8) These averages act before restriction to the physical phase map (6.3). They hold (r, z, t) fixed, or equivalently (R, Z, T) within a fixed normalized chart. The normalized Haar averages agree on the absolute, band, and common auxiliary tori by (6.19). In the wave construction, this composition specifies the entire fast average, including its normalization. For an angular mean coefficient, we define its auxiliary zero-mean part by
272. u272 (3.9) The profile construction also uses a separate auxiliary loop angle θ′ of period 2π: its average is ⟨f ⟩θ′ = (2π)−1 ∫︁2π 0
273. u273 f (θ′) dθ′, as in Lemmas C.1 and 4.11. Bare brackets in that local profile convention refer to this loop average. We also use the radial average
274. u274 (3.10) The auxiliary torus parametrizes oscillations; the torus in Corollary 10.6 is the spatial domain of the periodic solution.
275. u275 Normalized operators and coefficient classes. In a dyadic chart, Q = 2−ℓis fixed, q ≍Q, and The star on a spatial (R, Z, operator T) (r/ includes Q, the τ/Q), factor √Q ε converting S∗= physical spatial differentiation to chart units. Physical differentiation includes the chain rule for the auxiliary phase entiation map. to Using chart units. the radial Physical operator differentiation r of (6.4), we includes put Dr = the √Q chain r, Dz rule = √Q for ∂z the = auxiliary ε∂Z, and
276. u276 Dθ = R−1∂θ, as in (6.6). For a scalar f and a vector a = (ar, aθ, az) in cylindrical components, our conventions are
277. u277 div∗a = (Dr + R−1)ar + Dθaθ + Dzaz,
278. u278 (3.11)
279. u279 Dθaz −Dzaθ, Dzar −Draz, (Dr + R−1)aθ −Dθar The factors normalizing the fields themselves are recorded separately in (8.11). The physical momentum residual is R from (3.1).
280. u280 The classes Wα, Mα, Sα record wave amplitudes, angular mean coefficients, and radial moments in normalized units. A coefficient derivative differentiates the amplitude with its oscillatory exponential factored out, holding the band, label, and harmonic fixed. For each fixed correction stage and coefficient derivative order, ∂I denotes ordinary derivatives in (R, Z, T, H), with H the common auxiliary coordinate of (6.17); for moments, only (Z, T) are differentiated. The defining bounds have the following forms. The first two hold on Xa < X < Xb, and the wave bound is restricted to its labelled rectangle:
281. u281 Here ζ, δ are the radial-edge weights of (6.23), and Pv is the pulse envelope of (7.12), used as a pointwise weight. The constants Cj,I > 0, bj,I, dj,I ≥0 may differ by row and depend on the fixed stage, derivative order, and profile choices; they are uniform in the band, label, rectangle copy, and point. The full definitions, including the support, compatibility, and smooth-extension conditions, are Definitions 6.4 and 6.5. Angular mean coefficients may still depend on the auxiliary torus.
282. u282 Definition 3.2 (Regularity at the axis). A leading field is regular at the axis if its velocity and pressure extend smoothly across r = 0 in Cartesian coordinates, for t < 1. For scalar auxiliary profiles, “regular at the axis” means a smooth extension in (X, η) to X = 0. Definition 3.3 (Parameter smallness and order of choices). For α ≥0 and β > 0, we write α ≪β when α/β is required to be sufficiently small, with the allowed size depending on all data already fixed. In a chain of constants, choices proceed from right to left; small reciprocals specify large parameters. For expressions depending on profile variables, this smallness is uniform on the stated domain and is ensured by those parameter choices.
283. u283 The following table collects the principal symbols used in the construction, with references to their definitions. The different background fields are grouped together. Repeated letters are identified by their role or the section in which they are used. Orders in the coefficient classes refer to normalized representatives, with the weights and derivative bounds specified in their definitions.
284. u284 Table 1: Guide to the principal symbols.
285. u285 Symbol Physical fields and the background construction u, p, f, ν
286. u286 Velocity, pressure, external force, and physical
287. u287 viscosity in the Navier–Stokes equation. The local R(u, p) u(0), p(0)
288. u288 Leading axisymmetric velocity and pressure,
289. u289 including radial, azimuthal, and axial flow. The
290. u290 E, U, V0, Π
291. u291 Leading profiles of swirl, axial velocity, radial
292. u292 ϕ, F, H in the profiles
293. u293 m, (M, I, J, S, Cp)
294. u294 Five cumulative radial integrals preserving
295. u295 pressure, radial velocity, and stress across profile
296. u296 joins; Cp is the pressure increment from the axis.
297. u297 Qs, Ns
298. u298 Normalized radial primitives of the inviscid
299. u299 azimuthal and axial momentum residuals, used s = (a, −bs), ps
300. u300 inviscid-stress vector, with T0 = F(ps −s). Here ts, vs, Pc, Jc
301. u301 Shear direction and strength parameters, and
302. u302 (4.20); Equation (4.21)
303. u303 unnormalized components of ps along and across
304. u304 En, Un, Vn, Πn,
305. u305 Formal background coefficient profiles at order λn = 2nh
306. u306 q2nh. The index n counts background expansion uB, pB
307. u307 Smooth corrected axisymmetric background,
308. u308 Proposition 5.5; (5.41)
309. u309 obtained by summing the coefficients with
310. u310 cutoffs on vector potentials, direct swirl terms, (b, V, G)
311. u311 Radial, azimuthal (swirl), and axial components
312. u312 of the fixed background in normalized chart T0, T = q−A−1/2T0
313. u313 Leading stress profile and physical stress pair, in
314. u314 (4.11); Proposition 4.2
315. u315 (rθ, rz) order. The negative cylindrical divergence
316. u316 of T gives the leading tangential residual,
317. u317 retaining radial viscosity and omitting axial
318. u318 Continued on the next page.
319. u319 Guide to the principal symbols (continued).
320. u320 Symbol Tphys, EB
321. u321 Summed annular background stress and the flat
322. u322 (5.41); Proposition 5.5
323. u323 residual remaining after its negative divergence Σθ, Σz
324. u324 Fixed base stress components in chart units: Sn, An
325. u325 Physical background Stokes streamfunction and
326. u326 its azimuthal vector potential An = (Sn/r)eθ.
327. u327 Taking its curl gives radial and axial velocity.
328. u328 K(r, τ), Hext
329. u329 Exterior swirl heat velocity and its profile:
330. u330 K = r−1−2hHext(τ/r2). The exterior velocity is u[j], p[j]
331. u331 Full fields at finite correction stage j: the fixed
332. u332 background, waves, and mean corrections.
333. u333 Zj = (Aj, Bj, pj) in
334. u334 Physical correction tuple from stage j −1 to j, summation
335. u335 j ≥1: vector potential Aj, direct azimuthal
336. u336 velocity vector Bj, and pressure increment
337. u337 pj = p[j] −p[j−1]. Its velocity increment is A, B, uloc, ploc
338. u338 representation uloc = curl A + Beθ; B is an
339. u339 axisymmetric scalar. The local pair is written u, p after localization
340. u340 Whole-space velocity and pressure obtained by
341. u341 applying spatial and time cutoffs to the local
342. u342 potential representation. Their residual supplies
343. u343 Coordinates, scales, and auxiliary variables (r, θ, z), t, (er, eθ, ez)
344. u344 Physical cylindrical coordinates, time, and unit
345. u345 vectors. The directions tangent to a cylinder are τ, q
346. u346 Remaining time τ = 1 −t and concentration
347. u347 scale q = q(z, t), determined by τ = q(1 −η2), s, X, η
348. u348 Squared radial variable s = r2/2, radial similarity
349. u349 variable X = s/q, and axial similarity variable
350. u350 h, A, D
351. u351 Fixed small exponent h, tangential velocity
352. u352 growth exponent A = 1/2 + h, and axial length
353. u353 Fixed radial decay exponent in the reserved λ
354. u354 Continued on the next page.
355. u355 Guide to the principal symbols (continued).
356. u356 Symbol
357. u357 Dyadic band index, fixed band scale Q = 2−ℓ, ℓ, Q, ε, S∗in charts
358. u358 small parameter ε = Qh, and slow scale S∗= ℓ2.
359. u359 On a band, q ≍Q; these chart parameters are R in profiles; (R, Z, T)
360. u360 The dyadic profile chart, radius (R, Z, is T) R = = (r/√Q, 2X = z/QD, τ/Q). In a in charts u∗, p∗, g∗
361. u361 Normalized physical velocity, pressure, and
362. u362 QA, Q2A, Q2A+1/2, respectively. Moment
363. u363 normalizations also include radial integration.
364. u364 Y, Jg, vr, vt, dr
365. u365 Absolute auxiliary torus variable, its integer
366. u366 covering matrix, radial/time eigendirections, and
367. u367 radial phase exponent. Physical fields are
368. u368 Yi, H, y on auxiliary tori
369. u369 coordinate H = Yi0 for overlapping bands. The
370. u370 mean construction calls that common coordinate r, t; Dr, Dz, Dθ, t∗
371. u371 Physical radial/time derivatives represented on
372. u372 extended fields before auxiliary evaluation, and
373. u373 their normalized spatial/time counterparts in the ∇∗, div∗, curl∗
374. u374 Normalized gradient, divergence, and curl,
375. u375 differentiation of the auxiliary phase map.
376. u376 Xa, Xb
377. u377 Edges of the active stress annulus Xa < X < Xb,
378. u378 Ipos, Imean
379. u379 Disjoint fixed intervals inside the active annulus
380. u380 Theorem 4.6(vi); (4.30)
381. u381 corrections and mean-velocity corrections,
382. u382 The same two patches expressed in r/√q: √ the Jpos, Im
383. u383 Ca(ε), Cb(ε)
384. u384 Fixed profile collars at the two radial annular
385. u385 edges. Here ε denotes their logarithmic width,
386. u386 Waves, amplitudes, and momentum flux γ = (ℓ, a, σ) ∈Γ
387. u387 Wave label specifying a band, slow box, and one
388. u388 of two wave families. This label is distinct from
389. u389 the axial mean-velocity component γ below.
390. u390 Continued on the next page.
391. u391 Guide to the principal symbols (continued).
392. u392 Symbol ηγ, Kγ, Rγ
393. u393 Slow localization cutoff, its support in physical
394. u394 (r, z, t), and the auxiliary support rectangle on k, m, Φ, nΦ
395. u395 Carrier frequency k = ⌈ε−1/2⌉, nonzero integer
396. u396 harmonic, label phase, and its normalized spatial tm, πm, fm
397. u397 Principal transverse velocity amplitude, pressure
398. u398 amplitude, and prescribed source for a harmonic; B, Bℓ, K, AΦ
399. u399 the background shear/rotation matrix and the
400. u400 projected inviscid evolution operator on that
401. u401 Viscous damping coefficient d = εk2|nΦ|2, its d, dref, λ(v) in pulses
402. u402 reference approximation, and the positive
403. u403 reference growth rate in the frame B, with Pv = P(v), v in pulses
404. u404 Pulse envelope and its auxiliary time coordinate.
405. u405 The coordinate v advances at unit speed under t∗.
406. u406 H, y, aσ, W0
407. u407 Covariance matrix, positive squared pulse
408. u408 Proposition 7.5; (7.24)
409. u409 weights, their square-root amplitudes, and the Cm, Am, rm, am
410. u410 Wave-potential coefficient, oscillating vector
411. u411 potential, curl remainder, and full harmonic Was 0
412. u412 0 = wtan 0
413. u413 Assembled leading wave transverse to the phase
414. u414 normal, and the actual wave velocity formed by
415. u415 curls, including their remainders and later wave Wab = ⟨wawb⟩θ
416. u416 Full angularly averaged wave covariance in
417. u417 cylindrical components; its radial off-diagonal
418. u418 entries transport azimuthal and axial momentum.
419. u419 C(w), B(w, v)
420. u420 Doubly averaged radial momentum-flux pair and
421. u421 its symmetric cross term, used to prescribe Σ, LΣ in the wave
422. u422 Prescribed signed covariance increment and its correction
423. u423 linearized principal-wave correction. The base
424. u424 Mean velocity and its compatibility conditions (β, v, γ)
425. u425 Angularly invariant radial, azimuthal, and axial
426. u426 velocity corrections. They may depend on the pm, pw
427. u427 Continued on the next page.
428. u428 Guide to the principal symbols (continued).
429. u429 Symbol gr, Eθ, Ez
430. u430 Radial pressure source and exact azimuthal/axial
431. u431 mean residuals. The radial mean residual is P, Jθ, Jz
432. u432 Three slow compatibility defects: the radial
433. u433 source integral and two momentum-flux integrals Mθ, Mz
434. u434 Angular-momentum and axial-flux moments of
435. u435 the auxiliary-averaged mean correction. Both are γd, Ψ∗, ∆γ
436. u436 vector-potential coefficient, and the actual axial Hθ, Hz
437. u437 Supported covariance targets obtained from the
438. u438 averaged tangential residuals after subtracting
439. u439 Averages and inversion operators ⟨f ⟩θ, ⟨f ⟩Y, ⟨f ⟩, f ◦
440. u440 Angular average, auxiliary Haar average, and
441. u441 Equations (3.7) to (3.9)
442. u442 auxiliary zero-mean part f −⟨f ⟩Y; bare brackets
443. u443 for wave fields denote the combined angular and
444. u444 auxiliary average. These act before evaluation at AX( f ), DX I, J , Ic
445. u445 Phase-following prefix integral, full radial
446. u446 integral, and compactly supported primitive De, Te, Ae
447. u447 Weighted radial derivative, supported inverse,
448. u448 and cutoff remainder: DeTe f = f −Ae f, N−1, N−1 abs
449. u449 Zero-Haar-mean inverse of the auxiliary-time
450. u450 derivative, in chart and absolute auxiliary
451. u451 Weights, coefficient classes, and correction orders ζ, δ, κs
452. u452 Flat radial-edge weight, capped logarithmic
453. u453 distance to the radial edges, and fixed small Wα
454. u454 Normalized wave amplitudes of order εα, with
455. u455 radial Normalized weight wave √ζ, pulse amplitudes envelope, of order and prescribed εα, with
456. u456 Normalized angular mean coefficients of order εα, Mα
457. u457 with radial weight ζ; auxiliary dependence is Sα
458. u458 Normalized radial moments depending on
459. u459 (Z, T), of order εα, with the radial integration
460. u460 Continued on the next page.
461. u461 Guide to the principal symbols (continued).
462. u462 Symbol G[j], F [j]
463. u463 Supported residual part and separately retained
464. u464 flat residual at stage j. Flatness concerns every
465. u465 fixed Cartesian space-time derivative and every σj, Bj, C∗ j
466. u466 Stage decay parameters: σj = 1/5 + j/10,
467. u467 wave and mean residual orders in powers of ε.
468. u468 χ(ajq), aj
469. u469 Cutoffs and their increasing scale parameters
470. u470 shrinking neighborhoods of q = 0. Cutoffs are
471. u471 applied to potentials before taking curls.
472. u472 4. Constructing the leading order flow
473. u473 The concentrating flow of Section 3.1 requires an axisymmetric velocity with prescribed growth near the origin and a heat flow in the exterior. We construct its profiles E, U, Π, which define the leading velocity and pressure (u(0), p(0)) in (4.3). Incompressibility holds exactly. The leading tangential momentum residual is minus the cylindrical divergence of the two-component stress with profile T0, defined in (4.11). This stress must vanish near the axis and in the exterior, leaving a cylindrical annulus on which the waves can act: an interval of the scaled radius X at each fixed axial parameter η.
474. u474 The stress direction is also constrained. As explained in Section 3, it must admit the positive covariance representation of Proposition 7.5. Theorem 4.6 gives profiles satisfying this condition, the required regularity and support conditions, and the exact heat exterior. Their remaining momentum residual is treated in two stages: Section 5 adds higher-order axisymmetric corrections to form the background (uB, pB), and Section 7 supplies waves whose leading covariance supplies the annular stress.
475. u475 We first derive the stress from the profiles, then identify the five cumulative radial integrals in (4.15), whose preservation allows profiles on different radial intervals to be joined without changing the prescribed exterior (Lemma 4.4).
476. u476 Next we describe the permitted stress directions and state Theorem 4.6. We state the required construction results in Section 4.5 and then prove the theorem in Section 4.6. The proofs of those construction results are in Sections A to C.
477. u477 4.1. Similarity variables and radial integration of the residual. We begin by expressing
478. u478 (u(0), p(0)) and their derivatives in similarity coordinates, then recover the tangential stress by radial integration in Proposition 4.2. We use the right-handed cylindrical coordinates (r, θ, z), with θ ∈R/(2πZ), and the exponents A, D from Section 3.1. Along with the relations (3.2), we set d = 1 −η2, L = 1 −2hη2, and s = r2/2:
479. u479 (4.1) The coordinate identities in this subsection hold for 0 < h < 1/2; Theorem 4.6 will choose h < 1/100, as in Theorem 3.1. As noted in Section 3.1, for τ > 0 the similarity relations
480. u480 determine a unique q > |z|1/D. Indeed, q −z2q2h is zero at that lower endpoint, tends to infinity, and has derivative 1 −2hz2q2h−1 = L ≥1 −2h > 0 on the indicated interval. Thus |η| < 1; the parameter endpoints η = ±1 describe one-sided limits. Throughout the profile construction, we write DX f = X∂X f for differentiation in the logarithmic radial coordinate, with η held fixed. To compute the residual of the field described in Section 3.1, we now derive the operators converting time and axial derivatives into derivatives in (X, η). Lemma 4.1. For a smooth profile f and any real b,
481. u481 ∂t(qb f ) = qb−1Tb f,
482. u482 (4.2) Proof. Differentiating the equation for q, followed by η = zq−D and X = s/q, gives
483. u483 The middle entry in the second row uses L −2Dη2 = d. The chain rule proves the claim.
484. u484 □
485. u485 To construct the leading field of Section 3.1 smoothly across the axis, we factor out the linear radial vanishing of its azimuthal component. We introduce a fixed amplitude normalization C > 1 and a smooth scalar ϕ, and write
486. u486 (4.3) The normalization C is chosen in √ the proof of Theorem 4.6. As in Section 3.1, E, U, V0, Π are functions of (X, η). The factor
487. u487 2X accounts for the vanishing of the azimuthal velocity at the axis; smoothness is imposed on ϕ. For the profiles constructed here, regularity at the axis in the sense of Definition 3.2 follows from the conditions
488. u488 F = ϕ/C, U, v0, Π ∈C∞([0, Xc] × [−1, 1])
489. u489 (4.4) for some Xc > 0. Smoothness on this closed rectangle means that all mixed X, η derivatives have continuous one-sided extensions to its boundary. To see the Cartesian smoothness, write x1 = r cos θ, x2 = r sin θ, so that X = (x2 1
490. u490 (4.5)
491. u491 Here all profiles are evaluated at (X, η), and q, η are smooth functions of (t, z) with q > 0 for t < 1. Thus √ the coefficients in (4.5) are smooth functions of (x1, x2, z, t). In particular, E itself contains a
492. u492 X factor; smoothness at X = 0 is imposed on F.
493. u493 We choose E, U; incompressibility determines V0, and the leading radial momentum balance determines Π once its axis value is fixed. For a smooth radial scalar, recall the radial average from the notation paragraph at the end of Section 3; we specify its smooth extension to the axis by
494. u494 (4.6)
495. u495 The incompressibility and centrifugal-pressure balances stated in Section 3.1 give the following explicit profile identities:
496. u496 (4.7) To see the first identity, use A + D = 1 in ∂s(ru(0) r
497. u497 = 0. It gives (V0)X = L−1(2AηU − dUη + 2ηXUX); integration with zero axis value gives the displayed √ formula. For √ the second identity, the coefficients of ∂rp(0) and (u(0) θ
498. u498 2X, with common power q−2A−1/2. The remaining radial terms, and axial viscosity relative to radial viscosity in the tangential equations, carry an additional factor at least q2h. This viscosity comparison explains the powers q2nh used in the background corrections of Section 3. Here these are comparisons at fixed profile coordinates; physical derivative bounds will be proved separately for the completed fields. The higher-order terms just identified are retained in the full residual and treated in Section 5.
499. u499 With incompressibility and the leading radial balance imposed, it remains to express the tangential residual as a radial stress divergence, as required in Section 3. We first integrate its inviscid terms, denoting the resulting scalar profiles by Qs, Ns, and then add the contribution from radial viscosity. The definitions below depend only on E, U, Π; Proposition 4.2 verifies the resulting stress identity.
500. u500 For profiles satisfying (4.7), with ϕ, U, Π smooth at the axis and ϕ > 0, we set
501. u501 (4.8)
502. u502 Here H is the profile of angular momentum ru(0) θ
503. u503 2X from E, and l measures the logarithmic radial slope of H. The coefficients W, Hc collect the transport terms in (4.14). We define Qs, Ns by the following radial equations, selecting the solutions that extend smoothly to X = 0:
504. u504 Sq = −Wl −h(1 −2ηU) −Hc(log E)η,
505. u505 (4.9)
506. u506 Sn = −WDXU −A(1 −2ηU)U −HcUη
507. u507 The integrating factors are XH and X, respectively. Thus the integration constants, which may a priori depend on η, enter as
508. u508 (4.10) The required smooth extensions select CQ = CN = 0. Indeed, H = 2Xϕ/C near the axis, a nonzero constant would produce an X−2 singularity in Qs, or an X−1 singularity in Ns. With both constants zero, changing variables x = sX gives
509. u509 The sources are smooth at X = 0, since l = 1 + XFX/F and (log E)η = Fη/F. These formulas give the smooth extensions Qs(0, η) = Sq(0, η)/2 and Ns(0, η) = Sn(0, η). We define the two
510. u510 stress coefficients in the order (rθ, rz) by
511. u511 (4.11)
512. u512 Thus ps is a two-component vector of integrated inviscid contributions, distinct from the pressure p(0). The vector s = (a, −bs) records the cylindrical radial shear, with its normalization specified by
513. u513 In (4.11), Fps is the inviscid contribution and −Fs is the radial-viscosity contribution, as the calculation below shows. In the notation of Section 3, the physical stress representation to be proved is T = q−A−1/2T0. Its two components are the required rθ and rz entries of the wave covariance, the tensor of averaged quadratic velocity products constructed in Proposition 7.5. To verify the stress identity used in Section 3, we must separate the tangential terms at the leading order from axial viscosity. Define the leading tangential residuals by
514. u514 (4.12) where eθ, ez are the cylindrical unit vectors and R is the full momentum residual defined in the introduction. Adding ∂2
515. u515 removes axial viscosity from that residual; radial viscosity is retained.
516. u516 Proposition 4.2. The residuals R(0) θ
517. u517 in (4.12) are minus the cylindrical divergences ∂r + 2/r and ∂r + 1/r of q−A−1/2T0. In particular the equations for vanishing leading residual stress are
518. u518 (4.13) Proof. The angular equation is simplest in angular momentum ru(0) θ
519. u519 = q−hH. Equations (4.2) and (4.7) give
520. u520 {WDXH + HcHη + h(1 −2ηU)H} = −q−h−1 L
521. u521 (∂t + u(0) r
522. u522 (4.14) The axial material derivative plus the pressure gradient is similarly −q−A−1Sn/L. The integrating factors for the two radial divergences are r2 and r. Integrating the negative inviscid residual from the axis therefore produces the coefficients FXQs/L and FXNs/(LE). Radial viscosity contributes
523. u523 Adding these viscous contributions to the radially integrated inviscid residual proves (4.11), including its sign. If that stress is zero then XQs/L = a, Ns/L = −2UX; inserting these relations in (4.9) proves (4.13). At the axis l = 1 + XϕX/ϕ and (log E)η = ϕη/ϕ are smooth,
524. u524 □ so Qs, Ns are regular at the axis in the scalar-profile sense of Definition 3.2.
525. u525 4.2. The cumulative radial integrals. Note that the stress profile T0 in (4.11) depends on the profiles E, U at every smaller radius through the integrals in (4.10). To join an inner profile to a prescribed outer one, as in Section 3, we must therefore match the accumulated integrals as well as the profile values. Five integrals suffice to preserve the outer pressure, radial velocity, and stress (Lemma 4.4). Integrating (4.9) expresses Qs, Ns in terms of these quantities:
526. u526 (4.15)
527. u527 Each integral is a function of (X, η); at a fixed matching radius it is a scalar function of η. Here Cp is the pressure increment from the axis to X, and Π(0, η) is the prescribed value of the pressure at X = 0.
528. u528 Lemma 4.3. For the solutions of (4.9) with CQ = CN = 0,
529. u529 Qs = −W + (1 −h)I −DηIη XH −dJη + 2(h −D)ηJ
530. u530 (4.16)
531. u531 Ns = −WU + D(M −ηMη) X + 4hηS −dSη Proof. Use ∂X(XW) = 1 −2DηU −dUη to integrate the terms −WDXH and −WDXU. The angular terms −dUηH −dUHη combine to −d(UH)η. For the pressure contribution, integration source integrals by parts are in ΠX = E2/(2X) gives ∫︁X 0
532. u532 0 E2/2 dx. Consequently the source integrals are
533. u533 HSq dx = −XWH + (1 −h)I −DηIη −dJη + 2(h −D)ηJ,
534. u534 Sn dx = −XWU + D(M −ηMη) + 4hηS −dSη + X(4AηΠ −dΠη).
535. u535 □ Division by XH and X gives the result.
536. u536 When two profiles agree beyond a joining radius, matching the five cumulative integrals there also preserves the quantities obtained by radial integration. Their common pressure value at the axis is held fixed. Small changes in the profiles, integrals, and their parameter derivatives also give small changes in Qs, Ns, ps, without requiring radial derivatives of the profiles to be close. We use the exact identity to join profiles and the estimate to alter their shear.
537. u537 Lemma 4.4. Fix h ∈(0, 1/2). For i = 1, 2, let (Ui, Ei) be smooth on (0, ∞) × [−1, 1], with Ei > 0, and suppose the five integrals in (4.15) define smooth functions there, including one-sided η-derivatives at η = ±1. Write
538. u538 Fix one smooth function Πax(η) on [−1, 1] and set
539. u539 Thus the pressures have the same integration constant at the axis; the velocity profiles need not agree near the axis. Define V0,i by (4.7), Qs,i, Ns,i by (4.16), and ps,i, ai, bs,i, T0,i by (4.11). We use the explicit formulas (4.16), previously derived with CQ = CN = 0. For any of these quantities f, let ∆f = f2 −f1. Then the following two conclusions hold.
540. u540 Then m1 = m2, and the corresponding quantities Πi, V0,i, Qs,i, Ns,i, ps,i, ai, bs,i, T0,i agree for all
541. u541 (ii) Fix 0 < X0 < X1 < ∞, an integer k ≥0, and emin > 0. On R = [X0, X1] × [−1, 1], put
542. u542 using the maximum of the component norms for tuples. If Ei ≥emin on R, then
543. u543 (4.17)
544. u544 The constant Ck depends only on k, h, X0, X1, emin, the Ck+1-norm of Πax, and a common bound
545. u545 for ∥(Ui, Ei, mi)∥k+1, i = 1, 2. No radial derivative of a difference occurs in this estimate. Proof. By the definitions of the five cumulative integrals,
546. u546 (4.18)
547. u547 The quantities on the left are evaluated at (X, η), and each integrand on the right at (x, η). For part (i), agreement of Ui, Ei on X ≥Xh makes each integrand in (4.18) zero there. Consequently This is an identity of functions of η, so their η-derivatives agree as well. The common pressure datum gives ∆Π = ∆Cp = 0. Equation (4.7) then gives ∆V0 = 0, and (4.16) gives ∆Qs = ∆Ns = 0. The profiles agree on a radial interval, so their radial derivatives also agree. The definitions in (4.11) therefore give equality of ps, a, bs, T0.
548. u548 For part (ii), the quantities entering (4.16) can be written as
549. u549 It follows that
550. u550 The last identity is the use of the common axis pressure datum. On R, the denominators The satisfy last X identity ≥X0, L is ≥1 the −2h use > of 0, the and common Hi ≥√2X0emin. axis pressure For example, datum.
551. u551 Apply the product rule and these reciprocal identities to (4.16), and then to the formula for ps in (4.11). After at most k η-derivatives, each term contains a difference of U, E, or of
552. u552 a component of m, through order at most k + 1. The remaining factors are controlled by the stated common bounds. This proves (4.17) and the dependence of its constant. These
553. u553 □ formulas contain no radial derivatives of the inputs.
554. u554 Part (ii) applies to the radial modulation in Proposition C.2. The profiles U, E, the five integrals m, and their η-derivatives remain close, even though the radial shears a, bs change substantially. The estimate controls ps; the full stress T0 = F(ps −s) also depends on those shears.
555. u555 To use η), and this comparison define the remaining at a large joining hatted quantities radius ρ, we by set ˆ︁U(x, η) = U(ρx, η) and ˆ︁E(x, η) = E(ρx, η), and define the remaining hatted quantities by
556. u556 (4.19) Here unhatted functions are evaluated at (ρx, η), and hatted functions at (x, η). We put
557. u557 Substitution x. Thus on a into fixed (4.16) rectangle and (4.11) [x0, x1] gives × [−1, the 1], same with formulas 0 < x0 < for x1, ˆ︁Qs, ˆ︁Ns, ˆ︁ps with X replaced by x. Thus on a fixed rectangle [x0, x1] × [−1, 1], with 0 < x0 < x1,
558. u558 where the norms use this fixed x-rectangle. The constant is independent of ρ provided the positive common lower Πax are bound independent for ˆ︁Ei, the of ρ, bounds with h for fixed. ∥( ˆ︁Ui, ˆ︁Ei, ˆ︁mi)∥k+1, and the Ck+1-bound for the common Πax are independent of ρ, with h fixed.
559. u559 The moment corrections used below add finite linear combinations of smooth functions supported on short radial intervals. Their changes to m are linear or quadratic in the coefficients. The invertibility of the moment matrices and the solution of these finite systems are proved in Lemmas A.1 and A.2; at each application we verify the required bounds for the moment discrepancies and inverses.
560. u560 Moment conditions at infinity serve a separate purpose. The quantities M, J, S control, respectively, the radially integrated axial momentum, the axial transport of angular momentum, and the axial momentum flux including pressure. The angular moment I is made convergent by subtracting the prescribed exterior power Hpow before integration, as in (4.28). The resulting total identities make the two weighted integrals of the tangential residuals vanish. Without them, a zero exterior residual could leave stresses proportional to r−2 and r−1 in the angular and axial components: the integrated inner residual may remain nonzero. Lemma A.8 uses the total moment identities to remove these tails and prove that the stress vanishes in the exterior.
561. u561 4.3. The cone imposed on the leading stress. Since the annular stress T0 must be produced by the chosen waves with real amplitudes, it must lie in the positive span of their covariance directions (Proposition 7.5). We now express the required cone condition in the profile variables; Lemma 4.5 gives the equivalent inequalities. At each (X, η), the relevant profile data are the radial shear (a, −bs) and the integrated inviscid contribution ps = (ps,1, ps,2) from (4.11). Where a > 0, we set
562. u562 (4.20)
563. u563 Thus Pc, Jc are the scalar products of ps with the orthogonal, unnormalized vectors (1, ts) and (−ts, 1), respectively, and s = a(1, ts). We define
564. u564 (4.21) For a > 0, the two inequalities in (4.21) are the relaxed cone condition. The admissible stress cone condition consists of these same inequalities together with vs > 2. We use the relaxed condition while joining the axis profile to the outer profile. The additional inequality is required by the viscous waves. Under the admissible condition, Proposition 7.5 constructs positive squared amplitudes whose wave covariances supply the required stress. The next lemma gives an equivalent test and a sufficient condition used in the outer construction of Proposition A.4.
565. u565 Lemma 4.5. Let a > 0, bs ∈R, and ps = (ps,1, ps,2) ∈R2, and define ts, vs, Pc, Jc by (4.20). If vs > 2, the admissible stress cone condition, namely the inequalities in (4.21) together with vs > 2, is equivalent to
566. u566 (4.22)
567. u567 For every nonempty compact set
568. u568 such that, for every (a, bs, w) ∈K,
569. u569 there exists PK > 0 with the following property. For every (a, bs, w) ∈K and every ps,1 ≥PK, setting ps,2 = wps,1 gives the relaxed cone condition (4.21). If also vs > 2, these data satisfy the admissible stress cone condition.
570. u570 Proof. Both conditions in the first assertion imply Pc > 2. On this range, the polynomial 2(Pc −v)2 −(v −2)J2 c
571. u571 For Pc > 2, 2 < v−≤Pc. Hence on 2 < v < Pc the polynomial is positive exactly when v < v−. Since U(Pc, Jc) = v−, this proves the first assertion.
572. u572 For the second assertion, put
573. u573 Then Pc = ps,1c, Jc = ps,1j, and
574. u574 The hypotheses give c > 0 and G := 2c2 −(v −2)j2 > 0 on K. By compactness there are constants cK, γK > 0 and BK ≥1 such that
575. u575 Choose
576. u576 For every ps,1 ≥PK, we have Pc ≥BK + 2 > max{2, v} and
577. u577 If v > 2, the first assertion gives the relaxed condition. If v ≤2, it follows from Pc > 2 and
578. u578 □ U(Pc, Jc) > 2. Thus one threshold works throughout K, including points with vs = 2.
579. u579 In stress coordinates, the cone condition takes the form
580. u580 (4.23) Since F > 0, for vs > 2 the admissible stress cone condition is
581. u581 (vs −2)(T0,z −tsT0,θ)2 < 2(T0,θ + tsT0,z)2.
582. u582 These inequalities are homogeneous in T0. Thus, as the stress tends to zero at an annular edge, its unit direction can retain a strict margin in the cone; Theorem 4.6(iii) specifies this boundary condition. Proposition 7.5 represents the nonzero stress by positive coefficients multiplying the allowed wave covariances. Differentiating this representation gives the signed amplitude changes used for higher-order stress corrections in Proposition 7.6. 4.4. Properties of the leading profile. We seek profiles E, U satisfying both the radial matching conditions of Lemma 4.4 and the stress cone inequalities of Lemma 4.5. At the annular edges the stress T0 must tend to zero smoothly, while its direction remains within the cone. Theorem 4.6 makes these edge conditions quantitative and supplies the regular axis, heat exterior, and moment identities required in Sections 2.1 and 3. It also reserves intervals for the later background and mean corrections in (4.30).
583. u583 To specify the edge estimates, we use inner collar and outer collar for fixed neighborhoods of the annular edges in profile coordinates. For 0 < Xa < Xb and 0 < ε < log(Xb/Xa), they are the rectangles
584. u584 (4.24) Thus at fixed η, an inner collar has 0 ≤log(X/Xa) ≤ε, and an outer collar has 0 ≤ log(Xb/X) ≤ε. Their widths are fixed independently of η and physical q. A smaller collar means one of these sets with a smaller positive ε. When a condition is imposed only inside the annulus, we intersect the collar with {Xa < X < Xb}.
585. u585 A smooth scalar or vector g is flat at the radial edge Xe ∈{Xa, Xb} if, componentwise,
586. u586 The stress will be flat at both edges. Its normalized direction, initially defined only for Xa < X < Xb, will extend to the closed collars with the positive margin in Theorem 4.6(iii). The construction results used in the proof are stated after the theorem.
587. u587 Theorem 4.6. There exist fixed h ∈(0, 1/100), λ > 0, C > 1, and 0 < Xa < Xb, together with profiles E, U, Π for X ≥0, −1 ≤η ≤1, having the following properties. The quantities F, H, V0, Qs, Ns, ps, a, bs and T0 are defined by Equations (4.7) to (4.11); where a > 0, ts, vs are defined in (4.20). We use A, d from (4.1). The closed annulus below means the profile rectangle [Xa, Xb] × [−1, 1] in (X, η). All assertions hold for every η ∈[−1, 1].
588. u588 (i) For every finite R > 0, the scalars F = ϕ/C, U, Π, V0/X are smooth on [0, R] × [−1, 1], with
589. u589 uniform bounds there for every fixed mixed √ X, η-derivative; derivatives at the boundary are
590. u590 2XF > 0 for X > 0. These factors give a smooth
591. u591 Cartesian field across the axis by Equations (4.4) to (4.5). There is a fixed Xan ∈(Xa, Xb)
592. u592 such that on [0, Xan] × [−1, 1] these scalars and their fixed radial derivatives are analytic in η
593. u593 on one common complex neighborhood of [−1, 1]. The inner collar [Xa, Xan] × [−1, 1], in the
594. u594 sense of (4.24), has the positive directional margin in part (iii). The pressure is normalized to
595. u595 (4.25)
596. u596 (ii) The radial pressure balance ΠX = E2/(2X) in (4.7), with its smooth extension ΠX = F2 at
597. u597 X = 0, and the tangential residual identities of Proposition 4.2, with physical stress q−A−1/2T0,
598. u598 hold exactly. The stress profile T0 is zero for 0 ≤X ≤Xa and X ≥Xb, and is nonzero at every
599. u599 Xa < X < Xb. For 0 ≤X ≤Xa, the profiles solve (4.13).
600. u600 (iii) The quantities F, a, vs −2 have positive lower bounds on the closed annulus [Xa, Xb] × [−1, 1].
601. u601 The unit direction n = T0/|T0|, initially defined in its radial interior (Xa, Xb) × [−1, 1],
602. u602 extends smoothly to the closed annulus, including the radial boundaries X = Xa, Xb. There is
603. u603 a fixed κ ∈(0, 2) such that throughout this rectangle,
604. u604 (vs −2)(nz −tsnθ)2 ≤(2 −κ)(nθ + tsnz)2.
605. u605 (4.26)
606. u606 For each η, n(Xa, η) is parallel to (a(Xa, η), −bs(Xa, η)); at X = Xb, n(Xb, η) = (1, 0) and
607. u607 (iv) There is a smooth weight ζ(X), positive for Xa < X < Xb and zero for X ≤Xa and X ≥Xb.
608. u608 Every radial derivative of ζ vanishes at Xa, Xb. Put δ = min{1, log(X/Xa), log(Xb/X)}
609. u609 in the radial interior. There exists a constant c > 0, independent of α, and for every fixed
610. u610 multi-index α = (α1, α2) there are finite constants Cα, mα, such that, with ∂α = ∂α1 X
611. u611 X ∂α2 η
612. u612 η ,
613. u613 (4.27)
614. u614 (v) The limits M(∞, η), J(∞, η), S(∞, η) of the cumulative integrals in (4.15) exist. The angular
615. u615 moment is made convergent by subtracting the exterior power Hpow below from H before
616. u616 integration. These four convergent quantities satisfy
617. u617 (4.28)
618. u618 where c∞> 0 is independent of η. For X ≥Xb,
619. u619 (4.29)
620. u620 The physical swirl there is the exact radial heat solution c∞s−AH(2τ/s). The profile H is
621. u621 positive and smooth for Z ≥0, including its one-sided endpoint at zero. Moreover, for some
622. u622 fixed Xv ∈(Xa, Xb), U = V0 = 0 throughout [Xv, ∞), including the outer collar [Xv, Xb].
623. u623 (vi) Two fixed disjoint intervals Ipos and Imean remain available for later positive-order background
624. u624 corrections in Lemma 5.2 and corrections to the averaged flow in Lemma 8.7, respectively,
625. u625 with sup Ipos < inf Imean. Here positive order means a relative power q2nh, n ≥1, in the
626. u626 background expansion (5.1). Both intervals lie in (Xa, Xv). For every X in either interval,
627. u627 (4.30)
628. u628 with a positive constant cpatch independent of X, η. The construction of the leading profile All constants in the theorem depend only on the fixed profile choices and, for the derivative bounds, on the derivative order. They are independent of physical q and of all later dyadic bands and correction stages.
629. u629 4.5. Construction results used in the proof. Constructing the profile requires three compatible choices: an outer flow that fixes the axis pressure, a regular inner flow that matches its five cumulative integrals, and a shear on the connecting interval that satisfies the admissible cone. We first give the finite-dimensional moment solve in Lemma 4.7, then the outer and inner constructions in Lemma 4.8 and Proposition 4.10, and finally the periodic shear in Lemma 4.11. These results supply the inputs to the proof of Theorem 4.6; their detailed constructions are proved in the appendices at the references below.
630. u630 A parameter derivative in these statements means an η-derivative, with all construction constants held fixed. For functions on [−1, 1], we write ∥f ∥Ck = max0≤j≤k supη |∂j
631. u631 η f |, with componentwise maximum for vectors. Only finitely many such norms must be small when choosing parameters. Once those parameters are fixed, every other fixed derivative order has a finite bound.
632. u632 Lemma 4.7. Let α1, . . . , αm be distinct real numbers, and let β1, . . . , βm be nonnegative, nonzero smooth functions supported in ordered disjoint compact intervals of (0, ∞). Then
633. u633 is invertible. Its inverse and every fixed parameter derivative are bounded for a smooth compact family preserving these hypotheses.
634. u634 More generally, let B(η) be a smooth invertible matrix on [−1, 1], let Qη be a smooth bilinear map, and let d ∈C∞([−1, 1]; Rm). For a fixed integer k ≥0, denote by νj the norm of multiplication by B−1 on Cj, and by qj the bilinear norm of Q on Cj, for j = 0, k. If
635. u635 then the equation has a smooth solution, unique in ∥c∥C0 ≤2ν0∥d∥C0, and ∥c∥Ck ≤2νk∥d∥Ck. When Q = 0, the solution is c = B−1d without any smallness condition.
636. u636 Proof. These are Lemmas A.1 and A.2. For the first assertion, multilinearity gives
637. u637 The determinant in the integrand has constant nonzero sign when x1 < · · · < xm: a nonzero linear combination of m distinct powers has at most m −1 positive zeros, by induction and Rolle’s theorem. Thus the integral is nonzero. For the second assertion, on the ball of radius rj = 2νj∥d∥Cj, the iteration c ↦→B−1(d −Q(c, c)) satisfies
638. u638 It is a contraction in C0 and Ck; iteration from zero selects the same solution in both spaces.
639. u639 □ The pointwise implicit function theorem gives its smoothness.
640. u640 We next prepare the exterior before solving near the axis. Normalizing its pressure to vanish at infinity fixes the pressure datum for the axis problem. The family below includes the heat exterior and reserves intervals on which subsequent modifications can restore the required moments.
641. u641 Lemma 4.8. There are finite parameters Md, P∗, λ, h and a constant R∗> 0 with the following properties. The parameter Md is fixed first; P∗may then depend on Md, λ on Md, P∗, and h on Md, P∗, λ. These choices satisfy
642. u642 For every XR ≥R∗, there are smooth profiles (Uo, Eo) on (0, ∞) × [−1, 1], with Eo > 0, satisfying the conditions below. Put x = X/XR and f (η) = (1 + η2)−1.
643. u643 (4.31)
644. u644 is independent of XR, analytic on one complex neighborhood of [−1, 1], even, and satisfies
645. u645 For this temporary profile, define the five cumulative integrals by (4.15), V0,o by (4.7), and
646. u646 Qs,o, Ns,o by the explicit formulas (4.16). Its shear, stress, and cone coordinates are then given
647. u647 by (4.11) and (4.20). These are the definitions used in Lemma 4.4; they apply on X > 0 even
648. u648 though this temporary inner branch need not be regular at the physical axis.
649. u649 (ii) There are four nonempty ordered disjoint intervals Ij = XR(αj, βj) and radii
650. u650 whose endpoints divided by XR are fixed independently of XR, η. On all four intervals Uo = 0,
651. u651 The heat compensation uses only I2; the other three intervals remain available for corrections.
652. u652 2Xc∞X−A. Both cpatch, c∞> 0 are independent of X, η.
653. u653 (iii) The relaxed cone condition holds on [e−5XR, e1/2Xtail] × [−1, 1], and the admissible condition
654. u654 holds on [Xgood, e1/2Xtail] × [−1, 1], with a positive minimum for each defining gap on its
655. u655 closed rectangle. These minima may depend on XR. For X ≥Xb,
656. u656 where H is defined in (4.29). The physical swirl K = c∞(r2/2)−AH(4τ/r2) satisfies ∂tK =
657. u657 (∂rr + r−1∂r −r−2)K and Kr < 0 for r > 0.
658. u658 Proof. Proposition A.4 and Lemma A.5 supply the reference profiles and the common pressure datum in part (i); the four intervals defined after (A.9) are those in part (ii). Here Md controls the axial decrease, Td its logarithmic radial duration, P∗the initial azimuthal amplitude, and λ, h the intermediate and exterior exponents. We apply Lemma A.6 and Proposition A.7 to replace the tail by the heat profile in part (iii). The correction on I2 restores the pressure integral in part (i), and the total S-moment and renormalized angular moment in part (ii). Those results preserve the cone inequalities in part (iii) for all sufficiently large XR, with the
659. u659 □ other parameters already fixed.
660. u660 The prepared exterior has the correct moments and heat flow. Its stress is defined by integration from the axis, so the outer edge conclusions also require a regular inner extension with the same moments. Under that condition, the stress can be integrated from infinity and its limiting direction can be determined explicitly.
661. u661 Lemma 4.9. The family in Lemma 4.8 can be chosen with the following additional property for every XR ≥R∗. Suppose a leading field is regular at the axis, agrees with that family for X ≥Xtail, satisfies the four moment identities in (4.28) as functions of η, and has the canonical pressure (4.25). Then its physical stress T = q−A−1/2T0 satisfies
662. u662 so T0 = 0 for X ≥Xb. On e1/2Xtail ≤X < Xb, the admissible cone holds and
663. u663 For yb = log(Xb/X), there are smooth functions bθ, bz on a closed outer collar, with inf bθ > 0, such that
664. u664 (4.32) All assertions are uniform in η ∈[−1, 1] for the fixed profile. Their constants may depend on XR, but not on physical q.
665. u665 Proof. These are Lemma A.8 and Proposition A.10. The four moment identities cancel the total residual integrals, giving the backward primitives above. The prepared terminal profile
666. u666 □ then gives the cone inequalities and (4.32).
667. u667 We now construct the missing inner extension. It must be smooth at the axis and match the five cumulative integrals of the prepared exterior at a specified joining radius. We use the function Π0(η) from Lemma 4.8 as the pressure value at X = 0.
668. u668 Proposition 4.10. Fix the outer-profile data of Lemma 4.8 and any prescribed finite collection of η-derivative orders. One can choose finite axis parameters j0, δ∗, σ∗, Λ, a logarithmic transition length Tsh, and an amplitude threshold C0 such that every C ≥C0 admits the following construction. Set
669. u669 Xa = 4/Λ, After C, choose a positive activation width t1 in the logarithmic coordinate log(X/Xa), √ a positive shear factor κ0, and the remaining transition widths. There are profiles E =
670. u670 2X ϕ/C, U, and Π = Π0 + Cp on [0, Xh] × [−1, 1] with these properties.
671. u671 (i) The scalar ϕ is positive, and ϕ, U, Π, V0/X are smooth on [0, Xh] × [−1, 1], including X = 0.
672. u672 Thus Equations (4.4) to (4.5) give a smooth Cartesian velocity and pressure across the axis.
673. u673 For some Xan ∈(Xa, Xh), these scalars and every fixed radial derivative are analytic in η on
674. u674 one common complex neighborhood throughout [0, Xan] × [−1, 1]. The profiles solve (4.13) for
675. u675 (ii) The inequalities a > 0, Pc > 2, and vs < U(Pc, Jc) hold on (Xa, Xh]. On the first collar
676. u676 (Xa, Xan], one also has vs > 2. At its inner endpoint, a(Xa, η) > 0 and vs(Xa, η) > 2 + cex
677. u677 for a constant cex > 0. Writing ya = log(X/Xa), the first collar has the factorization
678. u678 (4.33)
679. u679 The functions ga, Ba are smooth through ya = 0, and for every fixed multi-index α there are
680. u680 All assertions in these two items hold for every η ∈[−1, 1].
681. u681 (iii) Near Xh, the fields equal U0 = 4η, E0 = P∗f x1/10, and m(Xh, η) = m0(Xh, η). Here
682. u682 m0 = (M0, I0, J0, S0, Cp,0) denotes the five integrals of this reference pair from zero. These are
683. u683 equalities of functions of η, and both pressures use the same prescribed value Π0(η) at X = 0. Thus XR may exceed any further prescribed lower bound without altering the earlier choices of Λ, Tsh. Proof. We first choose axis data for Proposition B.2, then use Proposition B.3 and Corollary B.10 to continue the profile and match its moments. Set U∗= 4η + j0, where 0 < j0 ≤.05 is the small axis-velocity offset, and define
684. u684 The pressure bounds in Lemma 4.8 imply that Z∗is positive at the unique zero of H∗. Choose a threshold δ∗> 0, then a regularization parameter σ∗> 0, so that
685. u685 These choices specify the positive analytic axis value
686. u686 The parameter Λ sets the radial scale Y = ΛX. Propositions B.2 and B.3 provide the stressfree solution through Y = 4.1 and the exit inequality ps,1 + p2
687. u687 s,2/ps,1 > 2 + cex at Y = 4. Proposition B.5 and Corollary B.6 give the axis and collar regularity in part (i) and the first-collar assertions in part (ii); κ0 is the factor reducing the reference shear, and t1 is the width of its initial transition in log(X/Xa).
688. u688 For part (iii), we compute the reference integrals m0. In the normalization
689. u689 direct integration of the reference pair gives
690. u690 (4.34)
691. u691 The first continuation supplied by Proposition B.5 ends at
692. u692 The next transition, (B.34), holds U = Gi while matching E to the azimuthal reference profile. The value U = Gi is retained until the restoration interval below.
693. u693 To specify a matching tolerance before C, put R = [e−8, e−5], and take minima and maxima over R × [−1, 1]. The reference fields have Qmin = min Qs,0 > 0, e∗= min E0 > 0, and
694. u694 finite w∗= 1 + max |Ns,0/(E0Qs,0)|. Here restoration means changing U = Gi(η) to 4η on e−8 < x < e−7, followed by the five moment corrections on e−6 < x < e−5. At the entrance prescribed x = e−6 to finite the correction order k, choose interval, write > 0 so ∆ˆ︁m that = ˆ︁m −ˆ︁m0, using (4.19) with ρ = XR. For the prescribed finite order k, choose εm > 0 so that
695. u695 ∥Gi −4η∥Ck+1 + ∥∆ˆ︁m(e−6, ·)∥Ck+1 < εm is sufficient for restoration and correction to satisfy
696. u696 Consequently vs = a + b2
697. u697 for every sufficiently large XR. Hence the relaxed condition holds throughout restoration and moment matching with one fixed tolerance, as required in part (ii).
698. u698 For each required order k, Lemma B.7 gives
699. u699 where ωfin is the sum of the two final widths in log X: the width for setting DXU = 0 and that for interpolating a to 4/5 before X = Xi. It also bounds log(CE(Xi, ·)) independently of the later amplitude, fixing the length Tsh of the transition to E0. Its endpoint satisfies
700. u700 Choose j0, Λ to control the first two errors, then C large, and finally the widths and κ0 small. The bound (B.39) makes the normalized moment error at x = e−6 smaller than the prescribed tolerance; the extra derivative required by (B.35) is included among the prescribed orders. Proposition B.8 restores U0 and solves all five moment equations before x = e−5, preserving the relaxed inequalities above. Direct integration of (U0, E0) gives (4.34),
701. u701 □ completing part (iii).
702. u702 The inner and outer constructions leave an interval on which only the relaxed cone condition is known. We will replace its shear by a periodic family with the same mean, every value of which satisfies the admissible condition. To measure its distance from the four constraint boundaries, for a > 0, b ∈R, and p = (p1, p2) ∈R2, define
703. u703 (4.35)
704. u704 a, v −2, c −v, 2(c −v)2 −(v −2)j2)︁ For the profile data (a, bs, ps), positivity of all four components is exactly a > 0, vs > 2, and (4.22). The map Ψ is smooth on {a > 0}.
705. u705 Lemma 4.11. Let I = [X−, X+] ⋐(0, ∞), and let a, bs, ps be smooth on I × [−1, 1], with a > 0. Define ts, vs, Pc, Jc by (4.20). Suppose Pc > 2 and vs < U(Pc, Jc) throughout this rectangle, and vs > 2 in neighborhoods of both radial endpoints. There is a smooth period-one family (aL, −bL)(X, η, φ), φ ∈R/Z, such that
706. u706 0 (aL, −bL)(X, η, φ) dφ = (a, −bs)(X, η).
707. u707 Every loop shear satisfies the admissible cone condition with the given vector ps(X, η) held fixed. More precisely, there is µL > 0 such that
708. u708 (4.36) Near both radial endpoints the loop is independent of φ and equals (a, −bs). Smoothness includes one-sided derivatives at η = ±1; the constant µL may depend on all the fixed input data. Proof. The variance construction in Lemma C.1 supplies a smooth 2π-periodic ratio tℓ(X, η, θ′) and a smooth ρℓ(X, η) ≥0 such that, writing ⟨f ⟩= (2π)−1 ∫︁2π 0
709. u709 It also gives, for every θ′,
710. u710 These are (C.10) and (C.8), together with the cutoff choice (C.9); their construction is smooth also where ρℓ= 0. Near the radial endpoints it has ρℓ= 0 and tℓ= ts. Define the lifted parameter φ, with φ(0) = 0, by
711. u711 (4.37) We verify that the new parameter has period one and that the shear has the prescribed mean in this parameter. Both follow by direct integration:
712. u712 The positive derivative in (4.37) has a positive minimum on the compact parameter set, so the inverse change of parameter is smooth. The resulting shear satisfies −bL/aL = tℓand aL(1 + t2
713. u713 ℓ) = vℓ. Thus Lemma 4.5 gives positivity of every component of Ψ(aL, bL, ps). Their common minimum over I × [−1, 1] × R/Z is a positive number µL, proving (4.36). Where
714. u714 □ ρℓ= 0, formula (4.37) reduces to (aL, −bL) = (a, −bs), proving the endpoint assertion.
715. u715 4.6. Proof of the leading profile theorem.
716. u716 Proof of Theorem 4.6. We first attach the two profiles in Lemma 4.8 and Proposition 4.10, preserving their five cumulative integrals at the joining radius. We then change the shear using Lemma 4.11 on the remaining interval where only the relaxed cone holds, estimate the resulting errors, and use Lemma 4.7 to solve five equations restoring the integrals. All estimates below concern η ∈[−1, 1].
717. u717 Step 1:
718. u718 choose the parameters and join the profiles. Choose the outer parameters Md, Td, P∗, λ, h and Π0 from Lemma 4.8, choosing the family with the endpoint property in Lemma 4.9. Apply Proposition 4.10 with this same function Π0. In that result Λ and the logarithmic transition length Tsh are fixed before the amplitude normalization C. Let R∗be the radius threshold in Lemma 4.8 and C0 a lower bound from the axis construction after these choices. We may take
719. u719 The tolerance εm and the total logarithmic width ωfin are defined in the proof of Proposition 4.10. With the smallness convention in Definition 3.3, the profile parameters may be chosen with
720. u720 sh ≪Λ−1 ≪σ∗≪δ∗≪j0 ≪εm ≪h ≪λ ≪P−1 ∗
721. u721 The second line continues the choices in the first.
722. u722 In particular, Md, P∗are large, λ, h, εm, j0, δ∗, σ∗are small, Λ, Tsh, C are large, and the three final transition parameters are small. The exact derived scales and the additional amplitude requirement remain
723. u723 After these choices, we fix the periodic shear and the correction bumps in Steps 2–3. We then choose N−1 ≪ωfin: with the same convention, take N sufficiently large after fixing ωfin, those functions, and all constants in the estimates below. Thus every radial scale and width is fixed before the final integer N.
724. u724 Put Xh = XRe−5. Write (Uin, Ein) for the axis connection and (Uout, Eout) for the prepared outer pair, and define
725. u725 Both definitions equal (4η, P∗(1 + η2)−1(X/XR)1/10) on a neighborhood of Xh, so U0, √ E0 are smooth there. These are the profiles before the shear modification in Step 2. Set H0 =
726. u726 2XE0, and define m0 = (M0, I0, J0, S0, Cp,0) from U0, E0 by (4.15). Retaining the prescribed axis pressure Π0(η), put Πprof 0
727. u727 (X, η) = Π0(η) + Cp,0(X, η). Compute V0, Qs,0, Ns,0, ps,0, a0, bs,0, T0,0 from U0, E0, Πprof 0
728. u728 by (4.7), (4.16), and (4.11); write p0 = ps,0, b0 = bs,0, and s0 = (a0, −b0). The five equalities in Proposition 4.10(iii) give
729. u729 For X ≥Xh the five integrands also agree, and hence
730. u730 m0(X, η) −mout(X, η) = m0(Xh, η) −mout(Xh, η)
731. u731 By Lemma 4.4(i), the pressures and the coefficients V0, Qs,0, Ns,0, p0, s0, T0,0 agree with the outer ones on this interval. In particular the pressure normalization and all four total moment identities in Lemma 4.8 hold for the joined pair. The angular identity follows from
732. u732 0 (H0 −Hout) dX = I0(Xh) −Iout(Xh) = 0.
733. u733 The joined pair is regular at the axis, so the conditional stress conclusions of Lemma 4.9 now apply. It has zero stress outside [Xa, Xb], the stated inner and outer edge factorizations, and the admissible cone near both edges. It satisfies the relaxed cone throughout the intervening interval.
734. u734 We specify the interval that still needs a shear change. Let I1 be the first reserved correction interval from Lemma 4.8, and choose a closed interval I = [X−, X+] ⊂(Xa, inf I1) containing
735. u735 every point of (Xa, Xb) where the joined pair may fail the admissible condition. Choose its left endpoint strictly between Xa and the end of the analytic collar, and its right endpoint to satisfy Xgood < X+ < inf I1. Shrink the analytic collar endpoint to a number Xan ∈(Xa, X−). Let Y0 < Y1 lie inside I1, with X+ < Y0, and set J = [X−, Y1]. The original pair is admissible near the ends of I and throughout [X+, Y1]. Apply Lemma 4.11 on I × [−1, 1] to its shear (a0, −b0) and inviscid vector p0. For each modified pair (Uj, Ej), j = N, f, below, define mj by (4.15), set Πj = Π0 + Cp,j, and compute its remaining coefficients by (4.7), (4.16), and (4.11). We abbreviate pj = ps,j, bj = bs,j, and write Vj for its radial-velocity profile. Step 2: realize the periodic shear at a finite frequency. The periodic shear has the required cone direction.
736. u736 To obtain it from actual profiles, we integrate its difference from the original shear and evaluate the resulting antiderivatives at a large radial frequency. Let (aL, −bL)(X, η, φ) be the loop just obtained. Its mean identities give periodic primitives A , B with zero mean, satisfying
737. u737 They vanish near both ends of I because the loop there is constant. Extend them by zero off I, and for an integer N ≥1 define
738. u738 (4.38) In the following shear identities, DXA and DXB differentiate X with φ held fixed; all loop expressions are then evaluated at φ = N log X. The chain rule gives exactly
739. u739 All radii are now fixed and E0 has a positive minimum on J × [−1, 1]. Since the phase has no η dependence, for every fixed k there are constants independent of N such that
740. u740 (4.39) Here and until the end of Step 3, ∥· ∥k takes the componentwise maximum of η-derivatives of orders 0 ≤j ≤k on J × [−1, 1], or on the indicated subinterval of J . The second bound is on I; the shear is unchanged on J \ I.
741. u741 Write uN = UN −U0, eN = EN −E0, and ∆mN = mN −m0. There is no change below X−. For X ∈J , the five exact increment formulas are
742. u742 The interval is fixed and bounded away from zero. The product rule and (4.39) therefore give
743. u743 Use Lemma 4.4(ii) with the common axis value Π0(η), taking N large enough that EN ≥ 1 1 2 minJ ×[−1,1] E0. It yields
744. u744 ∥pN −p0∥k ≤Ck(Ak+1 + Dk+1)/N =: Pk/N.
745. u745 (4.40) Thus the large radial shear change has only a small effect on the integrated vector ps; the estimate uses no smallness of radial derivatives of EN −E0 or UN −U0.
746. u746 We have controlled the change in ps, while the shear follows the prescribed loop to order N−1. It remains to show that these errors fit within the loop’s strict cone margin. Use the map Ψ in (4.35). With T = R/Z, let
747. u747 Both minima are positive by Lemma 4.11 and the choice of I. Fix a compact neighborhood of the values of (aL, bL, p0) and (a0, b0, p0) over the two displayed domains, on which a > 0, and let CΨ be an upper bound on the Lipschitz constant of Ψ on that neighborhood. By (4.39) and (4.40), for all sufficiently large N the perturbed triples lie in that neighborhood and satisfy, componentwise,
748. u748 Ψi(aN, bN, pN) ≥µL −CΨ(B0 + P0)/N ≥µL/2
749. u749 (4.41)
750. u750 Ψi(aN, bN, pN) ≥µR −CΨP0/N ≥µR/2 These four positive quantities assert aN > 0, vN > 2, and (4.22). The modulated pair is therefore admissible throughout J . Its five cumulative integrals still differ from the original ones by O(N−1); we now restore them exactly.
751. u751 Step 3: solve the five moment equations. On (Y0, Y1) ⊂I1, modulation has not changed the profiles, and
752. u752 Choose two nonnegative nonzero smooth functions β1, β2 with ordered disjoint compact supports in (Y0, Y1), and three such functions γ1, γ2, γ3. Set
753. u753 ∑
754. u754 ∑ For Uf = UN + uc, Ef = EN + ec, the exact changes at Y1, ordered as (M, J, I, S, Cp), are
755. u755 (4.42)
756. u756 All five integrals are over (Y0, Y1), and dN is −∆mN(Y1, η) in the same row order. The linear map has two diagonal blocks:
757. u757 The weights in BU have exponents 0, −λ, and those in BE have exponents 1/2, −1/2 − λ, −3/2 −λ. They are distinct since λ > 0. By Lemma 4.7, both blocks are invertible,
758. u758 smoothly in η. Their inverse bounds are finite because λ, the radii, and all bump functions were fixed before N.
759. u759 Let νk be the norm of multiplication by B−1 on Ck([−1, 1]), and let qk be the bilinear norm of Q : Ck × Ck →Ck, as in Lemma 4.7. Since ∥dN∥Ck ≤Dk/N, it suffices to choose
760. u760 to obtain a smooth solution of (4.42) with ∥c∥C2 ≤2ν2D2/N. The fixed bump shapes then give
761. u761 Integrating the five correction integrands in (4.42) only up to X ∈[Y0, Y1] gives
762. u762 By the shear formulas and Lemma 4.4(ii), increasing N if necessary gives
763. u763 ∥a f −aN, bf −bN, p f −pN∥0 ≤Crep/N. Here Crep is independent of N. Increase N so that the final triples stay in the fixed compact neighborhood used to define CΨ. Consequently (4.41) remains strict on the repair interval: for N ≥4CΨCrep/µR,
764. u764 Ψi(a f , bf , p f ) ≥µR/2 −CΨCrep/N ≥µR/4 > 0.
765. u765 No profile or cumulative integral before Y0 is affected by this correction.
766. u766 The equations (4.42) give mf (Y1) = m0(Y1). All profile changes are supported in (X−, Y1); thus
767. u767 (4.43)
768. u768 The last two equalities follow from Lemma 4.4(i). Here Vf and V0 denote the radial-velocity profiles computed from the final and joined pairs. All lower bounds on N encountered above are finite and depend only on already fixed data; one finite integer can therefore satisfy them simultaneously. We henceforth write (U, E, Π) for the final profiles and T0 for their stress. Step 4: verify the six conclusions. The final profiles satisfy the cone inequalities, and their five cumulative integrals agree exactly with the joined pair beyond the correction interval. We now use these two facts to verify the regularity, support, and endpoint estimates of the theorem. For the regularity in part (i), the changes in Steps 2–3 have compact support above Xan. The analytic axis rectangle from Proposition 4.10 is therefore preserved. All remaining profile pieces and their joins are smooth. Positivity of E follows from positivity of E0, √ the exponential in (4.38), and the bound for Ef on the repair interval. Near the axis E =
769. u769 2Xϕ/C; there F = ϕ/C, U, Π, V0/X have the regularity asserted in part (i). On any finite rectangle this gives finite bounds for every fixed mixed derivative.
770. u770 The preserved pressure increment satisfies Cp(∞, η) = −Π0(η). Hence
771. u771 Π(X, η) = Π0(η) + Cp(X, η) = Cp(X, η) −Cp(∞, η) = − This proves the pressure normalization in part (i) and the radial balance in part (ii). In particular ΠX = E2/(2X) = F2 extends smoothly to the axis. Proposition 4.2 gives the exact
772. u772 tangential residual identities. The first edge is unchanged, so the stress is zero for X ≤Xa and (4.13) holds there. By (4.43), the outer stress is unchanged and zero for X ≥Xb.
773. u773 On Xa < X < Xb, the cone holds: it was preserved near the two edges and beyond Y1, and was proved on [X−, Y1] in Steps 2–3. In particular, (4.23) gives
774. u774 Thus the stress is nonzero at every interior point, completing part (ii).
775. u775 We next prove the uniform assertions in part (iii). Set ya = log(X/Xa), yb = log(Xb/X). The factors (4.33), (4.32) give the following extensions of n = T0/|T0|:
776. u776 Their denominators stay positive on smaller collars. Since Ba(0, η) = F(Xa, η)s(Xa, η),
777. u777 The smooth functions F, a, vs −2 are positive in the interior; the endpoint estimates in Lemma 4.9 and Proposition 4.10 make them positive also at Xa, Xb. They therefore have positive minima on the closed annulus.
778. u778 Define, on that rectangle,
779. u779 In the interior, (4.23) implies
780. u780 At the two edges, Bn = 0, while An =
781. u781 1 + t2s at Xa and An = 1 at Xb. Thus An > 0 on the closed rectangle, and we may define
782. u782 c /(Pc −vs)2 > 0 in the interior, while Gn = 2 at both edges. By (4.22), Gn = 2 −(vs −2)J2 c Hence An, Gn are smooth and positive throughout. Taking
783. u783 gives (4.26), with κ < 2. This proves part (iii) including its endpoint directions.
784. u784 For part (iv), let ca = t2 1
785. u785 1 > 0 be the inner exponential constant from (4.33), and put
786. u786 This function is smooth and flat at both edges. On a fixed inner collar, the ratio ζ/e−ca/y2 a
787. u787 a is bounded above and below by positive constants; on a fixed outer collar the same holds for ζ/e−4/y2
788. u788 b. The smooth nonzero factors in (4.33), (4.32) therefore give
789. u789 Each derivative of an exponential factor contributes only finitely many inverse powers of ya or yb. On the compact interval between the collars, ζ > 0, |T0| > 0, and all fixed derivatives are bounded. With δ = min{1, ya, yb}, these estimates combine to give (4.27), proving part (iv).
790. u790 Finally, the total moment identities in part (v) survive both exact connections. For the last correction this follows explicitly from
791. u791 (M, J, S) f (∞) = (M, J, S)0(∞) = (0, 0, 0),
792. u792 0 (Hf −H0) dX = If (Y1) −I0(Y1) = 0.
793. u793 Together with Step 1, these equalities give (4.28). The exterior pair remains the heat profile supplied by Lemma 4.8, with the same c∞. The identity
794. u794 q−Ac∞X−AH(2d/X) = c∞s−AH(2τ/s) proves the physical exterior formula. Since Y1 < sup I4 < Xv, equation (4.43) preserves U = M = J = 0 for X ≥Xv with the same Xv as in Lemma 4.8. Equation (4.7) yields
795. u795 The radius Xv precedes the terminal collar and lies after all four reserved intervals. This proves part (v). For part (vi), the heat compensation used I2, and Step 3 used only I1. Thus Ipos = I3 and Imean = I4 retain the exact profile (4.30), proving part (vi). All choices were made in profile coordinates before physical q or any later correction stage is introduced, as
796. u796 □ required.
797. u797 The profiles now give the leading field from Section 3 with its prescribed annular stress. The next section corrects the higher-order residual terms identified after (4.7).
798. u798 5. Correcting the base flow to every order
799. u799 Near the axis, the leading profiles (E, U, V0, Π) of Theorem 4.6 satisfy the balances in Equations (4.7) and (4.13), but axial viscosity and the remaining radial terms still contribute to the residual. We add axisymmetric corrections to obtain the background (uB, pB) described in Section 3 and Proposition 5.5. The velocity uB is exactly divergence-free, and its momentum residual is minus the cylindrical divergence of a physical annular stress Tphys, as in (5.41), plus an error EB bounded by every power of q as q ↓0. The same bounds hold for every fixed physical derivative on each compact profile range. The annular stress is the input to the wave construction in Section 7.
800. u800 These remaining terms enter at successive powers of q2h in Equations (5.3) to (5.6). At each order, Lemma 5.1 solves for a correction near the axis, and Lemma 5.2 extends it radially and imposes the five integral conditions that keep its stress supported in the prescribed annulus. The induction in Proposition 5.3 completes the current order before using its coefficients to construct the next order.
801. u801 The leading profiles and their construction constants remain fixed throughout. The resulting coefficient sequence is the formal expansion (5.1): convergence of the unmodified infinite series is not asserted. After constructing this sequence, Lemma 5.4 produces smooth fields for q > 0 with the same asymptotic coefficients. The proof of Proposition 5.5 checks that summation preserves incompressibility, the exterior velocity, and the required residual estimates.
802. u802 5.1. Higher coefficients near the axis. We first solve the coefficient equations Equations (5.2) to (5.6) near the axis, where the stress must remain zero. For n ≥0, we set λn = 2nh, E0 = E, U0 = U, and Π0 = Π. Each coefficient profile is a function of (X, η). The subscript zero now denotes expansion order: Π0(X, η) is the entire leading pressure profile, whose trace
803. u803 Π0(0, η) is the axis pressure function defined in (4.31). We use the similarity variables of (4.1) and write the physical coefficients as
804. u804 uθ,n = q−A+λnEn = rq−A−1/2+λnϕn/C,
805. u805 (5.1) Here ϕn, rather than En, is the azimuthal coefficient that is smooth at X = 0. The radial average is the operator from (4.6), applied at fixed η: AX(Un)(X, η) = X−1 ∫︁X 0
806. u806 0 Un(x, η) dx.
807. u807 The spacing 2h in (5.1) accounts for the two lower-order effects identified after (4.7). We recall A = 1 2
808. u808 1 2 −h from (4.1), so 1 −2D = 2A −1 = 2h. First, the axial derivative identity 2 in (4.2) gives 2 the factor q−2D = q−1q2h for two axial derivatives. The radial relation X = r2/(2q) in (4.1) gives q−1 for two radial derivatives. Thus axial viscosity applied to order n −1 enters the angular and axial equations at order n; these are the order-n −1 viscous terms in Equations (5.3) and (5.4) below.
809. u809 Second, we recall the leading radial balance r∂rp(0) = (u(0) θ
810. u810 θ )2 underlying the pressure identity in (4.7). With the physical powers in (4.3), both sides have power q−2A. The other terms in the radial momentum equation multiplied by r start at q−1 = q−2Aq2h, as noted after (4.7). This gives the order shift in the pressure equation (5.5): the remaining radial terms are collected in (5.6) and enter one order later.
811. u811 To include differentiation of the powers of q, we recall the operators Ta, Za from (4.2) and abbreviate The order of composition in the last expression accounts for the change of power after the first axial derivative. Negative field indices mean zero. Incompressibility, with the integration constant fixed by regularity at the axis, is equivalent to
812. u812 X = 2ηUn −2η(D + λn)AX(Un) L −d∂ηAX(Un)
813. u813 (5.2) We fix n ≥1. At this point all coefficients of orders j < n are known, including their radial cutoffs and the integral corrections constructed in Lemma 5.2. We solve for ϕn, Un, Πn on a common inner interval; Vn is determined from Un by (5.2). The pressure coefficient Πn must be solved together with ϕn, Un.
814. u814 We set b = −A −1 2
815. u815 2 and c = −A. Primes in the following system mean ∂X, with η fixed. The equations that cancel the order-n momentum residual on the inner interval are
816. u816 2(Xϕ′′ n
817. u817 (5.3)
818. u818 2(XU′′ n
819. u819 (5.4)
820. u820 (5.5)
821. u821 (5.6)
822. u822 Every occurrence of Ωn−1, ϕn−1, or Un−1 in these source terms uses already constructed coefficients. For example, the pressure equation separates into
823. u823 ∑ The sum is empty when n = 1. Its terms and Ωn−1 are known, while the first term is linear in the current unknown. Splitting the transport sums into (i, j) = (0, n), (n, 0) and 1 ≤i, j < n gives the same conclusion for the other equations. Thus the system is linear at each positive order. The pressure equation will also be imposed when the current profiles are extended beyond the inner interval.
824. u824 The powers of q and the terms arising from the cylindrical basis are obtained as follows. On rqb+λnϕn/C, angular radial transport includes ur(∂r + r−1)uθ and therefore yields Vi(∂Xϕj + ϕj/X). The angular vector Laplacian gives 2(X∂XXϕn + 2∂Xϕn) after removing rqb−1+λn/C. The corresponding axial scalar Laplacian gives 2(X∂XXUn + ∂XUn). Every transport product has the same power because A + D = 1, and axial viscosity increases the order in the q2h expansion by one because 1 −2D = 2h. In the radial equation, multiplication by r places pressure and centrifugal acceleration at power q−2A+λn, whereas the other radial terms start at q−1+λk; hence k = n −1. Finally, applying the radial operator ∂rr + r−1∂r −r−2 to V(X)/r gives 2X∂XXV/(qr), giving the last two terms of (5.6). Since (5.2) gives Vj = Xvj with smooth vj, every term of Ωk is divisible by X. For example, Vi(∂XVj −Vj/(2X)) = Xvi(vj/2 + X∂Xvj). Thus Ωk/X is regular at the axis, including all fixed parameter derivatives.
825. u825 The linear equations must be solvable on one radial interval at every order. This requires control of the η-derivatives in the recursion even as the coefficient bounds grow. The analytic dependence of the leading inner profile supplies that control.
826. u826 Lemma 5.1. There exists an interval 0 ≤ξ ≤a, ξ =
827. u827 X, extending from the axis into the inner collar of Theorem 4.6(i), where the stress cone inequalities hold with a uniform positive margin, on which every positive order has a unique solution of Equations (5.2) to (5.6) with ϕn(0, η) = Un(0, η) = Πn(0, η) = 0. The profiles are smooth in X and, for each fixed radial derivative, holomorphic on a neighborhood of [−1, 1] in the parameter η. That neighborhood and its bounds may depend on the order and the radial derivative; a does not. Uniqueness holds among solutions whose profiles extend holomorphically in η to a neighborhood of [−1, 1], with bounds uniform for 0 ≤ξ ≤a.
828. u828 The zero axis values leave the leading traces of ϕ, U, and Π unchanged by every positiveorder coefficient.
829. u829 Proof. Step 1: write a linear system on the fixed inner interval. The analytic axis solution and its first collar, supplied by Proposition B.2 and Theorem 4.6, have the stated parameter regularity on a fixed radial rectangle. We choose a inside this rectangle, beyond the shorter collar that will remain uncut. At order n, we use the previously constructed lower-order coefficients, including their cutoffs and moment corrections. The corrections are supported to the right of this rectangle, and the radial cutoffs preserve parameter analyticity there. A sufficiently small complex neighborhood Sρ = {η ∈C : dist(η, [−1, 1]) < ρ} therefore contains no zero of L and supplies bounded holomorphic coefficients and sources throughout 0 ≤ξ ≤a.
830. u830 We introduce the auxiliary variable Kn and the vector Wn to write the system as
831. u831 (5.7)
832. u832 ∂ξWn + ξ−1 diag(0, 0, 2, 0, 3, 1)Wn = A0Wn + A1∂ηWn + fn,
833. u833 The matrices A0, A1 are 6 × 6 coefficient matrices in (ξ, η), with n fixed; fn is a known six-component source formed from the lower-order profiles. The additional unknown Kn replaces the radial average by a first-order equation, so the system has no unevaluated radial integral of the current unknowns. Indeed, ∂ξKn + 2Kn/ξ = −∂ξUn, and the two radial second-order operators become (∂ξξ + 3ξ−1∂ξ)/2 and (∂ξξ + ξ−1∂ξ)/2. At positive order all terms containing an order-n unknown are linear. The pressure row is ∂ξΠn = 4ξϕ0ϕn/C2+ known terms; we also use this row for X∂XΠn in the axial equation. All apparent divisions by X are then regular.
834. u834 Step 2: solve without shrinking the radial interval. Iteration of the system differentiates in η. We control the number of such derivatives before estimating the successive radial integrals. The coefficient of ∂η has the following sparse block form. With Hc = Dη + dU0, its only possibly nonzero entries are (A1)51 = 2Hc/L,
835. u835 (A1)52 = (A1)53 = −2d(ϕ0 + X∂Xϕ0)/L, (A1)62 = 2(Hc −dX∂XU0)/L,
836. u836 (A1)64 = 2d/L. Thus A1, at every radius and after every parameter derivative, maps the first four coordinates into the last two and annihilates the last two.
837. u837 For ci = (0, 0, 2, 0, 3, 1)i, we invert the diagonal part of (5.7), with zero axis data, by the integral operator
838. u838 Its kernels are independent of η and preserve the two coordinate subspaces just described. If D0 is any intervening diagonal kernel, then
839. u839 Consequently two adjacent derivative operators vanish even when the left derivative falls on a variable matrix coefficient. A nonzero composition of k factors contains at most pk = ⌈k/2⌉ parameter derivatives.
840. u840 We bound A0, A1, fn on the larger neighborhood Sρ, uniformly on the whole radial interval. On Sρ′, 0 < ρ′ < ρ, we split the Cauchy radius loss ∆= ρ −ρ′ among the at most pk derivatives. The diagonal kernels have norm at most one, and the ordered integration simplex has volume ak+1/(k + 1)!. Summing the at most 2k compositions gives
841. u841 (5.8) The kth root of the right-hand side of (5.8) tends to zero. The Picard series Wn = ∑k≥0 KkG fn therefore converges on Sρ′ for every finite a, regardless of the size of Cn. Iterating the homogeneous equation for the difference of two solutions and applying the same bound proves uniqueness. In particular, the radial interval is independent of the coefficient norms at later orders.
842. u842 Step 3: recover smooth profiles at the axis. Radial smoothness follows without differentiating a singular expression:
843. u843 The second identity is continuous at zero. Repeated use of it, with a smaller parameter neighborhood when necessary, gives every fixed radial derivative. We extend the integral
844. u844 equation across ξ = 0. The original equations preserve even parity of the first four coordinates and odd parity of the last two; uniqueness gives precisely these parities. Taylor’s formula for an even smooth function of ξ shows, to every finite order, that it is smooth as a function of X = ξ2 on the half-interval. This also verifies the regularity required to form the next source
845. u845 □ Ωn/X.
846. u846 5.2. Canceling the total radial residual integrals. The inner profiles (ϕn, Un, Vn, Πn) from Lemma 5.1 are not yet coefficients that can be used at the next order. We must extend En, Un to all X ≥0, recompute Vn, Πn from Equations (5.2) and (5.5), and control the residual created by that extension. A radial cutoff of En, Un alone does not give compact support for the fields obtained by radial integration. In particular, the pressure may acquire a nonzero exterior constant and the stress Tn defined below in (5.9) may acquire a nonzero exterior tail. We impose vanishing of the five moments defined below in Equations (5.10) and (5.11) to remove these terms, as proved in Lemma 5.2.
847. u847 We √ specify how to collect the residual coefficients before imposing the moments, using R =
848. u848 2X. To extract order n, we substitute (5.1) into the differential polynomial R, perform the physical differentiations, remove the common tangential power q−A−1, and collect the coefficient of q2nh. This is a finite algebraic operation: multiplication uses the sums i + j = n, and axial viscosity uses order n −1. Explicitly,
849. u849 These expressions vanish where the inner equations are solved. We define the two required stress components by weighted radial integration of the negative residuals:
850. u850 (5.9) All profiles under these integrals have the same parameter η. The two physical stress components are q−A−1/2+λnTn. For the forward integrals to vanish beyond the support of their sources, the total integrals of R2rθ,n and Rrz,n must vanish. The five conditions below imply those two identities through the conservative momentum equations; they also remove the radial-velocity and pressure terms left by integration.
851. u851 We also write Fn = ∫︁X 0
852. u852 0 Un(x, η) dx = XAX(Un) for the Stokes streamfunction √ coefficient; this indexed notation is distinct from the leading azimuthal profile F = E/
853. u853 For a tentative extension at order n, we denote the five total moments by
854. u854 (5.10)
855. u855 (5.11)
856. u856 These are functions of η, with every lower-order factor held fixed. We write
857. u857 Radial supports below are uniform over |η| ≤1. For the stress estimates, we use the flat weight ζ of Theorem 4.6 (Item (iv)) and set
858. u858 δ(X) = min{1, log(X/Xa), log(Xb/X)} Lemma 5.2. There are radii Xa < X−< X+ < Xb, independent of n, for the following induction. Fix n ≥1. For each 1 ≤j < n, assume that Pj is smooth, satisfies Equations (5.2) and (5.5) globally, and obeys (5.12) with n = j. Assume also that these profiles agree with their inner solutions on [0, X−] and retain the parameter regularity of Lemma 5.1 throughout [0, a2]. For n = 1, there are no lower positive-order hypotheses. Then the inner order-n solution of Lemma 5.1 has a smooth extension to X ≥0, |η| ≤1, agreeing with it on [0, X−] and retaining its parameter regularity throughout [0, a2], satisfying Equations (5.2) and (5.5) globally, and obeying
859. u859 (5.12) Both Vn/X and Fn/X extend smoothly to X = 0. The stress defined by (5.9) satisfies
860. u860 and, for every fixed profile derivative ∂I,
861. u861 (5.13) Here Cn,I, Nn,I may depend on the order and derivative; for n ≥2 one may take Nn,I = 0. Proof. Step 1: extend the current profiles while preserving the inner solution. We preserve the inner solution on [0, X−], then choose the extension so that mn(η) = 0 for every |η| ≤1. The conditions mn,1 = mn,3 = 0 remove the exterior streamfunction and pressure terms. The conditions mn,1 = mn,2 = 0 cancel the integrated time and viscosity terms; mn,4 = mn,5 = 0 cancel the integrated momentum fluxes, including the pressure contribution. Steps 3 and 4 verify these implications.
862. u862 n ) be the inner profiles, defined for 0 ≤X ≤a2 by Lemma 5.1. Let
863. u863 Let (ϕin n Ipos be the reserved positive-order patch of Theorem 4.6 (Item (vi)). We choose X−in the inner collar of Theorem 4.6(i), where the stress cone inequalities have a uniform positive margin, and fix, independently of n,
864. u864 The initial extensions are
865. u865 with the products extended by zero past X = a2.
866. u866 In the R coordinate, the reserved patch Ipos is Jpos = {R > 0 : R2/2 ∈Ipos}. We choose nonnegative smooth bumps
867. u867 with mutually disjoint ordered supports, fixed across orders. For five unknown scalar functions αn,j(η), βn,j(η), we set
868. u868 ∑
869. u869 (5.14)
870. u870 ∑ At R = 0, the quotient defining ϕn is its smooth extension: near the axis the bumps vanish profiles and ϕn = by ˜︁ϕn. For each choice of these five coefficient functions, we define the remaining profiles by
871. u871 (5.15)
872. u872 All lower-order profiles and Ωn−1 remain fixed in these formulas. The quotient Ωn−1/x is smooth at the axis, as established after (5.6). Equations (5.2) and (5.5) hold globally by construction. On [0, X−], the cutoff is one and the bumps vanish, so forward integration from the zero axis values gives
873. u873 The radial cutoff is independent of η, and the bumps vanish on [0, a2]. The reconstructed profiles therefore retain the parameter regularity of Lemma 5.1 on that whole interval, as required at the next order.
874. u874 Step 2: solve the five moment equations. On Ipos,
875. u875 where f is smooth and bounded away from zero. With the lower-order profiles fixed, all five moments are affine in the five unknown coefficients. For the fifth moment, the pressure equation gives
876. u876 ∑ The term Ωn−1 depends only on lower orders and is unchanged by the five coefficients. Using R2 = 2X, the pressure equation also gives
877. u877 ∑ We define the constant moment matrices
878. u878 Here BU is 2 × 2 and BE is 3 × 3. Let m0
879. u879 n,j be the moments obtained by setting all five coefficients in (5.14) to zero and reconstructing the profiles by (5.15). We put
880. u880 Writing αn = (αn,1, αn,2)T and βn = (βn,1, βn,2, βn,3)T, the five conditions are exactly
881. u881 Lemma A.1 applies because the exponents in each block are distinct and the bump supports are ordered. Both matrices are therefore invertible, and the required coefficients are
882. u882 (5.16) The discrepancies are smooth functions of η, and all fixed derivatives of 1/ f are bounded on [−1, 1]. Thus these coefficients are smooth for discrepancies of any finite size. This proves mn = 0. The support assertion in (5.12) remains to be checked.
883. u883 Step 3: close the velocity and pressure outside the correction region. We choose X+ beyond the leading axial-velocity perturbation U0 and every positive-order patch, and before the terminal outer collar. The first moment gives
884. u884 Hence AX(Un) = Vn = 0 there at positive order, while the leading axial-flux moment correction gives U0 = V0 = 0 there as well. Each term of every Ωk contains a V factor or a derivative of a V field, and hence vanishes on this same exterior region. At positive order each product ϕiϕj, i + j = n, contains a positive-order factor. Thus ∂XΠn = 0 beyond X+, and the third moment gives
885. u885 Thus ϕn, Un, Vn, Πn, Fn all vanish for X ≥X+. Their smoothness follows from the cutoff ansatz, the finite solve (5.16), and the integrals (5.15). At the axis,
886. u886 is smooth, and (5.2) gives the same conclusion for Vn/X. Compactness of [0, X+] × [−1, 1] now yields the coefficient bounds
887. u887 (5.17) The support radius is common to all orders; these constants may grow with n and m.
888. u888 On the reserved interval Imean for later angular-mean corrections, the higher-order velocity coefficients vanish identically. The support of the inner cutoff and the patch Ipos both lie to the left of Imean. Thus En = Un = 0 there, and the first moment gives Fn = 0 already beyond the positive-order patch; (5.2) then gives Vn = 0. In particular,
889. u889 on the reserved mean patch, for every n ≥1.
890. u890 (5.18) Placing X+ beyond the leading axial-velocity perturbation is essential for the pressure support: Ω0 can remain nonzero beyond the positive-order patches.
891. u891 Step 4: cancel the total tangential residual integrals. The inner equations make both positive-order stress components zero on [0, X−]. To prove that they also vanish beyond the outer radius, we use the conservative forms of the tangential residual for any smooth axisymmetric divergence-free field:
892. u892 r2Rθ = ∂t(r2uθ) + ∂r(r2uruθ) + ∂z(r2uzuθ) −∂r(r2∂ruθ −ruθ) −∂zz(r2uθ),
893. u893 Axis parity and compact positive-order supports make the radial boundary terms vanish. At order n, the remaining angular moments before differentiation are
894. u894 They vanish because mn,2 = mn,4 = 0. The product power is independent of the split i + j = n, so this cancellation holds before taking derivatives in the physical variables z, t. The lower angular moment used by axial viscosity vanishes because mn−1,2 = 0 when n ≥2. When n = 1, the compensated leading profile instead supplies the convergent identity
895. u895 (5.19) where P is the fixed pure-power exterior of (A.45), so ∂2
896. u896 zP(r) = 0. Differentiation under the integral is justified: beyond the varying terminal collar uθ,0 = K(r, t) is independent of z, while the difference moment itself converges. Thus the order-one angular viscosity term also has zero total moment.
897. u897 The axial moments are
898. u898 (5.20)
899. u899 ∑
900. u900 ∑ For the pressure contribution, integration by parts gives
901. u901 The identities mn,1 = mn,5 = 0 therefore make both integrals in (5.20) vanish. The axialviscosity term uses the order-n −1 axial-flux integral, which also vanishes, including at order zero. Integrating the conservative equations now gives, order by order,
902. u902 (5.21)
903. u903 ∑ Here Rj,n = q−A−1+λnrj,n, for j ∈{θ, z}, denotes the order-n tangential residual in physical variables. In (5.21) at n = 1, the final integral means the convergent integral of uθ,0 −P from
904. u904 (5.19); differentiation applies to this convergent subtracted integral. Removing the common physical powers yields
905. u905 Consequently (5.9) can also be written as
906. u906 This representation applies to the signed higher-order stress.
907. u907 Step 5: locate the stress support and bound the order-one outer term. For n ≥2 the residual itself vanishes beyond X+: its same-order terms have a positive-order factor and its axial viscosity acts on order n −1 > 0. The backward representation proves supp Tn ⊂[X−, X+]. At n = 1 the only remaining exterior source is −∂zzuθ,0 in the angular equation. Beyond Xb, the leading profile is the z-independent heat field, so T1 is still supported in the closed active annulus. For the quantitative edge estimate we restrict further to the terminal collar where the exterior heat-flow cutoff equals one. There uθ,0 = K(r, t) fo, where K(r, t) = c∞s−AH(2(1 −t)/s) is the physical heat field of (4.29). The smooth terminal multiplier fo is constant on its initial plateau and equals one for X ≥Xb; its exact construction is (A.12), with a translation of the logarithmic radial coordinate. In the local coordinate y = log X it has 1 −fo = e−4/δ2 b
908. u908 b times a smooth factor, with δb = log(Xb/X). Direct differentiation gives
909. u909 az(η) = −2η/L, All fixed η-derivatives of az, bz are bounded on [−1, 1]. The derivatives ∂y fo and ∂yy fo are bounded respectively by Ce−4/δ2
910. u910 b . Backward integration gains three powers of δb by Lemma A.9, applied with c = 4 and j = 3, 6. The radial weights and the conversion between R and δb are bounded smooth factors on this fixed collar. After factoring out the stress power q−A−1/2+λ1, Lemma A.9 gives the pointwise estimate
911. u911 (5.22) The second estimate in (5.22) follows by differentiating the backward integral; each derivative adds only a finite inverse power of δb. Near this edge the inner factor of ζ is bounded above and below by positive constants. This proves (5.13) for n = 1; the identity T1 = 0 on [0, X−] gives the estimate near the inner edge. For n ≥2 the common compact interior support
912. u912 □ makes every fixed derivative of Tn/ζ bounded.
913. u913 5.3. The coefficient induction and finite residual. The preceding lemmas construct and extend one coefficient from the completed lower orders. Lemma 5.1 solves the inner equations Equations (5.2) to (5.6), and Lemma 5.2 supplies the extension and support properties while retaining those equations on 0 ≤X ≤X−. In this subsection we apply these lemmas inductively and estimate the residual of each finite truncation in (5.25), including its physical derivatives. The gain in decay must grow with the truncation order, while the exponent loss caused by physical differentiation stays independent of that order, as quantified below in (5.26).
914. u914 For a tuple V of physical fields, we write
915. u915 (5.23) This is the pointwise maximum of its Cartesian space–time derivatives through order m. We write un = ur,ner + uθ,neθ + uz,nez for the physical velocity coefficient in (5.1), and T0 for the leading stress profile of (4.11). We define the finite tuple, before applying any cutoff in q, by
916. u916 ∑ The profiles in this sum already include their radial cutoffs and moment corrections. For a physical stress pair T = (Tθ, Tz), we define
917. u917 Fslow(u, p, T) = R(u, p) + (∂r + 2/r)Tθeθ + (∂r + 1/r)Tzez.
918. u918 (5.24) The added terms are the tangential divergence of the stress retained for the later wave construction.
919. u919 Proposition 5.3. Fix the leading profiles of Theorem 4.6. There is a sequence of positive-order profiles constructed at each order by Lemma 5.1 and then Lemma 5.2, using the finalized lower orders. Every finite velocity sum u[N]
920. u920 slow is divergence-free. On every fixed compact profile range 0 ≤X ≤Xmax, −1 ≤η ≤1, with 0 < Xmax < ∞, for each N, m ≥0 and 0 < q ≤1, its augmented momentum residual satisfies
921. u921 (5.25) The constants may depend on the fixed profile range, and CN,m may depend on N; the exponent loss Km does not.
922. u922 Proof. Step 1: construct the sequence and cancel the retained coefficients. At order n, we apply Lemma 5.1 with the already finalized orders 0, . . . , n −1. We then apply Lemma 5.2 to extend and repair the current profiles, reconstruct Vn, Πn from Equations (5.2) and (5.5), and define Tn by (5.9). These operations preserve the equations on the retained inner interval and the regularity needed to form Ωn/X at the next order. Both lemmas use the same fixed radial intervals, so the induction continues for all n.
923. u923 Equation (5.2) gives exact incompressibility at each order, hence for every finite sum. The pressure equation holds globally. Differentiating the stress primitives gives
924. u924 Thus the tangential residual is exactly minus the stress divergence at each retained order. The same leading identities hold at order zero by Proposition 4.2. After truncation at N, every uncancelled product or shifted viscous term has exponent increment at least 2h(N + 1). For example, at N = 1 the product of two order-one velocity coefficients and axial viscosity applied to order one first enter at order two.
925. u925 Step 2: estimate physical derivatives of the finite residual. The powers of q introduced Step by physical 2: estimate differentiation physical derivatives follow directly of the from finite the coordinates. residual. The For powers a profile of q g(x⊥/√q, introduced η) smooth in its Cartesian arguments on a compact profile range, a derivative with a = |β| transverse, k axial and l time differentiations satisfies⃓⃓⃓
926. u926 t[qbg(x⊥/√q, η)]⃓⃓⃓≤Cm(1 + |b|)m∥g∥Cmqb−a/2−Dk−l,
927. u927 (5.26)
928. u928 This is the chain rule with ∂t = q−1L−1(−q∂q + X∂X + Dη∂η) and ∂z = q−DL−1(2ηq∂q − 2ηX∂X + d∂η); transverse differentiation introduces a factor q−1/2 in these smooth Cartesian profiles.
929. u929 The powers of n generated by differentiating qλn enter the coefficient constants. The exponent reduction depends only on the number and type of physical derivatives. On the common stress support, X ≥Xa > 0, so r−1 = q−1/2(2X)−1/2 also has derivative losses independent of N; near the axis the velocity is estimated through its smooth Cartesian representative. There are only finitely many terms in a fixed truncated residual. Applying
930. u930 □ (5.26) to them gives (5.25), with Km independent of N.
931. u931 5.4. Divergence-preserving summation of the coefficients. The finite tuples U [N] have the
932. u932 improving residual decay in (5.25), but the constants Cn,m in their coefficient bounds (5.17) may grow too quickly for the full formal series (5.1) to converge. We obtain smooth fields by multiplying the nth positive-order term by χ(cnq), where χ = 1 near zero and χ = 0 beyond one, and choosing cn →∞, as in (5.34). Later terms then vanish outside successively smaller neighborhoods of q = 0. On a compact set with q bounded below, only finitely many terms remain. We must show that the finite residual bound of Proposition 5.3 survives these cutoffs. This is the conclusion (5.36) of Lemma 5.4, applied to the augmented residual (5.24).
933. u933 To preserve incompressibility, we apply these cutoffs to vector potentials before differentiating. An axisymmetric Stokes streamfunction S gives such a potential: if S/r2 is smooth in (r2, z, t), then
934. u934 The assumed smoothness of S/r2 as a function of (r2, z, t) makes the potential smooth in Cartesian coordinates. For the positive-order profiles constructed above, the physical streamfunctions and their vector potentials are
935. u935 (5.27) Here s = r2/2, as in (4.1). Indeed curl An = ur,ner + uz,nez gives the radial and axial velocity components. In Cartesian coordinates, using r2 = 2qX,
936. u936 Since Fn/X is smooth at X = 0, this expression extends smoothly across the axis for q > 0. Lemma 5.2 supplies common supports and bounds on every fixed coefficient derivative, with no bound uniform in n required.
937. u937 Cutting off An and taking its curl includes the extra velocity term from differentiating the cutoff, so the result remains divergence-free. We formulate the summation argument for tuples of physical fields, allowing its later application to the wave and mean correction sequence. For the present application its indices are j = n, its decay orders are gn = 2nh, and its entries are
938. u938 Its differential polynomial will be Fslow from (5.24). In the lemma, Uj denotes a tuple of physical fields, not the axial profile Un(X, η). We use the derivative notation (5.23) and set Λlog(q) = 1 + | log q|. All estimates in the lemma hold uniformly over its stated domain.
939. u939 The general argument needs a common domain, increasing decay orders, and derivative losses independent of the summation index. Let Ωbe a domain on which 0 < q < q0 ≤1. Assume that q = q(z, t) is smooth, is bounded below on compact subsets of Ω, and satisfies
940. u940 (5.28) Let U0 = (u0, p0, T0) be smooth, with div u0 = 0. Assume that for every m there are finite Cm, Km, Pm such that |U0|m ≤Cmq−KmΛlog(q)Pm. The entries T0 and Tj below may be omitted. For j ≥1, suppose that the smooth physical tuples
941. u941 (5.29) are defined on this same domain. Require smooth Cartesian representatives at the axis and smooth zero extensions at every lateral boundary of their spatial supports. Suppose that, for every j, m, there are finite constants Cj,m, Pj,m such that
942. u942 (5.30) where the nonnegative, nondecreasing losses ℓm are independent of j.
943. u943 For each finite J, we set U[J] = U0 + ∑J
944. u944 j=1 Uj. We write the scalar components of a tuple as U = (U1, . . . , UNU). A fixed differential polynomial of finite order and degree has, componentwise, the form
945. u945 ∑
946. u946 ∏
947. u947 (5.31) Here the output index runs through the fixed finite set 1 ≤a ≤NF, each Ma is finite, and the fixed integers s ≥0, d ≥1 bound the order and degree: 1 ≤daν ≤d, |βaνr| ≤s, and 1 ≤kaνr ≤NU. For β = (α, b) ∈N4
948. u948 t . The coefficient functions, indices, and order and degree bounds s, d are fixed independently of the summation and truncation indices.
949. u949 For each nonconstant monomial in (5.31), fix a common region Ωaν containing the supports of its field factors in the base and in every increment, both before and after applying cutoffs. For the term independent of U, we set Ωa0 = Ω. For every coefficient caν, 0 ≤ν ≤Ma, and every Cartesian space–time multi-index I ∈N4
950. u950 (5.32) The regions are independent of truncation. Since the coefficient list is finite, the finite constants CI, KI, PI may be chosen common to that list for each fixed I; they are independent of the truncation index. Thus the bounds hold throughout a comparison of partial sums and cutoff sums. The terms ca0, which are independent of U, cancel from that comparison. The following residual hypothesis concerns the full F, including those terms: assume that each finite partial sum satisfies
951. u951 (5.33) where KF m
952. u952 m is independent of J and, for every fixed J, m, N, 0 ≤EJ,m ≤CJ,m,NqN. These bounds, including the flat remainder, are uniform over the full domain at that fixed stage. Lemma 5.4. There are numbers aj+1 ≥2aj, with a−1 1
953. u953 < q0, and a fixed smooth cutoff χ, equal to one on [0, 1/2] and zero on [1, ∞), such that
954. u954 curl(χ(ajq)Aj) + χ(ajq)Bj, χ(ajq)pj, χ(ajq)Tj
955. u955 (5.34)
956. u956 is locally finite and smooth, and its velocity component u satisfies div u = 0. Its added terms extend smoothly by zero across the outer boundary q = q0. For every m and J ≥max(1, m) one has
957. u957 (5.35) where ℓ′
958. u958 Under the residual hypothesis (5.33), every Cartesian space–time derivative of F(U) vanishes to infinite order as q ↓0:
959. u959 (5.36) Proof. We bound the derivatives of each cutoff, choose the cutoffs so the tail is summable after any fixed number of derivatives, and compare the residual with one finite partial sum before cutoffs. In these bounds, each prescribed Cartesian derivative reduces the power of q by an amount independent of the truncation order.
960. u960 Step 1: bound the cutoff and curl derivatives. For a positive-order derivative of χ(aq), the chain rule gives terms of the form
961. u961 ∏
962. u962 ∑
963. u963 ∑ On their support, 1/2 ≤aq ≤1. Using (5.28), the factors qk compensate for ak, since aq is bounded on the cutoff support. Consequently
964. u964 with constants independent of a. Since q = q(z, t), ∂x1q = ∂x2q = 0. In particular the commutator in
965. u965 curl(χ(aq)Aj) = χ(aq) curl Aj + aχ′(aq)∇q × Aj is controlled by the assumed derivatives of the potentials. The product rule therefore bounds derivatives through order ℓ′ m of the jth summand in (5.34) by ˆ︁Cj,mqgj−ℓ′
966. u966 mΛlog(q) ˆ︁Pj,m, where the exponent reduction ℓ′ Step 2: choose the cutoffs and estimate the tail. We choose the scales aj recursively so that aj+1 ≥2aj, each cutoff χ(ajq) is supported in q < q0, and
967. u967 (5.37) Only finitely many requirements are imposed at step j, and each holds for all sufficiently small q because gj > 0. Derivatives through order m of the resulting cutoff increment are bounded by 2−jqgj/2−ℓ′
968. u968 m whenever m ≤j. On a compact subset where q ≥δ > 0, all terms with aj > δ−1 vanish. Thus the sum is locally finite. The curl of any vector field is divergencefree, and χ(ajq)Bj is divergence-free because its angular amplitude is independent of θ. Smoothness and the stated support extensions follow termwise.
969. u969 For fixed m and J ≥max(1, m), when q < (2aJ)−1 the first J cutoffs equal one. Since q < 1 and gj is nondecreasing, the remaining terms obey
970. u970 ∑ which proves (5.35).
971. u971 Step 3: compare the residual with one finite partial sum. The cutoff sum U is now smooth, and its velocity component is divergence-free. To prove flatness of F(U), we compare it with F(U[J]) for a finite partial sum with sufficiently high residual decay, and control the change caused by U −U[J]. The assumed bound on |U0|m and (5.30) give
972. u972 (5.38) with Km ≥0 independent of J: every increment has gj > 0, and taking the curl requires one additional derivative of its potential. The order and degree bounds s, d are those in (5.31). Subtracting two monomials one factor at a time, then applying the product rule and (5.32), gives for e = U −U[J]
973. u973 (5.39) where Hm is independent of J; the finitely many coefficient logarithms are absorbed into this fixed power of q−1. A polynomial independent of U has zero difference and needs no estimate.
974. u974 For fixed derivative order m and decay order N, we choose J ≥m + s large enough that
975. u975 We also take J large enough that gJ+1/2 −ℓ′
976. u976 m+s is nonnegative, and then keep J fixed. After shrinking the q-neighborhood, χ(ajq) = 1 for j ≤J, the constants and logarithms in (5.38) are absorbed by q−1, and |e|m+s ≤1 by (5.35). Formula (5.39) is then O(qN). The same holds for (5.33), after absorbing its logarithm by one power and using the flatness of this fixed EJ,m. This proves (5.36). The order of choices is (m, N), then J, then the neighborhood and
977. u977 □ constants; the comparison uses only the remainder EJ,m at that fixed truncation.
978. u978 A finite block of initial terms with nonpositive decay orders may receive one common cutoff, applied as in (5.34), and be included in the base U0. Its cutoff equals one near q = 0, preserving every comparison with a finite partial sum there.
979. u979 For smooth vector fields v, e and smooth scalar fields p, π, the physical residual comparison is
980. u980 R(v + e, p + π) −R(v, p) = ∂te −∆e + ∇π + (v · ∇)e + (e · ∇)v + (e · ∇)e. For Fslow, we add the difference of the linear tangential stress-divergence term in (5.24). On the common annular stress support, the cylindrical coefficients and each of their prescribed Cartesian derivatives are bounded by a fixed negative power of q. In the final correction sequence, F = R and there is no stress variable.
981. u981 The same cutoff choices can meet further normalized estimates when those estimates have been proved for the coefficients. More precisely, at step j one may add any finite list of requirements
982. u982 (5.40) to (5.37). For derivatives of the normalized expansion coefficients and boundary weights, we impose these additional coefficient estimates using the corresponding cutoff product rules. To recover a full expansion after a fixed truncation N and at a fixed derivative order m, we choose M ≥max(N, m) with gM+1/2 ≥gN+1: the finite block from N + 1 through M retains its original orders near q = 0, while (5.35) controls the remaining tail at order gN+1, with its fixed exponent reduction ℓ′
983. u983 5.5. The realized base field. The finite tuples U [N] from Proposition 5.3 now satisfy the residual estimates required by Lemma 5.4. In this subsection we apply that lemma to the coefficient fields and their potentials An in (5.27) to construct the smooth background (uB, pB). We must also retain the leading profile as in (5.42), the exact heat exterior (4.29), and the weighted stress bounds (5.43) used to choose the wave amplitudes and their corrections in Propositions 7.5 and 7.6.
984. u984 Proposition 5.5. There are smooth axisymmetric fields (uB, pB) for q > 0 and two physical tangential stress components Tphys, supported where Xa ≤X(r, z, t) ≤Xb, such that div uB = 0 and
985. u985 R(uB, pB) = −(∂r + 2/r)Tphys,θ eθ −(∂r + 1/r)Tphys,z ez + EB.
986. u986 (5.41) For every fixed 0 < Xmax < ∞, uniformly on the profile rectangle [0, Xmax] × [−1, 1] in (X, η), for every Cartesian space-time derivative ∂α and every M > 0, |∂αEB| ≤Cα,MqM as q ↓0. The constants may depend on Xmax. The formal expansion of (uB, pB, Tphys) has the finite truncations U [N] from Proposition 5.3, with the physical Stokes streamfunctions Sn from (5.27) summed before differentiation. Fix radial endpoints 0 < Xlo < Xa < Xb < Xhi < ∞. The rectangle [Xlo, Xhi] × [−1, 1] is the fixed radial enlargement of the closed active annulus [Xa, Xb] × [−1, 1] from Theorem 4.6(ii), with η ranging over its full closed parameter interval. Uniformly on this enlarged rectangle, for every fixed composition D I of q∂q, ∂X, ∂η, as q ↓0,
987. u987 (5.42) The the weighted constants comparison in these estimates uses may from depend Theorem on the 4.6(iv) fixed and radial holds endpoints. on Xa < With X < ˆ︁T = −1 qA+1/2Tphys, ≤η ≤1: the weighted comparison uses ζ, δ from Theorem 4.6(iv) and holds on Xa < X < Xb, −1 ≤η ≤1:
988. u988 |D I( ˆ︁T −T0)| ≤CIq2hζδ−NI ≤CIqhζδ−NI
989. u989 (5.43) Every fixed physical space-time derivative of uB, pB, Tphys, EB extends continuously to η = ±1 on compact sets with q > 0. The positive-order corrections uB −u(0) and pB −p(0) vanish for X ≥X+, with X+ from Lemma 5.2; in particular the leading profile’s exact exterior (4.29) is preserved. On the reserved interval X ∈Imean ⊂(Xa, Xv) from Theorem 4.6(vi), for every −1 ≤η ≤1, the base velocity agrees exactly with the leading profile:
990. u990 qAuθ,B = E0 = cpatch(1 + η2)−1X−1/2−λ.
991. u991 (5.44) Here u(0) denotes the velocity determined by the leading profile in (4.3).
992. u992 Proof. Step 1: verify the summation hypotheses and choose the cutoffs. We use the potentials (5.27) and the physical tuples specified before Lemma 5.4, keeping order zero intact. The Cartesian formula for An there verifies regularity at the axis, and Lemma 5.2 supplies the common supports and all fixed coefficient derivative bounds. By (5.26), since D < 1/2, the tuple of coefficients in (5.27) and (5.1), including the stress components, has a reduction in the power of q bounded by ℓm = 2A + m. The coefficient constants absorb the powers of n generated by differentiating qλn. For the cutoffs, we set σ = cnq. For every integer j ≥0,
993. u993 Thus the same derivative estimates hold uniformly after applying the cutoffs. The stressdivergence coefficient bounds were checked in Proposition 5.3, whose estimate (5.25) is the lemma’s finite-residual hypothesis with ρN = 2h(N + 1) and no additional remainder. The hypotheses of Lemma 5.4 are therefore satisfied on each fixed compact profile range. We apply it to obtain uB, pB, Tphys, with residual EB = Fslow(uB, pB, Tphys) bounded by every power of q with all prescribed physical derivatives.
994. u994 Step 2: preserve divergence, support, and the mean-patch velocity. The radial contribution from differentiating the cutoff appears explicitly in
995. u995 (5.45) The term containing χ′(cnq)Fn has the same q2nh prefactor and the common radial support supplied by Lemma 5.2. Commutation of physical derivatives proves that the divergence is identically zero across every cutoff transition. Equation (5.18) makes both terms in the radial bracket zero on the mean patch; the positive-order axial and azimuthal terms also vanish there. This proves (5.44), including the Stokes streamfunction cutoff contributions. Step 3: prove the normalized velocity and stress estimates. We use the additional diagonal bounds in (5.40) to control, through order n, all normalized coefficient and Stokes streamfunction derivatives, the extra streamfunction derivative, and, for n ≥2, all derivatives of Tn/ζ. The normalized velocity estimates are taken on the fixed enlarged rectangle in the statement; the weighted stress estimates are on Xa < X < Xb, −1 ≤η ≤1. For estimates reaching the axis we use the profiles expressed in smooth Cartesian coordinates. The finite constants include the powers of n and the fixed cutoff derivatives. Thus, in addition to the lemma’s physical requirements, the cutoff choice gives
996. u996 (5.46) where an denotes one of En, Un, Πn, Vn/
997. u997 2X on the enlarged rectangle, or a component of Tn/ζ for n ≥2 on the active radial interior. The same estimate holds for the expression in brackets in (5.45) after multiplication by q2nh. The diagonal choice in the summation lemma accommodates these finitely many additional seminorms at each order. The analytic neighborhoods may depend on the order.
998. u998 At a fixed derivative order, we retain a finite initial block, whose positive-order terms are all O(q2h), and sum (5.46) from an index at least two and larger than the selected derivative order. This proves (5.42); in the common normalization qA, the positive-order radial correction is even O(q3h). For the stress components, the same argument for n ≥2 carries the factor ζ. The single term n = 1 is controlled by (5.22), with a finite inverse power of δ in each derivative. This proves (5.43). The wave construction can therefore use the fixed positive covariance representation of T0 and treat the higher-order stress by signed corrections.
999. u999 Step 4: recover the full expansion and the endpoint extensions. For a fixed formal order N and fixed derivative order m, we split the summed remainder at J = max{2N + 2, m + 1}: the finitely many terms N < n < J have their original powers at least q2h(N+1), and the tail satisfies the displayed geometric estimate
1000. u1000 ∑
1001. u1001 The coefficient constants for the finite intermediate block may depend on N, m. The cutoffs of the finitely many initial terms are identically one for sufficiently small q. This recovers every fixed original formal order, including after physical differentiation with its fixed exponent loss. The geometric bounds used for summation therefore do not change the asymptotic coefficients. The residual conclusion was established in Step 1 by comparison with a sufficiently long finite partial sum.
1002. u1002 Finally, the cutoff sequence grows at least geometrically, so only finitely many positiveorder terms remain on any compact set with q bounded below. For each fixed physical space-time derivative, we intersect the finitely many parameter neighborhoods of the contributing positive-order coefficients. Their profiles and Stokes streamfunctions extend to this neighborhood. The order-zero term has the smooth one-sided extension in η supplied by the leading profile and heat construction. Since L > 0, the coordinate map transfers these extensions to the physical variables, including all the specified derivatives. This proves the
1003. u1003 □ asserted extensions at η = ±1.
1004. u1004 6. Auxiliary torus and separation of oscillatory supports
1005. u1005 Fix the background velocity uB, pressure pB, and annular stress Tphys supplied by Proposition 5.5. Its momentum residual is the divergence of that stress, with the sign in (5.41), plus a flat remainder. The leading stress will be supplied by the waves of Proposition 7.5, whose amplitudes evolve under the local background shear through (7.5). We must first localize the waves to regions where that shear varies little and arrange their supports so that distinct waves do not produce additional quadratic interactions. The localization must also permit rapid temporal variation without introducing radial derivatives large enough to spoil the residual estimates.
1006. u1006 We introduce the periodic variable Y ∈T2 = R2/Z2, independent of the physical coordinates (r, θ, z, t), then obtain physical fields by evaluating it at the fixed map of (r, t) in (6.3). This map allows rapid temporal variation with much smaller additional radial derivatives.
1007. u1007 The temporal variation drives the pulse equation in Section 7; the radial derivative bounds control the errors caused by localization. Oscillations whose supports in the remaining variables overlap receive disjoint auxiliary supports, so their cross products vanish after evaluation. Harmonics of the same localized oscillation still interact.
1008. u1008 We first derive the evaluated derivative operators in (6.6). Lemma 6.1 gives the support separation, and Lemma 6.2 supplies a common representation for overlapping bands. We finish with the coefficient classes in Section 6.4 and their product and derivative rules in Proposition 6.6.
1009. u1009 All preliminary geometric choices are made on one domain 0 < q < qbig. The number qbig may be decreased when the fixed base and pulse coefficients are chosen, before the correction iteration begins.
1010. u1010 6.1. Dyadic charts and derivatives after phase evaluation. Recall q = q(z, t) and A = 1/2 + h, D = 1/2 −h from (4.1). We work on dyadic regions where q is comparable to a fixed scale Q. The radial, axial, and temporal coordinates are rescaled by Q1/2, QD, Q, respectively, so that the concentrating annulus has bounded size in each region. For a dyadic index ℓ, we set Q = 2−ℓ,
1011. u1011 τ = 1 −t. (6.1)
1012. u1012 The √ radius R now uses the fixed band scale Q, whereas in the profile calculations it denoted √
1013. u1013 2X radius = r/√q; now the uses two the representatives fixed band scale satisfy whereas Rchart =
1014. u1014 q/Q Rprofile. Throughout a chart, Q, ε, S∗are constants. A physical velocity, pressure, and residual have chart representatives obtained by multiplication by QA, Q2A, Q2A+1/2, respectively. An integer matrix acts on the torus while preserving periodicity. Its distinct eigenvalues will give different rates of variation in the radial and temporal directions. Fix
1015. u1015 (6.2)
1016. u1016 These vectors satisfy Jgvr = Λgvr and Jgvt = Tgvt, with 1 < Λg < Tg. We define the physical evaluation map by
1017. u1017 (6.3) For a field on the extended domain with independent variable Y, the chain rule differentiates its restriction to this map by the operators
1018. u1018 (6.4) Here ∂t and ∂r on the extended domain hold Y fixed.
1019. u1019 Thus physical time and radial differentiation become t and r, respectively, before restriction.
1020. u1020 We call Y the absolute auxiliary variable. Its band representative for band ℓis the image Yi under the following integer covering:
1021. u1021 (6.5) For a function f on the band torus, its pullback to the absolute torus is the composition f (Ji
1022. u1022 gY). A function descends to that band torus when its value is unchanged by every translation Y ↦→Y + a with Ji
1023. u1023 ga ∈Z2; such translations are called deck translations. Only large ℓ, for which i(ℓ) ≥0, will be used. Writing Ni = vt · ∂Yi and Li = vr · ∂Yi, the eigenvector identities give Nabs = Ti
1024. u1024 gLi on band pullbacks. The chart versions of the physical derivatives are therefore
1025. u1025 (6.6)
1026. u1026 The velocity and residual normalizations following (6.1) will be used throughout. In (6.6), the slow derivative terms act on R, Z, T while holding the auxiliary variable Yi fixed. The fast terms act on Yi and arise from the chain rule along the physical phase map. For example, −ε∂T is the slow time term and ciNi is the fast time term. The velocity and residual normalizations give the time factor Q2A+1/2Q−A = Q1+h and the viscous factor Q2A+1/2Q−AQ−1 = ε.
1027. u1027 The integer i(ℓ) in (6.5) satisfies
1028. u1028 These inequalities prove both comparisons in (6.6): the choice of covering index makes ci
1029. u1029 ∗, and the choice of dr leaves only the factor ε−κs in Mi, apart from its comparable to S−1 power of S∗. These are the two derivative scales used below.
1030. u1030 The absolute coordinate derivatives r, ∂z, ∂θ, t commute: the only variable coefficient is a function of r multiplying a constant torus direction. Cylindrical frame coefficients, such as 1/R, must still be retained when differentiating vector components. Every correction depending on the auxiliary variable is supported away from the axis, so no smoothness of rdr at r = 0 is required. Auxiliary periodicity imposes no spatial periodicity after restriction to (6.3).
1031. u1031 We will also integrate along each of the two torus directions. On a Fourier mode of frequency n, this requires division by the scalar product of that direction with n. Both directions satisfy the bound
1032. u1032 (6.7) Indeed vr · n = (n1 + n2) −
1033. u1033 2 n2; multiplication by its algebraic conjugate gives the nonzero integer (n1 + √ n2)2 −2n2
1034. u1034 2. The conjugate has absolute value at most C|n|. For vt, we use (n2 −n1) +
1035. u1035 2 n1 and the same argument. The inverse directional operators on zero-mean torus functions therefore have only a finite loss of torus derivatives; they are used in Lemma 8.2 and in the mean correction equations.
1036. u1036 6.2. Slow cutoffs and disjoint auxiliary rectangles. We next assign supports to the localized
1037. u1037 oscillations. The squared partition of unity formed from the slow cutoffs in (6.9) will recover the target stress when their covariances are summed in (7.30). To eliminate cross terms between distinct local oscillations, we place them in disjoint auxiliary rectangles whenever their slow supports meet, as proved in Lemma 6.1.
1038. u1038 We choose a smooth squared partition ∑ℓχℓ(q)2 = 1, with χℓsupported where q/Q ∈ [1/2, 2]. In each band we choose a product squared partition ∑a χℓ,a(R, Z, T)2 = 1 whose mesh is S−3 ∗
1039. u1039 in each coordinate and whose supports extend at most one mesh length on either side of a grid point. Normalizing translates of a smooth bump by the square root of their squared sum constructs both partitions. To specify which boxes and labels we use, we write
1040. u1040 Fix a sufficiently large lower band index ℓ0, and decrease qbig so that qbig ≤2−ℓ0. The closed active part of band ℓat τ ≥0, expressed in this chart, is
1041. u1041 We select the indices, representatives, and labels by
1042. u1042 (6.8)
1043. u1043 Γ = {(ℓ, a, σ) : ℓ≥ℓ0, a ∈Iℓ, σ ∈{+, −}}.
1044. u1044 For γ = (ℓ, a, σ) ∈Γ, we define its slow cutoff and its closed support in physical slow coordinates by
1045. u1045 (6.9) The supports are taken in the coordinate domain, including its smooth extension near τ = 0. The signs duplicate the cutoff, so on the active shell the partition identity is
1046. u1046 ∑ Representatives and all other discrete choices are fixed before taking derivatives.
1047. u1047 On these supports, R lies in a fixed compact subinterval of (0, ∞) and Z, T in fixed bounded intervals. These bounds may depend on the fixed profile scale C. Derivatives of any fixed order in (R, Z, T) of these cutoffs are bounded by powers of S∗. The same statements hold on fixed small enlargements within the coordinate domain, using the smooth base extension from Proposition 5.5.
1048. u1048 Each label receives a small rectangle in its band torus. Its coordinates are chosen along vr, vt, so that the pulse coordinate will advance at unit speed under t∗. For each label we will choose a center cγ ∈T2. We write πj(Y) = Jj
1049. u1049 gY (mod Z2) for the covering in (6.5). Given r0 > 0, we define a band rectangle, its enlargement, and their preimages on the absolute torus by
1050. u1050 (6.10)
1051. u1051 We take r0 small enough that the enlarged parametrizations are injective. We choose a representative of cγ in R2. On a local lift of Rγ, the coordinates are
1052. u1052 (6.11)
1053. u1053 Here kcopy ∈Z2 selects the local lift; the same coordinates extend to R+ γ
1054. u1054 γ . The dual coordinates satisfy Liηg = Niξg = 0 and Niηg = Liξg = 1. Since ci is constant within the chart, (6.6) gives
1055. u1055 (6.12) Thus v measures normalized time along a rectangle while staying constant under the normalized spatial derivatives. We now choose the centers so that these local coordinates can be used without cross products between different local oscillations.
1056. u1056 Lemma 6.1. For the labels Γ and slow supports Kγ in Equations (6.8) and (6.9), there exist centers cγ and a radius r0 > 0, common to all labels and independent of the band, such that the enlarged rectangles in (6.10) are injectively parametrized and
1057. u1057 (6.13) Proof. Step 1: bound the number of possible interactions. If two dyadic supports meet, then q/Q, q/Q′ ∈[1/2, 2], so |ℓ−ℓ′| ≤2. We enlarge each mesh box by a fixed factor. We join two distinct labels if these enlarged slow boxes intersect in physical coordinates and their band indices differ by at most four; we also join opposite signs of one box. This graph has
1058. u1058 bounded degree. In fact, for |ℓ−ℓ′| ≤4, the changes of physical scale in each coordinate are bounded, as are the ratios ℓ2/(ℓ′)2. A box can consequently meet only a bounded number of enlarged boxes of any such band. The degrees and the number of colors required are independent of the band.
1059. u1059 There is also a fixed bound on the differences of covering indices i(ℓ). If all bands have ℓ≥ℓ0, then, for |ℓ−ℓ′| ≤4,
1060. u1060 |i(ℓ) −i(ℓ′)| ≤1 + 4(1 + h) log 2 + 8/ℓ0
1061. u1061 (6.14) after taking an integer upper bound.
1062. u1062 This follows by subtracting ((1 + h)ℓlog 2 − 2 log ℓ)/ log Tg and using the mean value theorem; flooring adds at most one.
1063. u1063 Step 2: separate the centers under the relevant coverings. We greedily color the countable graph with finitely many colors. We choose one rational center cν ∈T2 for each color ν, and assign cγ = cν when label γ has that color. The color centers are chosen subject to all the ordered constraints
1064. u1064 (6.15) Each excluded equality is a proper closed constraint on the finite tuple of centers. For a self-pair and ∆> 0, this uses that J∆ g
1065. u1065 g −I is invertible. Their complement is open and dense and contains a rational tuple. Thus all the finitely many forbidden differences have a positive separation from the integer lattice.
1066. u1066 Step 3: choose one rectangle size. We take rectangles in the eigenvector coordinates around these centers, small enough to be injective on the band torus and to retain that separation after a fixed enlargement. If a point belonged to the lifted rectangles of adjacent labels of levels i and i + ∆, it would give cµ −J∆ g
1067. u1067 g eν −eµ modulo the lattice, with |eν| + |eµ| ≤Cr0. The finite bound on ∆contradicts (6.15) for one sufficiently small fixed r0. By Step 1 every pair with overlapping slow supports was joined in the graph. This proves
1068. u1068 □ (6.13), including the separation needed for supports of derivatives.
1069. u1069 For the remainder of the construction, we fix the centers cγ and radius r0 supplied by Lemma 6.1, together with one integer ∆max satisfying (6.14). In particular, for a family (Fγ)γ∈Γ, with supports taken in (r, z, t, Y) uniformly in θ,
1070. u1070 The same identity holds for derivatives of smoothly extended fields and after evaluation on (6.3).
1071. u1071 We choose a nonzero smooth transverse cutoff χg ∈C∞ c
1072. u1072 c ((−r0, r0)), with 0 ≤χg ≤1. The time cutoff 0 ≤ψ ≤1 is rescaled from a fixed function and has
1073. u1073 (6.16) These cutoffs lie strictly inside the enlarged rectangles. Multiplication by ψ will give temporal support inside the rectangle after the pulse equation has been solved there. 6.3. A common torus for overlapping bands. The rectangle Rγ in (6.10), for a label in band ℓ, is expressed in the band coordinate Yi = πi(ℓ)(Y) of (6.5). To add fields from overlapping bands, we first express these rectangles and fields on the same torus H, without changing their physical evaluations. Lemma 6.2 gives uniformly bounded derivative factors
1074. u1074 for this change of representation and preserves normalized Haar averages. Fix (z, t) with 0 < q(z, t) < qbig. For a sufficiently small neighborhood U of this point, we set
1075. u1075 (6.17)
1076. u1076 We choose U so that its band indices differ by at most four. Then (6.14) gives the factorization
1077. u1077 (6.18) We define RH,+ γ
1078. u1078 by the same preimage formula with R+ γ
1079. u1079 γ . Since πi0 is surjective, the separation in (6.13) also holds for these common-torus rectangles. A sum of band fields now has the representation
1080. u1080 ∑ Lemma 6.2. For U, i0, H as in (6.17), each change H ↦→Yi has uniformly bounded derivative factors,
1081. u1081 γ consists of 14∆ℓ≤14∆max disjoint lifts of Rγ. Haar averages on the absolute, common, and and RH γ band tori agree whenever the function is a pullback from the respective torus. Radial integration at fixed (z, t), torus translation, averaging, and directional Fourier inversion on zero-mean functions preserve common-torus descent and introduce no new dyadic band.
1082. u1082 Proof. Step 1: compare the covering maps and their averages. We write ∆= ∆ℓ. The matrix norms of J∆ g
1083. u1083 g are bounded because 0 ≤∆≤∆max. The preimage rectangles are indexed by a finite quotient:
1084. u1084 Here cγ denotes the chosen representative in R2. Haar compatibility can be checked on Fourier characters:
1085. u1085 (6.19) Indeed a character of frequency n pulls back to frequency (J∆ g
1086. u1086 g )Tn, which is zero exactly when n = 0.
1087. u1087 Step 2: preserve the common representation under the stated operations.
1088. u1088 Radial integration fixes (z, t), and hence q = q(z, t). It leaves every band cutoff unchanged even though it can pass through many radial grid boxes. Torus translations commute with deck translations; averaging and directional Fourier multipliers preserve the lattice of pullback
1089. u1089 □ frequencies. These facts prove the preservation claims.
1090. u1090 Fields are defined on the extended domain with the independent variable Y, using the original physical normalization; the common torus is a local representation. On overlapping neighborhoods U, V, fix the same normalized band chart and write fU, fV for the two common-torus representatives. Their compatibility is the identity
1091. u1091 (6.20) at points over U ∩V. Both sides equal the original field in that chart, so the identity also holds on the support closures where the set of overlapping dyadic supports changes.
1092. u1092 For a fixed band ℓ∈L(U), we write ∆= ∆ℓ.
1093. u1093 On its chart, a field descending to H = Ji0g Y has the operators in (6.6) with i replaced by i0, while Q remains fixed. In particular, ci0 = T−∆ g
1094. u1094 Mi. Since ∆is bounded by (6.18), the change of torus leaves the powers of ε in the derivative bounds unchanged.
1095. u1095 Radial integration can collect more terms than pointwise multiplication. The mesh gives the following bounds for sums over slow labels, with Sℓ= ℓ2 when several bands are compared.
1096. u1096 Lemma 6.3 (Number of relevant labels). There is a constant C, independent of the band and point, such that
1097. u1097 For a fixed band ℓ, compact chart set B ⊂R3, and bounded normalized radial interval IR,
1098. u1098 #{(ℓ, a, σ) ∈Γ : Bℓ,a ∩B̸ = ∅} ≤CBS9
1099. u1099 #{(ℓ, a, σ) ∈Γ : Bℓ,a ∩(IR × {Z} × {T})̸ = ∅} ≤CIRS3 The same bounds hold for fixed-factor enlargements of the mesh boxes.
1100. u1100 Proof. At a point, the dyadic partition and each one-dimensional grid partition have bounded overlap; the two signs add a factor of two. On a bounded interval, a mesh of size S−3 ℓ
1101. u1101 has O(S3
1102. u1102 ℓ) positions. A compact chart varies three coordinates, giving O(S9
1103. u1103 ℓ) boxes; a radial line
1104. u1104 □ varies only one, giving O(S3
1105. u1105 ℓ). Fixed-factor enlargement changes only the constants.
1106. u1106 During radial integration the possible bands are fixed:
1107. u1107 The selected slow supports lie in a uniformly bounded normalized radial interval. Thus Lemma 6.3 bounds the number of labels met by the integral by O(S3
1108. u1108 ref for any ℓref ∈L(U). All these Sℓare comparable. Counting the common-torus lifts adds at most the fixed factor 14∆max. The sum over these labels therefore permits polynomial growth in S∗, without changing the bands or their covering-index differences.
1109. u1109 To solve an amplitude equation from the beginning of a pulse to its current coordinate v, we need the earlier points on the same lifted rectangle. The slow and transverse coordinates remain fixed along this path. A source formed from several bands may be a function of H without being a function of a particular Yi: it can take different values at the distinct preimages of that band-torus point. We therefore describe the path separately on each lifted rectangle. The linear functional λt(w) = vt · w/(1 + b2
1110. u1110 g) satisfies λt(vr) = 0, λt(vt) = 1. Within one lifted band rectangle, we write ηg(H) = λt(J∆ g
1111. u1111 g H −cγ −kcopy). For w ∈[0, Ls], we define
1112. u1112 (6.21) This is the point with the same slow coordinates and band transverse coordinate ξg, and with pulse coordinate w. Thus a source f (R, Z, T, H) is evaluated along the path as f (R, Z, T, Hw), without averaging over the other preimages. Since λtJ∆ g
1113. u1113 (6.22)
1114. u1114 Higher derivatives of these affine maps vanish. Equivalently, in the original auxiliary coordinate Y, the path is Y′ = Y + T−i g
1115. u1115 g −ηg)vt. A deck translation preserving H translates this path by the same amount and reindexes the rectangle copy. A uniquely specified solution with zero initial data therefore descends to the common torus. The precise differentiated inverse estimate is proved in Proposition 7.2; (6.22) supplies its coordinate change without a new power of ε.
1116. u1116 6.4. Coefficient bounds and their algebra. The residual estimates must account for both the size of a correction and its vanishing at the boundary of its support. Let f, g denote angular mean coefficients and a, b denote labelled wave amplitudes. Their products and normalized derivatives appear in the correction equations, for example
1117. u1117 We define three classes in Definitions 6.4 and 6.5 to record the size of a mean coefficient, a wave amplitude, and a radial moment, respectively:
1118. u1118 The exponent α records a power of ε. We measure radial vanishing by the weight ζ in (6.23) and temporal decay by an envelope Pv satisfying 0 < Pv ≤1 on 0 ≤v ≤Ls. Its formula will be given in (7.12). The estimates are imposed at every fixed derivative order so that the shell extensions are smooth and the bounds survive the products and derivatives in Proposition 6.6.
1119. u1119 Recall the flat edge weight of Theorem 4.6. On the fixed active interval Xa < X < Xb, it may be written ζ(X) = exp
1120. u1120 δ(X) = min{1, log(X/Xa), log(Xb/X)},
1121. u1121 (6.23) where aa, ab > 0 are fixed; we extend ζ by zero outside the interval. For every b > 0 and finite N, ζbδ−N tends to zero faster than any power at either boundary. Both ζ and δ depend only on X, so they are constant along (6.21).
1122. u1122 The rapidly varying phase and its amplitude have different derivative scales. A coefficient derivative acts on the amplitude with the oscillatory exponential factored out. Fix a finite stage j, and use normalized chart representatives. For a coefficient a, we write
1123. u1123 These derivatives hold the band, label, representative, and harmonic integer fixed. The auxiliary coordinate derivatives measure each representative separately and transform by the chain rule when i0 changes. In the definitions below, Cj,I > 0 and bj,I, dj,I ≥0 may depend on j, I and the fixed profile data, and are uniform in band, label, rectangle copy, and point. Definition 6.4 (Mean and moment coefficients). A smooth coefficient f = f (R, Z, T, H) belongs to Mα if it descends to each applicable common torus with representatives satisfying (6.20), extends smoothly by zero outside the active radial shell, and on Xa < X < Xb satisfies
1124. u1124 (6.24)
1125. u1125 It may depend on H; it has no auxiliary rectangle restriction or pulse-envelope bound. For a function F = F(Z, T), define
1126. u1126 (6.25)
1127. u1127 Radial moment normalization. The moments to be estimated in Sα include the radial measure and weight in their normalization: if f∗= Qa f fphys, then
1128. u1128 (6.26) The auxiliary Haar average removes the dependence on Y; by (6.19), it can be taken on any applicable common torus. The same scaling identity holds before averaging. Phase, envelope, and support of a wave. For each label γ, we prescribe local data
1129. u1129 (6.27) Here pγ is fixed for each label, ϕγ is defined on its enlarged lifted rectangles, and the harmonic set Hj is independent of the band. The pulse construction specifies the phases and envelopes in Equations (7.3) and (7.12).
1130. u1130 We write x∗= (R, Z, T) and K∗ γ
1131. u1131 γ = Cℓ(Kγ). On each component of RH γ
1132. u1132 γ , we use the coordinates ξg, v from (6.11). The domains and allowed supports are
1133. u1133 (6.28)
1134. u1134 The slow coordinates range over the chart domain, and X = QR2/(2q). Replacing the closed rectangle in Eγ by RH,+ γ Definition 6.5 (Labelled wave coefficients). For γ ∈Γ and m ∈Z \ {0}, a smooth amplitude aγ,m on E+ γ
1135. u1135 γ belongs to Wα if its local formulas agree whenever lifts represent the same common-torus point, satisfy (6.20) on overlapping neighborhoods, and
1136. u1136 It extends smoothly by zero across the slow and transverse support boundaries and both shell edges. On Eγ ∩{Xa < X < Xb}, its bounds are
1137. u1137 (6.29)
1138. u1138 These are amplitude derivatives; Pv is only a pointwise weight. The continuation beyond v = 0, Ls has no envelope bound or temporal zero-extension condition.
1139. u1139 Cut-off fields and sums. An actual cut-off coefficient additionally has
1140. u1140 No divisibility by ηγ, χg, ψ is assumed.
1141. u1141 Let aν be a finite or locally finite family of amplitudes, with labels γν ∈Γ and harmonics mν ∈Hj, representing a real wave
1142. u1142 Here kγ is the integer k in (6.27) for the band of γ. We collect equal label–harmonic pairs before testing membership in Wα:
1143. u1143 ∑ Then w ∈Wα means that every Aγ,m belongs to Wα, with the uniform bounds in (6.29).
1144. u1144 The bounds at every fixed derivative order justify smooth zero extension at the shell edges: The bounds ζ and √ζ at absorb every fixed every derivative allowed inverse order power justify of smooth δ, including zero extension those introduced at the shell by differentiation. We can now determine the size and support of the products that occur in the momentum residual.
1145. u1145 Proposition 6.6. Each class in Section 6.4 is closed under finite sums at a fixed exponent. For α, β ∈R, products satisfy
1146. u1146 (6.30) For aγ,m ∈Wα and bγ,m′ ∈Wβ with the same label,
1147. u1147 (6.31) Before cutoff, a zero harmonic satisfies the mean bounds on its labelled rectangle. Actual waves with distinct labels satisfy, for every pair of derivative multi-indices I, J,
1148. u1148 For C ∈{M, W}, every coefficient derivative in a fixed common-torus representative preserves the ε exponent in its defining bounds. The operators below also preserve (6.20) and therefore act on the classes by
1149. u1149 (6.32)
1150. u1150 These mappings concern amplitudes with the phase factored out. The local coordinate derivative bounds, including those for ∂H, retain the prescribed supports. The wave rules are local to the labelled rectangles before cutoff, and hold for smoothly extended fields on the common torus after cutoff. Products, differentiation, and angular averaging preserve finite band-independent harmonic sets at every fixed stage.
1151. u1151 Proof. The triangle inequality gives closure under finite sums at a fixed exponent.
1152. u1152 Step 1: multiply the coefficient bounds. Leibniz’ formula bounds each fixed derivative of a product using finitely many derivative bounds for its factors. Powers of S∗and δ−1 add
1153. u1153 and are allowed to increase. The weight bounds in Equations (6.30) and (6.31) follow from 0 ≤ζ ≤1, 0 < Pv ≤1, and
1154. u1154 A mean coefficient may have larger radial or auxiliary support than a wave coefficient. Their product is supported within the wave’s support and retains its envelope bound.
1155. u1155 Step 2: separate labels and select angular harmonics. The support of a derivative of a smoothly extended function is contained in its original closed support. After the time cutoff has provided this smooth extension, Lemma 6.1 therefore eliminates all cross-label products of actual fields. Before that cutoff, the coefficient algebra is used only on its labelled band rectangle.
1156. u1156 Recall that ⟨·⟩θ is the physical angular average and ⟨·⟩Y is normalized Haar averaging in the auxiliary variable. By Lemma 6.2, the latter can be computed in any applicable torus coordinates. Within one label the product harmonic is m + m′. Because kpγ is a nonzero integer, angular averaging gives
1157. u1157 In particular,
1158. u1158 The second expression recalls the auxiliary zero-mean projection in (3.9); it is different from both the physical angular mean and the radial prefix average AX( f ) of (4.6). A mean coefficient may therefore have a nonzero auxiliary projection f ◦. Differentiation preserves each harmonic integer and products take pairwise sums, so finitely many operations preserve a finite harmonic set at a fixed stage.
1159. u1159 Step 3: apply the normalized derivative operators. The coordinate coefficients of Dr and all their coefficient derivatives are bounded by CIε−κsSCI
1160. u1160 ∗on the active annulus. Together with (6.6), these bounds prove the exponent estimates in (6.32); changing from a band to a common torus contributes only the bounded matrices of Lemma 6.2. The local ∂H bounds follow directly from (6.24) and (6.29).
1161. u1161 In a fixed band chart, ∂R,Z,T preserves (6.20), since the covering maps are independent of the slow coordinates. The auxiliary terms Mi0Li0 and ci0Ni0 represent the same absolute operators Qdr/2Labs and Q1+hNabs on every common torus. Thus the remaining operators in (6.32) also preserve compatibility. These derivative rules concern amplitude coefficients; differentiating eikmΦγ also differentiates the phase and introduces its explicitly retained factor km.
1162. u1162 Differentiation, integration, translation, averaging, and Fourier multipliers in (R, Z, T, H)
1163. u1163 □ preserve independence of θ.
1164. u1164 The wave support convention also fits the initial-value equations used to construct pulses. The path (6.21) fixes the same slow and transverse variables. If a source vanishes there along the whole rectangle, its linear solution with zero initial data is identically zero; after multiplication by ψ, it is supported in the original rectangle. Smooth source extension and smooth dependence of the linear equation give smooth extension at those boundaries. The mapping estimates at every fixed derivative order are proved in Proposition 7.2. The
1165. u1165 correction cycle of Proposition 9.6 preserves containment in these prescribed supports; its estimates require no divisibility by the original cutoff factors.
1166. u1166 7. Oscillatory realization and correction of the residual stress
1167. u1167 We fix the background (uB, pB) of Proposition 5.5 and the charts, slow cutoffs, and auxiliary rectangles of Equations (6.1), (6.9) and (6.10). We construct waves whose averaged quadratic velocity products supply the prescribed radial fluxes of azimuthal and axial momentum in the leading stress T0, with the physical normalization in (7.30). Their principal linearized evolution must also agree with the background shear and viscous diffusion in (7.5). The supports allow us to solve these problems locally and sum the waves without cross terms between distinct labels, by Lemma 6.1. For subsequent corrections, we solve the same amplitude equation with a prescribed harmonic source using the inverse in Proposition 7.2. A second linear operator, L in Proposition 7.6, prescribes a signed change in the leading covariance.
1168. u1168 The common ingredient is a linear amplitude equation determined by the background shear and viscosity. We choose phases for which the homogeneous solution th σ
1169. u1169 σ in Lemma 7.4 grows and then decays. Its Gaussian bound (7.16) makes the solution exponentially small near both ends of its interval. We call this solution a pulse. We compute the covariance of two pulses with different phase gradients and choose their amplitudes to match the stress. The strict cone condition (7.1) permits positive squared amplitudes in Proposition 7.5; linearizing at those fixed amplitudes permits subsequent stress increments of either sign. Finally, we take curls of supported vector potentials to enforce exact incompressibility in Lemma 7.7. The additional velocity terms and cutoff errors remain in the covariance estimate of Corollary 7.8 and the residual identity (7.40).
1170. u1170 All calculations use normalized chart representatives until physical factors are displayed. A band, its representative, its discrete labels, and its integer frequencies are fixed before differentiation. Constants for derivatives of any fixed order may depend on the fixed profiles and that order; in applications they may also depend on the finite correction stage. They remain uniform over bands, labels, and rectangle copies.
1171. u1171 7.1. Phase construction and a frame orthogonal to its gradient. We first construct the phase Φ in (7.3) that approximately follows transport by the background, controlling the defect in the leading transport terms as in (7.9). For a harmonic eikmΦ, the leading incompressibility condition on its amplitude tm is ∇∗Φ · tm = 0. The frame B in (7.8) for this two-dimensional plane will compare the undamped principal equation with a diagonal system having one positive and one negative eigenvalue, as in (7.10).
1172. u1172 Fix a label γ = (ℓ, a, σ) from (6.8). On each of its lifted rectangles we use the slow point x∗= (R, Z, T), transverse coordinate ξg, and pulse coordinate v ∈[0, Ls], where Ls = 2r0/ci ≍S∗, as in (6.11). Recall Q = 2−ℓ, ε = Qh, and S∗= ℓ2 from (6.1). The normalized spatial derivatives are those in (6.6); write ∇∗f = (Dr f, R−1∂θ f, Dz f ). A prime means ∂v with x∗and ξg fixed. In coefficient estimates, DI denotes ordinary derivatives in the listed slow and auxiliary coordinates; the individual operators Dr, Dz retain their normalized meanings.
1173. u1173 We write the chart base velocity as ber + Veθ + Gez, and set F = V/R and g = (RFR, GR). Thus F is angular velocity, and g is the radial shear in the tangential component order (θ, z). We identify a tangential two-vector with the vector in span(eθ, ez) having those components. By (5.42), b = O(ε) and the tangential velocity differs from its order-zero value by O(ε2), in
1174. u1174 every fixed slow derivative. A subscript 0 denotes an order-zero quantity evaluated at the chosen representative x0
1175. u1175 The positive lower bounds in Theorem 4.6(iii) allow us to define the following tangential frame and growth parameters:
1176. u1176 We use the quarter turn K = (−Nz, Nθ), so N, K are an orthonormal pair in the tangential plane, and take λ0 > 0. The definitions of a, bs, vs in Equations (4.11) and (4.20) give g0 = F0(−a, bs) and λ2 0
1177. u1177 0 > 0 and c0 < 0 follow from the profile bounds; they are not additional choices.
1178. u1178 The phase choice must also retain the stress directions allowed by Lemma 4.5. Before freezing N, K, c0, that pointwise condition is equivalently
1179. u1179 (7.1) For this equivalence, the frame, c0, and target are evaluated at the same point. Indeed, c2 0 0 = (vs −2)/2, while the absolute quotient without c0 equals |Jc|/(Pc −vs). At an annulus boundary the stress vanishes; there the quotient means the one-sided limit obtained from the smooth unit stress direction in Theorem 4.6(iii).
1180. u1180 The strict margin on the closed annulus permits us to choose one u∗> 0 such that
1181. u1181 1 + u2∗exceeds the supremum of |c0(T0,∗· K)/(T0,∗· N)|, with this interpretation at the u∗/ edges. We now freeze N, K, c0 at each representative. Uniform continuity of the leading stress direction preserves the strict inequalities, with a uniform margin, throughout sufficiently small slow boxes. The frequency and shear bounds, the positive lower bound for λ0, and the frame bounds also hold on fixed small enlargements of the active slow supports. We take these slow neighborhoods to be enlargements of the mesh boxes by one fixed factor, relative to τ ≥0, so every point remains within CS−3 ∗
1182. u1182 of its representative. Derivatives at τ = 0 are understood one-sidedly. The uniform cone margin persists on each neighborhood’s intersection with the closed active annulus, using the limiting stress direction at its radial edges. All these bounds and margins are uniform over bands and labels.
1183. u1183 The stress has two components, so we use two phases with different covariance directions in each slow box. The phase gradient will change along the pulse until viscous damping exceeds shear amplification. On the rectangle of sign σ ∈{+1, −1}, we define
1184. u1184 (7.2) We take Bs > 0. The carrier frequency k keeps εk2 of order one, so viscosity remains in the leading amplitude equation. To choose the angular and axial wave numbers p, pz, we first set
1185. u1185 Angular periodicity requires kp to be an integer. We take kp to be a nearest nonzero integer for to k˜︁p, the with label. a fixed With rule x0 = at σBsu∗/2, ties, and we leave define pz unchanged. the phase All on a these lifted wave rectangle numbers by are constants for the label. With x0 = σBsu∗/2, we define the phase on a lifted rectangle by
1186. u1186 Φ = pθ + pzZ/ε + x0R −v(pF + pzG),
1187. u1187 (7.3)
1188. u1188 x0 −v(pFR + pzGR), p/R, pz −εv(pFZ + pzGZ)
1189. u1189 (7.4)
1190. u1190 For a source coefficient fm ∈C3 and a nonzero integer m, the principal amplitude problem is to find tm ∈C3 and a pressure coefficient πm ∈C satisfying nΦ · tm = 0 and
1191. u1191 m + Ktm + m2dtm + ikmnΦπm = −fm,
1192. u1192 (7.5) Here K is defined below. This equation retains transport in the pulse coordinate, the base tangential velocity and radial shear, and the viscous term in which both derivatives act on the exponential. The two factors 2F in K include the cylindrical connection terms: radial motion changes the tangential frame as well as the base swirl. Eliminating the normal pressure term while differentiating nΦ · tm = 0 gives the projected evolution operator AΦ:
1193. u1193 (7.6)
1194. u1194 The slow transport, the error in phase transport, and the remaining viscous terms occur in the residual beyond (7.5); Proposition 9.1 estimates them. For later use, we fix the geometric coordinates and frame explicitly. Write nΦ,tan = (nΦ,θ, nΦ,z). Since kp ∈Z \ {0} and R > 0, the component nΦ,θ = p/R is nonzero. We may therefore define
1195. u1195 (7.7) The corresponding basis of n⊥ Φ
1196. u1196 (7.8)
1197. u1197 The next lemma compares the actual phase normal and projected operator with their reference values. In the resulting coordinates, the amplitude equation has two scalar reference growth rates and a small matrix error.
1198. u1198 Lemma 7.1. For the labels and phases in Equations (6.8), (7.2) and (7.3), there exists a sufficiently small q∗> 0, common to all labels and fixed derivative orders, such that every rectangle on 0 < q < q∗ has the following properties. Every eikmΦ, m ∈Z \ {0}, is single-valued in θ, with nonzero angular frequency. On the rectangle 0 ≤v ≤Ls,
1199. u1199 Eik := (t∗+ bDr + F∂θ + GDz)Φ = O(εSC
1200. u1200 (7.9) The estimate for Eik in (7.9) holds in every fixed coefficient derivative, with a corresponding polynomial degree. The coefficients extend smoothly to the fixed enlargement. All fixed mixed derivatives in the slow, transverse, and pulse coordinates of nΦ, and of the frames used here, have polynomial bounds in S∗.
1201. u1201 The real 3 × 2 matrix B in (7.8) has columns forming a basis of n⊥
1202. u1202 Φ, and is also viewed as a map C2 −→{t ∈C3 : nΦ · t = 0}. It has a uniformly bounded left inverse Bℓand satisfies
1203. u1203 (7.10) For the damping coefficient d in (7.5), put dref = εk2B2
1204. u1204 (7.11) Proof. Step 1: periodicity and estimates for the phase. Because kp ∈Z \ {0}, the angular frequency kmp is a nonzero integer for every m̸ = 0. This proves the periodicity assertion.
1205. u1205 Write HΦ = pF + pzG. Since Drv = Dzv = 0 and the base is independent of the auxiliary variable,
1206. u1206 These are precisely the three entries of nΦ. Before rounding, (p/R0, pz) · g0 = −σBsu∗/Ls. The slow box has diameter O(S−3
1207. u1207 ∗), the base comparison contributes O(ε2), and rounding
1208. u1208 + ε2 + k−1). Multiplicacontributes O(k−1). Consequently the error in pFR + pzGR is O(S−3 ∗ tion by v = O(S∗), together with the tangential error O(S−1 ∗
1209. u1209 + εS∗), proves the first estimate in (7.9) after one choice of q∗. Differentiation in the pulse coordinate gives |n′
1210. u1210 Φ| ≤C/S∗. Differentiating any fixed number of times introduces only powers of S∗; the discrete rounding is not differentiated. For the phase transport we compute each term:
1211. u1211 Consequently
1212. u1212 Every term has a factor ε or b = O(ε), and the remaining derivatives of fixed order have polynomial bounds in S∗. This proves the differentiated phase-defect estimates as well as the value estimate. Since εk2 ≍1 and Bs is bounded above and below, (7.11) follows. Step 2: coordinates on the moving plane. It remains to bound the frame (7.8) and express the projected evolution in its coordinates. The phase comparison and the lower bound on Bs bound |nΦ,tan| away from zero. The vector Na in (7.7) is the perpendicular close to N. Let M be the 2 × 2 matrix in (7.8). Then
1213. u1213 The matrices U, M, B and these left inverses are uniformly bounded.
1214. u1214 Step 3: diagonalization at the reference normal. For nref = Bs(s, K), write t = x(er −sK) + yN ∈n⊥
1215. u1215 ref, and form K0 from the reference coefficients. Since g0 = |g0|N,
1216. u1216 nref · K0t = 2BsF0{Kθ(1 + s2)x −sNθy}.
1217. u1217 Let e = er −sK. The vectors e, N are orthogonal, with squared lengths 1 + s2, 1. Directly,
1218. u1218 where each ordered pair gives the radial and tangential parts. Therefore
1219. u1219 Dividing the first coordinate by 1 + s2 gives the reference undamped matrix in (x, y):
1220. u1220 Its eigenvectors are the columns of M, with eigenvalues λ, −λ. To compare this matrix with the actual evolution, first use the value estimates for the base coefficients and nΦ −nref. They change the algebraic projection −K + nΦnT
1221. u1221 ∗). Also Ka −K and sa −s are O(S−1
1222. u1222 ∗), so the actual frame and its left inverse differ from their reference values by that order. Expressing the algebraic projection in these frames therefore changes the reference
1223. u1223 Φ)T/|nΦ|2 has the same bound, since matrix by O(S−1
1224. u1224 ∗). The moving-normal term −nΦ(n′
1225. u1225 ∗) and |nΦ| is bounded below. Also s′ = O(S−1 |n′
1226. u1226 Φ| = O(S−1
1227. u1227 ∗), so the definitions of U, M give
1228. u1228 U′, M′ = O(S−1
1229. u1229 ∗). Hence B′ = U′M + UM′ = O(S−1
1230. u1230 ∗), and multiplication by the bounded Bℓgives −BℓB′ = O(S−1
1231. u1231 ∗). These contributions give the error E in (7.10).
1232. u1232 We now specify why this choice of q∗works for every fixed derivative order. The value comparisons in Step 1 require only finitely many smallness conditions. For example, we can require to bound the errors multiplied by v = O(S∗) by C/S∗. Since ε = 2−hℓ, k−1 ≤ε1/2, and S∗= ℓ2, this holds for all sufficiently large band indices. The bound 1 ≤εk2 ≤4 for ε ≤1 gives fixed positive upper and lower bounds for Bs. We make the phase-normal error smaller than half this lower bound. The estimate |nΦ,tan| ≥Bs −C/S∗then keeps the tangential phase normal √ uniformly nonzero; the same lower bound applies to |nΦ|. The formula
1233. u1233 1 + s2 and the fixed lower bound for |c0| likewise bound | det M| away from det M = −2c0 zero. Thus one lower bound on the band index, or equivalently one upper bound q∗on q, makes every denominator in the frame and its left inverse uniformly nonzero.
1234. u1234 With that domain fixed, any prescribed number of coefficient derivatives of the base coefficients and the phase normal nΦ has a bound CISbI∗. Applying the product and quotient rules to the displayed frame formulas gives bounds of the same form for U, M, B, Bℓand their pulse derivatives, using the lower bounds just established. Differentiating the phase defect retains its factor ε: the band is fixed, and every fixed derivative of b is O(ε). These arguments require bounds on the higher derivatives, with constants and polynomial degrees allowed to depend on I; they impose no further smallness condition on those derivatives.
1235. u1235 □ Consequently they use the same q∗.
1236. u1236 7.2. Solving the pulse equation and controlling derivatives. The frame B from Lemma 7.1
1237. u1237 reduces (7.5) to an ordinary differential equation along each pulse. In this subsection we will first solve it with zero initial amplitude and a prescribed source fm, obtaining the linear inverse used to remove later harmonics in Proposition 7.2. We will then solve the zero-source equation with a specified nonzero initial amplitude to produce the pulses for the leading covariance in Lemma 7.4. Localizing both solutions requires decay at both ends of the interval. Subtracting the reference damping from the growing eigenvalue defines the common envelope
1238. u1238 (7.12) We also write Pv = P(v), as in the coefficient class (6.29). The estimates (7.14) below show that a source bounded by this envelope has a response with the same envelope, at the cost of polynomial factors in S∗. Multiplying a solution by ψ will therefore change its principal equation only where the solution and source are exponentially small, as shown after (7.40). Proposition 7.2. Fix α ∈R, a finite set of harmonics 0 < |m| ≤M, and a label at level i on a common torus H = Yi0, with i −i0 bounded as in Lemma 6.2. For each harmonic, let fm(x∗, H) ∈C3 be a coefficient descending to H, smooth on the fixed enlargement of that label’s rectangles. Assume that it has a fixed closed slow support contained in the label’s slow support, a compact transverse support contained in supp χg, and support in the closed active shell Xa ≤X ≤Xb, with smooth zero extension across all these support boundaries. Assume also that on Xa < X < Xb and 0 ≤v ≤Ls, for every fixed multi-index I of derivatives in (R, Z, T, H),
1239. u1239 On the domain 0 < q < q∗fixed in Lemma 7.1, there is a unique amplitude tm satisfying nΦ · tm = 0, with zero initial data, and a pressure coefficient πm such that (7.5) holds. They are given by
1240. u1240 (7.13)
1241. u1241 Φ f = f −nΦ(nΦ · f )/|nΦ|2 is orthogonal projection. The source is evaluated along the Here projn⊥ Φ path Hw in Equation (6.21), with x∗= (R, Z, T) and the transverse coordinate fixed. The coefficients descend to the same common torus and preserve the specified slow, transverse, and shell supports, including smooth zero extension there. The solution is defined on the entire closed pulse interval 0 ≤v ≤Ls. On Xa < X < Xb, for every fixed I,
1242. u1242 (7.14)
1243. u1243 (7.15) Constants and degrees may depend on M, I, and the prescribed source bounds. The threshold q∗ does not depend on M, α, or the correction stage. If the source has uniform one-sided derivatives in the slow variables at τ = 0 on compact sets with q > 0, the solution and pressure have the same one-sided derivatives.
1244. u1244 Only descent to the common torus H is required in Proposition 7.2: the source may take different values at distinct preimages of a band-torus point. No orthogonality to nΦ or divisibility by either rectangle cutoff is assumed. The support conclusion concerns the slow, transverse, and shell variables; no temporal zero extension is asserted before multiplication by ψ.
1245. u1245 Proof. The moving frame gives a two-dimensional equation. We first bound its forward propagator by P(v)/P(w), then use the resulting Duhamel formula for values and derivatives. Finally, we check support, descent, and the normal pressure equation.
1246. u1246 Step 1: the envelope and forward propagator. The scalar reference growth rate is
1247. u1247 It vanishes at y = u∗, and its derivative in the pulse coordinate is
1248. u1248 Because u∗/2 ≤|s(v)| ≤3u∗/2, this derivative lies between −C/Ls and −c/Ls. By (7.12), (log P)′ = anet(|s(v)|), and both log P and its first derivative vanish at v = Ls/2. Integrating these derivative bounds twice from the midpoint gives
1249. u1249 e−C(v−Ls/2)2/Ls ≤P(v) ≤e−c(v−Ls/2)2/Ls ≤1.
1250. u1250 (7.16)
1251. u1251 The envelope therefore decays on both sides of the midpoint. To show that the sourced amplitude tm has the same decay, we write tm = Bzm using the actual moving frame. The coordinate vector zm satisfies
1252. u1252 (7.17)
1253. u1253 Let Vm(v, w) be the forward fundamental matrix for the homogeneous equation z′ = Amz, normalized by Vm(w, w) = I2; thus ∂vVm(v, w) = Am(v)Vm(v, w). The scalar nature of the damping gives the exact identity
1254. u1254 (7.18) The Euclidean norm inequality for the m = 1 equation is ∂v|z| ≤(λ −dref + C/S∗)|z|. As Ls/S∗is bounded, it follows that
1255. u1255 (7.19) This value bound is uniform in all nonzero integers m, since the exponential in (7.18) is at most one.
1256. u1256 Step 2: values and coefficient derivatives. The zero initial condition gives Duhamel’s formula. The bound on gm(w) contains P(w), which cancels the denominator in (7.19); integration contributes Ls = O(S∗):
1257. u1257 after enlarging b0 for the frame factors. To differentiate this formula, first use the slow variables, the transverse coordinate at v = 0, and the pulse coordinate. In the next equation, DI differentiates only the slow variables and initial transverse coordinate, holding v fixed. These parameter derivatives commute with ∂v, so differentiating the two-dimensional equation gives
1258. u1258 (7.20) All these parameter derivatives of the zero initial data vanish. If the coefficient derivatives obey |DJAm| ≤CJScJ
1259. u1259 ∗, an induction using (7.19) increases the polynomial degree by at most
1260. u1260 at the Ith step. The inverse-distance exponent can be enlarged to the maximum of the source and at the lower-order Ith step. The exponents. inverse-distance The slow exponent point is can fixed be enlarged in the integral, to the maximum so √ζ and of δ the are source fixed there. Pulse-coordinate derivatives then follow from (7.17). This proves every fixed-order derivative estimate with no decrease of α. Differentiation in the moving frame retains the constraint nΦ · tm = 0 and accounts for derivatives of the plane n⊥ Step 3: common-torus coordinates and support. The preceding estimates used ξg, v on one lifted rectangle. To express the derivative estimates in H, we use the directions vr, vt from (6.2) and recall the linear functional λt(w) = vt · w/(1 + b2
1261. u1261 g), so λt(vr) = 0 and λt(vt) = 1. For a current point H with pulse coordinate v(H), the point on the same path at pulse coordinate w is
1262. u1262 Since vt is an eigenvector of Jg with eigenvalue Tg,
1263. u1263 Thus the pullback H ↦→Hw at fixed w has bounded derivatives, and returning to the current common coordinate introduces only powers of S∗multiplying fixed derivatives in the pulse coordinate. Together with Step 2, this proves (7.14) in the stated coefficient variables. Deck translations reindex the copies and commute with this path. Uniqueness of the initial-value
1264. u1264 equation with zero data proves descent to H, without identifying sources on different preimages of the finer band torus. The path holds the slow and transverse points fixed. If the source vanishes along their whole rectangle, the solution does also. The differentiated equation, applied at their support boundaries, proves smooth zero extension and support containment.
1265. u1265 Step 4: the constraint and pressure. The definition of AΦ gives
1266. u1266 Zero initial data therefore preserve nΦ · tm = 0. Substitution in the projected equation yields
1267. u1267 Multiplication of (7.13) by ikmnΦ gives the negative of this normal vector. Thus (tm, πm) satisfies the full principal cancellation (7.5). The factor (km)−1 = O(ε1/2) and the fixed coefficient-derivative bounds prove (7.15). In particular, normal components of the source have been canceled as well. On a compact set with q > 0, the base has the one-sided endpoint derivative data of Proposition 5.5. The same parameter-dependent equations and differentiated Duhamel formulas extend those endpoint derivatives from the source to tm
1268. u1268 □ and then to πm.
1269. u1269 The harmonic factor in (7.18) only adds damping. Consequently the later appearance of higher harmonics does not force a smaller space-time domain for these inverses. Corollary 7.3. The solution operators fm ↦→(tm, πm) for the projected amplitude equation with tm(0) = 0 are defined at every finite correction stage on the same domain 0 < q < q∗. Increasing the finite harmonic set or the derivative order changes constants and polynomial degrees, but does not require shrinking this domain.
1270. u1270 Proof. The frame and damping comparisons require only the single finite choice in Lemma 7.1. The exact factor (7.18) makes every higher harmonic at least as damped in value as the first. In (7.20), the extra factors of m and coefficient derivatives enlarge only the constants and polynomial degrees in the source bounds. Thus the previously chosen threshold q∗remains
1271. u1271 □ valid.
1272. u1272 The zero-data inverse is now available for prescribed harmonic sources. For the leading stress, we instead need a nonzero real solution whose radial component stays comparable to P(v). We choose its initial value in the growing coordinate of the frame B; its amplitude and direction must remain controlled through the subsequent decay.
1273. u1273 Lemma 7.4. For each sign σ, let th σ
1274. u1274 σ with z+(0) = P(0), z−(0) = 0. This specifies the real amplitude of the homogeneous m = 1 problem, with pressure given by (7.13) at fm = 0. With Ka, Na, sa as in (7.7), write th σ
1275. u1275 (7.21) Every fixed derivative of this homogeneous solution in the coefficient variables or pulse coordinate is bounded by CISbI∗P(v). Its homogeneous pressure, given by (7.13) with f = 0, has the extra factor ε1/2.
1276. u1276 Proof. We control the ratio of the decaying and growing frame coordinates, then convert to the radial and tangential coordinates x, y. Write E = (Eab). While z+ > 0, the ratio r = z−/z+ satisfies
1277. u1277 r′ = E21 + (−2λ + E22 −E11)r −E12r2, Let cλ > 0 be a uniform lower bound for λ. For a sufficiently large fixed Kb and all sufficiently large S∗,
1278. u1278 r=±Kb/S∗≤−2cλKb + C(1 + S∗ Kb/S∗+ K2 Thus |r| ≤Kb/S∗. Since z+(0) = P(0) > 0, the envelope definition gives
1279. u1279 This prevents z+ from reaching zero and proves cP ≤z+ ≤CP on the whole √ interval. Multiplication √ by the matrix in (7.8) gives x = z+ + z−= z+(1 + r) and y = c0
1280. u1280 1 + s2(z+ − z−) = c0
1281. u1281 Differentiating the coordinate equation and using (7.19) proves the derivative estimates just as in (7.20); the initial value
1282. u1282 □ P(0) is fixed with the label. The pressure formula supplies the extra factor ε1/2. Energy transfer from the shear.
1283. u1283 The pulse equation also identifies the source of the homogeneous growth. For the real amplitude t = th
1284. u1284 σ, the constraint nΦ · t = 0 makes the normal term in AΦt orthogonal to t. Moreover,
1285. u1285 t · Kt = −2Ftrtθ + (2F + RFR)trtθ + GRtrtz = g · Taking the scalar product of t′ = AΦt −dt with t gives
1286. u1286 (7.22) The first term gives energy exchange with the base shear; the second gives viscous dissipation. The vector tr(tθ, tz) records transport of azimuthal and axial momentum across a cylinder. At the representative point, decompose such a flux vector as T = TNN + TKK. The energy transfer is −g0 · T = −|g0|TN; the covariance construction also prescribes the transverse component TK. We therefore need both components of the covariance to match the prescribed momentum flux.
1287. u1287 Viscosity remains a leading-order term in the pulse equation. At viscosity one, the base velocity and radial length give Reynolds number of order q−h. The carrier wavelength is velocity √Q/k ≍Q1/2+h/2, and radial whereas length give the Reynolds physical amplitude number of of order the leading q−h. The oscillations carrier wavelength constructed is below has ≍Q1/2+h/2, order Q−A√ε whereas = Q−1/2−h/2, the physical up amplitude to powers of of S∗. the The leading powers oscillations of Q in their constructed product cancel. Thus εk2 ≍1 in the pulse equation throughout the correction process.
1288. u1288 7.3. Covariance of the leading oscillations and its linearization. The two real homogeneous
1289. u1289 pulses th
1290. u1290 ± from Lemma 7.4 determine two covariance directions in each slow box. Their scalar amplitudes a± will be chosen in Proposition 7.5 so that a positive combination of these directions equals the leading stress. In this subsection we will compute the two covariances, solve for the squared amplitudes, and then differentiate this relation to obtain signed stress corrections using L from Proposition 7.6.
1291. u1291 Recall the angular and auxiliary averages in (3.7), with normalized Haar measure on the auxiliary torus. They are taken at fixed slow variables, before evaluation at the physical phase map. For real velocity fields w, v on the extended domain, we write wtan = (wθ, wz).
1292. u1292 The product wrwtan records radial transport of azimuthal and axial momentum. At each fixed slow point, we define the covariance C(w) and symmetric cross term B(w, v) by
1293. u1293 (7.23) The averages can be computed on any applicable common torus by Lemma 6.2. Expansion of the product gives DC(w)[v] = B(w, v).
1294. u1294 Write β = (ℓ, a) for a slow box and γ = (β, σ) for one of its two rectangle labels. The two signs have the same slow cutoff ηβ = η(β,+) = η(β,−); thus the squared partition sums over β, with each box counted once. On the slow neighborhood of β, we define
1295. u1295 and extend by zero off that sign’s auxiliary rectangle. The factors χg, ψ give smooth extension there. The field bσ includes neither the slow cutoff ηβ nor a scalar amplitude. The two signs have disjoint auxiliary supports.
1296. u1296 Let H = [ C(b+) | C(b−) ], a real 2× 2 matrix depending on the slow point. Its columns have component order (rθ, rz). For any real numbers a+, a−, disjointness gives C(a+b+ + a−b−) = H(a2
1297. u1297 −)T. Thus the nonnegative span of its columns is exactly the set of stresses obtainable by these two waves. The strict cone condition will give positive coefficients for the leading stress, whose square roots define the real wave amplitudes.
1298. u1298 Proposition 7.5. For each slow box β, use the homogeneous pulses of Lemma 7.4 to form b± and H as above, and use the order-zero target T0,∗= (Q/q)A+1/2T0. After a further single decrease of q∗, the matrix H is invertible throughout each enlarged slow neighborhood, and the components of H−1T0,∗ are positive on its intersection with Xa < X < Xb. We may therefore define, with componentwise square roots,
1299. u1299 (7.24) The squared amplitudes satisfy, for every fixed slow multi-index I on Xa < X < Xb,
1300. u1300 (7.25) The coefficients aσ extend smoothly by zero at the shell edges. On the label’s slow neighborhood, the harmonic coefficients of W0 obey the derivative and envelope estimates of W1/2. Multiplication by the slow cutoff gives ηβW0 ∈W1/2. Before that multiplication, the covariance is
1301. u1301 (7.26)
1302. u1302 This positive-coefficient decomposition uses only the order-zero target T0,∗. Proof. The proof of Lemma 7.1 applies throughout these neighborhoods: their diameter remains O(S−3
1303. u1303 ∗), and the base comparison holds on the enlarged annulus. The phase and homogeneous-pulse estimates, and hence the column estimates below, hold there with uniform constants.
1304. u1304 Step 1: compute and estimate the covariance columns. The angular frequency kp is a nonzero integer, so the angular average of cos2(kΦσ) is exactly 1/2. We may compute the auxiliary average on the band torus. There the rectangle parametrization has area element | det(vr, vt)| dξg dηg, with dηg = ci dv. Recall that x, y are the geometric coordinates of th σ
1305. u1305 σ in Lemma 7.4. This amplitude is independent of ξg and has radial component x. Therefore the column Hσ equals
1306. u1306 (7.27)
1307. u1307 Define the corresponding positive scalar by
1308. u1308 The Gaussian envelope and the region where ψ = 1 give
1309. u1309 Indeed the substitution w = (v −Ls/2)/√Ls reduces these to integrals of e−cw2 and |w|e−cw2.
1310. u1310 , and the normalized x2ψ2 average of |v −Ls/2|/Ls is O(S−1/2 ∗ Hence hσ ≍ci
1311. u1311 ). The definitions of Ka, Na, sa in (7.7) give the tangential part of the pulse: th
1312. u1312 σ,tan = −saxKa + yNa. By Lemma 7.4, its ratio to x differs by O(S−1
1313. u1313 1 + s(v)2N. The preceding weighted average concentrates near v = Ls/2, where s = σu∗. Thus, setting Ac = −c0
1314. u1314 (7.28) The derivatives of the covariance columns and homogeneous solutions have polynomial bounds by Lemma 7.4.
1315. u1315 Step 2: solve for positive squared amplitudes. First omit the errors eσ in (7.28). For a target T = TNN + TKK, the two scalar equations then give
1316. u1316 Thus the reference cone is TN < 0, |TK| < (u∗/Ac)(−TN). The two formulas above give the positive coefficient decomposition. Our choice of u∗and the uniform leading cone margin give |TK|/u∗≤(1 −ηc)(−TN/Ac) for some fixed ηc > 0 and T = T0,∗, by (7.1). Freezing N, K, c0 at the representative changes this normalized cone inequality by an error tending uniformly to zero. The smooth limiting stress direction gives the same comparison at the radial edges. The errors in (7.28) preserve this strict inequality and give
1317. u1317 (7.29) Step 3: derivatives, shell edges, and covariance. Apply the leading stress bounds (4.27) in the chart. Differentiating HH−1 = I2 gives polynomial bounds for all fixed derivatives of H−1; differentiating y = H−1T0,∗then proves (7.25). For |I| ≥1, the chain rule gives
1318. u1318 ∑
1319. u1319 ∑
1320. u1320 ∏
1321. u1321 The bounds in (7.25) therefore imply
1322. u1322 The same bound holds for I = 0. Its right-hand side tends to zero at each shell edge for every fixed derivative order, so the zero extension of aσ is smooth. The bounds for aσ, Lemma 7.4, and the cutoffs χg, ψ give the local derivative and envelope bounds for W0. Multiplication by ηβ then gives ηβW0 ∈W1/2. Disjointness of the two rectangles and (7.24) yield C(W0) = εHy,
1323. u1323 □ which is (7.26).
1324. u1324 Each slow box now has the required local covariance. Multiplication by the common slow cutoff ηβ multiplies (7.26) by η2
1325. u1325 β. Returning to physical velocities and summing slow boxes and bands gives the leading physical stress pair
1326. u1326 ∑
1327. u1327 (7.30) To check the scale explicitly, A = 1/2 + h gives
1328. u1328 Q−2AεT0,∗= Q−2A+h(Q/q)A+1/2T0 = Q−A+h+1/2q−A−1/2T0 = q−A−1/2T0. We then use ∑β η2 β
1329. u1329 β = 1; cross-label products vanish by the disjoint auxiliary supports.
1330. u1330 The sum supplies the leading physical stress. Later residuals require stress increments of either sign, which we obtain by varying the amplitudes to first order about this fixed positive representation. We keep the positive coefficients aσ fixed at their values in Proposition 7.5. Let α ∈R, and let Σ(R, Z, T) ∈Mα be a real two-vector independent of both angular and auxiliary variables. This auxiliary independence is an additional requirement here; the class Mα alone allows auxiliary dependence. On the label’s slow neighborhood, we define
1331. u1331 (7.31) The divisions are componentwise and are defined first on Xa < X < Xb, where aσ > 0. The estimates below justify smooth zero extension at the shell edges.
1332. u1332 Proposition 7.6. For every α ∈R and every real two-vector Σ(R, Z, T) ∈Mα independent of the angular and auxiliary variables, (7.31) defines a linear operation LΣ. Its harmonic coefficients on the slow neighborhood satisfy the local derivative and envelope bounds of Wα−1/2, with smooth zero extension at the shell edges. With the slow cutoffs shown explicitly, its bounds and covariance are
1333. u1333 (7.32) The denominator aσ remains this fixed coefficient at every correction stage.
1334. u1334 No sign or relative-size restriction is imposed on Σ in Proposition 7.6.
1335. u1335 Proof. Apply Leibniz’ rule to dΣ = H−1(Σ/ε), using (7.29) and the differentiated inverse bounds proved in Proposition 7.5. The weighted derivative bounds on Σ give |DI(dΣ)σ| ≤ CIεα−1SbI∗ζδ−aI. Every derivative of y−1/2 σ
1336. u1336 is a sum of terms of the form C y−k−1/2 σ
1337. u1337 ν=1 DIνyσ. The k numerator weights cancel all but σ ζ−1/2 in the denominator, by (7.25). σ Leibniz therefore gives
1338. u1338 (7.33) The remaining square-root edge weight proves smooth zero extension; Lemma 7.4 gives the local wave estimates.
1339. u1339 Multiplication by ηβ gives the supported wave class.
1340. u1340 Write a = (a+, a−)T and δa = (δa+, δa−)T, and let ⊙denote componentwise multiplication. Since these scalar coefficients are independent of the averaging variables, disjoint rectangles give the exact bilinear identity
1341. u1341 B(W0, LΣ) = εH(2a ⊙δa) = εHdΣ = Σ.
1342. u1342 (7.34) The factor 1/2 for the real cosine has already entered (7.27); the factor 2 in this cross identity explains the denominator in (7.31). In C(ηβLΣ), both wave factors have exponent α −1/2,
1343. u1343 □ so the class product rule gives the last assertion in (7.32).
1344. u1344 In particular, the covariance identity before the curl correction is
1345. u1345 Thus L is a right inverse of DC(W0). The quadratic remainder C(LΣ) is retained in the stress. The same squared partition used for the leading covariance also assembles this linear inverse across the slow boxes. Write W0,β and Lβ for the local constructions before multiplication by the slow cutoff. For a real two-vector stress σ(r, z, t), independent of the angular and auxiliary variables, whose normalized representatives Q2Aσ are in Mα, set
1346. u1346 (7.35)
1347. u1347 Each summand uses the scale Q = 2−ℓof its label β = (ℓ, a). The sums are locally finite for q > 0. For overlapping slow supports, disjointness of the assigned rectangles, the squared partitions, and the physical factor Q−2A give B(Was
1348. u1348 physσ) = σ. In a chart of scale Q, set Σ = Q2Aσ and define the normalized representatives by
1349. u1349 To test wave-class membership, collect all coefficients with the same label and harmonic, as in Definition 6.5. Bounded overlap of the slow cutoffs and the common-torus bounds give Was 0
1350. u1350 0 ∈W1/2 and LasΣ ∈Wα−1/2, and
1351. u1351 (7.36) These definitions are consistent across overlapping charts. Auxiliary independence of the input allows the scalar coefficients to be taken outside the covariance integral, as required for the identity in Proposition 7.6.
1352. u1352 7.4. Exact curls, retained tails, and the covariance remainder. For the harmonic velocity
1353. u1353 tmeikmΦ, orthogonality to the phase gradient, nΦ · tm = 0, cancels the leading term in its divergence. Derivatives of the amplitude tm still contribute. Taking the full curl of the vector potential Am = CmeikmΦ in (7.38) supplies the corrected amplitude tm + rm needed for exact incompressibility: the amplitude from differentiating the exponential is tm, while derivatives of the potential coefficient Cm and the cylindrical frame give rm. In this subsection we will estimate rm in Lemma 7.7, then the temporal cutoff residual in (7.40), and finally the covariance of a stress increment formed from the actual velocities in Corollary 7.8.
1354. u1354 For a normalized potential A = (Ar, Aθ, Az), we define
1355. u1355 R−1∂θAz −DzAθ, DzAr −DrAz, (Dr + R−1)Aθ −R−1∂θAr
1356. u1356 (7.37) The normalized derivatives include the chain-rule terms for evaluation at (6.3), so curl∗A is the normalized physical curl of the evaluated potential.
1357. u1357 Fix a label, a nonzero integer m, and a harmonic coefficient tm ∈Wα independent of θ, with nΦ · tm = 0. We require tm to be supported in the prescribed slow and transverse supports and compactly inside the pulse interval, with smooth zero extension across all these boundaries. We choose the potential so that the amplitude obtained when its curl
1358. u1358 differentiates the exponential is tm. The potential coefficient Cm and the oscillating potential Am are
1359. u1359 (7.38) The last identity defines rm, the difference between the curl amplitude and the prescribed transverse amplitude. Since Cm is independent of θ, the remainder rm is
1360. u1360 (︁−Dz(Cm)θ, Dz(Cm)r −Dr(Cm)z, (Dr + R−1)(Cm)θ Lemma 7.7. For every α ∈R and nonzero integer m, let tm ∈Wα satisfy nΦ · tm = 0. Assume it is supported in the prescribed slow and transverse supports and compactly inside the pulse interval, with smooth zero extension across all these boundaries. The construction (7.38) gives Cm ∈Wα+1/2 and rm ∈Wα+1/2−κs. The velocity (tm + rm)eikmΦ = curl∗Am is exactly divergence-free for the normalized operators and after evaluation at the phase map. Potential and velocity have smooth zero extensions across their support boundaries. For the full amplitude am = tm + rm, the scalar product with the phase normal satisfies
1361. u1361 (7.39) Physical potentials and pressures are obtained from chart ones by multiplication by Q1/2−A and Q−2A, respectively.
1362. u1362 Proof. The amplitude from differentiating the exponential in curl∗Am is
1363. u1363 Every remaining term in curl∗Am differentiates the potential coefficient Cm or the cylindrical frame. The factor (km)−1 increases the ε exponent by 1/2, and applying Dr decreases it by at most κs, by (6.32). The multiplier nΦ/|nΦ|2 has polynomial coefficient bounds by Lemma 7.1. These estimates prove the two class bounds with all derivatives of fixed order. The normalized divergence is div∗a = (Dr + R−1)ar + R−1∂θaθ + Dzaz. Since Dr(R−1) = −R−2, we have (Dr + R−1)(R−1 f ) = R−1Dr f. Substituting (7.37) gives
1364. u1364 div∗curl∗A = R−1Dr∂θAz −(Dr + R−1)DzAθ
1365. u1365 + Dz(Dr + R−1)Aθ −R−1Dz∂θAr = 0, where the three pairs cancel by commutation. The chain rule preserves this identity after evaluation at the phase map. Expanding the divergence of ameikmΦ, whose amplitude coefficient is independent of θ, gives (7.39); division by km and one application of Dr give the exponent improvement 1/2 −κs. The support and smooth extension statements follow
1366. u1366 □ from the coefficient bounds and cutoffs.
1367. u1367 To assemble real velocities, use conjugate pairs. For example, t cos(kΦ), with real t, has coefficients t/2 at harmonics m = 1, −1. The corresponding potential and pressure coefficients obey the same conjugacy condition. In particular, the pressure associated with a real cosine velocity mode retains the factor i/(km) in (7.13).
1368. u1368 The temporal = cutoff and its residual. For the solution of Proposition 7.2, we set ˆ︁tm = ψtm amplitude. and ˆ︁πm = ψπm, The and solution apply before Lemma multiplication 7.7 to ˆ︁tm. We by write may ˆ︁rm for reach the curl both remainder pulse endpoints. of this cutoff The amplitude. The solution before multiplication by ψ may reach both pulse endpoints. The
1369. u1369 cutoff makes the potential and pressure zero on an open neighborhood of each endpoint. On the labelled rectangle 0 ≤v ≤Ls, the product rule in (7.5) gives
1370. u1370 m + Kˆ︁tm + m2dˆ︁tm + ikmnΦ ˆ︁πm + fm = (1 −ψ) fm + ψ′tm.
1371. u1371 (7.40) By (6.16), ψ = 1 on |v −Ls/2| ≤Ls/5, where both terms on the right vanish. Wherever either term remains, |v −Ls/2| ≥Ls/5, so (7.16) gives an additional factor SC
1372. u1372 ∗e−cS∗in every fixed coefficient derivative estimate. In the iteration, fmeikmΦ is an actual supported harmonic source with smooth zero extension; thus this local identity describes its global uncancelled remainder as well.
1373. u1373 At a fixed correction stage, converting any fixed derivative estimate to physical variables introduces a fixed power of Q−1 and a polynomial in S∗. As S∗= ℓ2, Q = 2−ℓ, and q ≍Q, for each fixed M, C, N we have
1374. u1374 Thus the resulting physical derivatives are O(qN) for every fixed N. These terms remain separate additive flat residuals throughout subsequent corrections. The same argument treats homogeneous cutoff tails. The + pressure coefficient still satisfies ˆ︁πm ∈Wα+1/2, while the exact velocity amplitude is ˆ︁tm + ˆ︁rm, with ˆ︁rm ∈Wα+1/2−κs.
1375. u1375 Covariance of the actual velocities. The linear covariance inverse was defined at the fixed primary wave. During iteration, its increment is added to a divergence-free wave field that already contains earlier corrections, and the new increment has its own curl remainder. The exact expansion below identifies the prescribed stress increment and retains three further contributions: interaction with previous corrections, interaction with the new curl remainder, and the square of the new velocity.
1376. u1376 Corollary 7.8. Fix α, ρ ∈R, and use the normalized representatives of (7.35). Suppose U is a real divergence-free wave field, including its curl remainders, such that
1377. u1377 Let Σ(R, Z, T) ∈Mα be a real two-vector independent of the angular and auxiliary variables. Apply Lemma 7.7 to the signed amplitudes LasΣ, obtaining the divergence-free increment V = LasΣ + R, where R ∈Wα−κs is its curl remainder. Then
1378. u1378 C(U + V) −C(U) = Σ + B(E, LasΣ) + B(U, R) + C(V).
1379. u1379 (7.41) The three remainder terms satisfy
1380. u1380 (7.42)
1381. u1381 Taking the radial divergence of each remainder term decreases the ε exponent by at most a further κs. Proof. We expand the quadratic map and use B(Was 0
1382. u1382 C(U + V) −C(U) = B(U, V) + C(V)
1383. u1383 0 , LasΣ) + B(E, LasΣ) + B(U, R) + C(V)
1384. u1384 = Σ + B(E, LasΣ) + B(U, R) + C(V).
1385. u1385 The bounds for B(E, LasΣ) and B(U, R) are products of the stated classes. Since κs < 1/2, V ∈Wα−1/2, so C(V) ∈M2α−1. This proves (7.41) and (7.42) without any positivity condition
1386. u1386 on an accumulated stress. The radial-divergence assertion follows from (6.32); the zeroth-
1387. u1387 □ order cylindrical terms cause no further exponent loss.
1388. u1388 8. Compactly supported mean corrections
1389. u1389 The wave construction in Section 7 provides pulses whose averaged covariance realizes the leading stress target (7.26), and an inverse for the principal equation of each nonzero angular harmonic (Proposition 7.2). The angular mean of the residual still requires correction. It includes quadratic wave products and may depend on the auxiliary torus even though it is independent of the physical angle (Proposition 8.1). In this section we construct pressure, tangential stresses, and divergence-free velocity increments for this mean, with support inside the active annulus (Proposition 8.3 and Corollary 8.5).
1390. u1390 There are two different inversion problems. For an auxiliary-averaged source, a compactly
1391. u1391 Re f dR = 0; see Lemma supported 8.2. solution For a source φ of (Dr depending + e/R)φ = on f, the with auxiliary e ∈{0, torus, 1, 2}, requires inversion ∫︁ Lemma 8.2. For a source depending on the auxiliary torus, inversion of the fast time derivative requires zero auxiliary average at each slow point (Lemma 8.6). We first derive the mean momentum equations (Proposition 8.1) and construct the radial inverse (Lemma 8.2), including the remainder needed when its source depends on the torus. We then apply it to pressure and divergence-free axial increments (Proposition 8.3), and to tangential stresses (Corollary 8.5). The integrated equations (8.16) identify three scalar defects P, Jθ, Jz, defined in Equations (8.12) and (8.15). A final linear map chooses five velocity coefficients to cancel their linear changes while preserving two velocity moments (Lemma 8.7); we compute the nonlinear changes explicitly in Lemma 8.8. The correction cycle in Proposition 9.6 uses these constructions after each wave update.
1392. u1392 Unspecified radial integrals run from zero to infinity. We use the averaging and projection conventions of Equations (3.7) and (3.9). Recall that ⟨·⟩θ takes the angular average componentwise in the cylindrical basis, whereas ⟨·⟩Y takes Haar average on the auxiliary torus of Lemma 6.2, and f ◦= f −⟨f ⟩Y. In particular, an angularly invariant field may still depend on that torus. The coefficient classes Mα and Sα are those of Definition 6.4; their estimates hold for every fixed derivative order. Unless physical variables are displayed, we use the normalized chart variables and operators of (6.6).
1393. u1393 8.1. Conservative momentum equations and integral constraints. In this subsection we
1394. u1394 derive the mean momentum equations in divergence form (Proposition 8.1) and state the two velocity moments (8.2) that the corrections must preserve. The radial integrals of these equations determine which compactly supported corrections are possible (Proposition 8.4 and Corollary 8.5); divergence form makes the boundary contributions explicit. The calculation retains the quadratic products of the actual divergence-free field w, including its curl remainders, as in (8.1). In a common chart with the normalized operators of (6.6), the actual velocity has the decomposition
1395. u1395 (8.1) Here (b, V, G) is the slow base, (β, v, γ) is its angularly invariant correction, and w is the sum of the divergence-free fields obtained by taking curls of the wave potentials. The indices in Wab range over a, b ∈{r, θ, z}; thus W includes every curl remainder. The correction components and W extend smoothly by zero outside the active shell. The mean correction is divergence-free, meaning (Dr + R−1)β + Dzγ = 0, and we impose the two integral constraints
1396. u1396 (8.2)
1397. u1397 These identities hold at every (Z, T). They are the zero angular-momentum moment and zero axial-flux integral of the auxiliary-averaged correction, up to normalization; the base is treated separately. Compactly supported azimuthal potentials preserve the axial-flux constraint, and the finite-dimensional moment correction preserves both.
1398. u1398 Let pm be the perturbation mean pressure, and put ∆0 = D2 r
1399. u1399 z and Σa = Q2ATphys,a for a = θ, z, where the full residual stress tensor is that in (5.41). Proposition 8.1. Let u have the divergence-free decomposition (8.1), with slow base (b, V, G), angularly invariant correction (β, v, γ), and ⟨w⟩θ = 0. Let pm be its angularly invariant pressure correction and let Σa = Q2ATphys,a be the normalized base stress components from (5.41). Retain the base’s flat residual as a separate additive error: all its derivatives of every fixed order vanish faster than every power of q. Then the angular mean of the remaining normalized residual is (Drpm −gr, Eθ, Ez), where
1400. u1400 Eθ = t∗v + (Dr + 2/R)(bv + βV + βv + Wrθ)
1401. u1401 + Dz(Gv + Vγ + γv + Wzθ) −ε(∆0 −R−2)v −(Dr + 2/R)Σθ,
1402. u1402 Ez = t∗γ + (Dr + 1/R)(bγ + βG + βγ + Wrz)
1403. u1403 (8.3)
1404. u1404 + Dz(2Gγ + γ2 + Wzz + pm) −ε∆0γ −(Dr + 1/R)Σz,
1405. u1405 gr = −t∗β −(Dr + 1/R)(2bβ + β2 + Wrr)
1406. u1406 −Dz(bγ + Gβ + βγ + Wzr) + (2Vv + v2 + Wθθ)/R + ε(∆0 −R−2)β. Proof. Incompressibility puts the transport terms in divergence form. For cylindrical components of any divergence-free field, the three transport expressions are
1407. u1407 (u · ∇u)θ = (Dr + 2/R)(uruθ) + R−1∂θu2 θ
1408. u1408 (u · ∇u)z = (Dr + 1/R)(uruz) + R−1∂θ(uθuz) + Dzu2 The rotation of the cylindrical frame contributes the centrifugal term in the first row and the additional uruθ/R in the second. Angular differentiation integrates to zero; products with exactly one factor of w do also. Subtracting the pure base terms therefore leaves the fluxes in (8.3). The angular derivative terms in the vector Laplacian have zero angular average, leaving ∆0 −R−2 in the radial and angular components and ∆0 in the axial component. The residual stress tensor contributes the two displayed radial divergences by (5.41). Excluded pulse tails have nonzero angular harmonics and hence zero angular mean. The calculation retains the
1409. u1409 □ full covariance of the divergence-free wave field, including the curl remainders.
1410. u1410 8.2. A compactly supported primitive for the cylindrical weights. In this subsection we
1411. u1411 construct a radial primitive with support in the active shell (Lemma 8.2). A direct radial integral can remain nonzero beyond the source support. We will subtract its full radial integral with a fixed cutoff and retain the resulting error. Since the radial derivative after phase evaluation also differentiates the auxiliary variable (6.4), both integrals must follow the corresponding path on the torus. We include that path in the physical definition below.
1412. u1412 We define the radial operations first in physical variables, holding (z, t) and the independent angular variable fixed. We choose a slow cutoff χm(X) that is zero on an inner collar, one on an outer collar, and changes only in a fixed interior portion of the active shell. For a smooth source g(r, z, t, Y), periodic in Y and smoothly extended by zero outside that shell,
1413. u1413 set
1414. u1414 (8.4)
1415. u1415 The shift in Y follows the radial phase map while the physical (z, t) stay fixed. Below the shell, Ig = 0; above it, Ig = J g. Thus Icg vanishes for X ≤Xa and X ≥Xb, with smooth zero extension across both radial edges. The subtraction introduces an error supported where χm varies, which we retain in the radial equation.
1416. u1416 For the normalized operators, we use a common torus y = Yi0 and put M = Λi0g Qdr/2. At fixed (Z, T), the same symbols denote
1417. u1417 (8.5)
1418. u1418 Here χm = χm(QR2/(2q)) is the chart representative of the fixed physical cutoff. For e ∈{0, 1, 2}, define
1419. u1419 (8.6)
1420. u1420 At corresponding physical and chart points, with r = √Q R, the normalization is
1421. u1421 Qa−1/2r−eIc,phys(re fphys) = R−eIc(Re f∗) = Te f∗.
1422. u1422 Here Ic,phys is the physical operator defined above. We call Ae f the cutoff remainder introduced by making the primitive compactly supported.
1423. u1423 The inverse must preserve the decay order of its source. We also need its cutoff remainder to be flat when the weighted auxiliary-mean integral vanishes. The next statement provides both estimates.
1424. u1424 Lemma 8.2. Fix a correction stage j, an exponent α, a band, its common torus y = Yi0, and e ∈{0, 1, 2}. Let f ∈Mα be a smooth scalar coefficient. In particular, its smooth zero extension satisfies, for every fixed multi-index I of derivatives in (R, Z, T, y),
1425. u1425 Then Te f is supported in the same active radial shell and in the projection of the input support to (Z, T), and belongs to Mα under the primitive normalization just defined. Exactly,
1426. u1426 (8.7) If ∫︁
1427. u1427 Re⟨f ⟩Y dR = 0 at every (Z, T), then ⟨Ae f ⟩Y = 0, and for every fixed I and integer p ≥1,
1428. u1428 (8.8) Only finitely many input derivatives are used for each (I, p). Constants and powers may depend on these indices, the stage, and the fixed profile, but not on the band, label, or point. If f is independent of Y and ∫︁
1429. u1429 Re f dR = 0 at every slow point, the cutoff remainder is identically zero.
1430. u1430 Proof. We first estimate the primitive without differentiating its rapidly shifted argument. We then verify the exact radial identity and use torus Fourier inversion to estimate its cutoff remainder.
1431. u1431 Step 1: support and coefficient estimates. Work on a neighborhood with a fixed common index, as in Lemma 6.2, and put L = vr · ∂y and g = Re f. Multiplication by Re preserves the input estimate on the shell. To estimate Icg, set U = Rdr and smoothly extend F(U) = g(R)/(drRdr−1) by zero. The shell stays in a fixed compact subset of R > 0, so this change of variables and its inverse have bounded derivatives of every fixed order. The exact half-line representation is
1432. u1432 (8.9)
1433. u1433 At fixed u, the shift depends on neither U nor (Z, T). Consequently a coefficient derivative differentiates only F and χm, without a factor M.
1434. u1434 The input zero extension includes derivatives of the moving shell, so differentiation creates no boundary terms.
1435. u1435 Near the inner edge only the left integral occurs. Write da = log(X/Xa), db = log(Xb/X), so ζ = exp(−aa/d2 a
1436. u1436 b) and δ = min(1, da, db), with fixed positive aa, ab. For every fixed B, the function s ↦→e−aa/s2s−B is increasing for sufficiently small positive s. On the left integration segment this bounds the input weight by the same type of weight at its endpoint; the factors belonging to the opposite edge are bounded and comparable. Near the outer edge use the right integral and the analogous monotonicity for db. In the interior the output weight is bounded below. Integration takes place over bounded length. These observations apply after every fixed coefficient derivative, proving the Mα estimate. Formula (8.9) also makes the output zero below and above the shell. Multiplication by R±e costs only fixed constants there.
1437. u1437 Step 2: the radial equation. The operator ∂U + ML on the integrand in (8.9) equals its u-derivative. Thus it sends A−to F, A+ to −F, and J g to zero. Multiplication by drRdr−1, followed by the product rule for R−e, gives (8.7). In physical variables the same computation reads rIcg = g −(∂rχm)J g.
1438. u1438 Step 3: the cutoff remainder under the moment condition.
1439. u1439 The radial identity and the support bounds are now established. To estimate the remaining cutoff term, we use cancellation in the full radial integral. Haar measure is translation invariant, and hence zero-mean ⟨J g⟩Y = ∫︁⟨g⟩Y torus dR. function Under H, the the weighted Diophantine moment inequality condition (6.7) this gives is zero for g = Re f. For a zero-mean torus function H, the Diophantine inequality (6.7) gives
1440. u1440 (8.10) Indeed the Fourier multiplier is (2πivr · k)−p; after estimating m + p + 3 derivatives of H, the remaining series is bounded by ∑k∈Z2(1 + |k|)−3. Slow and radial derivatives commute with this multiplier. The zero-average part of the full-line integral therefore satisfies the exact identity
1441. u1441 It follows by integrating the u-derivative of L−1F◦and repeating; every endpoint term vanishes by compact support. The omitted zero Fourier mode has identically zero full-line
1442. u1442 integral at every slow point. Averaging, integration, and slow differentiation of coefficients commute on the smooth zero extensions, so all its derivatives have zero integral as well. The same fixed-shift argument handles further coefficient derivatives. For the output derivative indexed by I, radial input derivatives up to IR + p and torus derivatives up to |Iy| + p + 3 suffice, together with the slow derivatives specified by I. Since M−1 ≤CεκsS
1443. u1443 ∗, ρg and ∂Rχm has interior support, this proves (8.8). Fix a physical normalization and a physical derivative order, and let L ≥0 bound the resulting loss of powers of q, including differentiation after phase evaluation. The graph operators give such a finite L, independent of p. For any prescribed flatness order N ≥0, choose p with h(α + pκs) > N + L. Since ε = Qh and q ≍Q, the strict excess absorbs the fixed powers of S∗and gives an O(qN) bound for that physical derivative. No estimate uniform in p is asserted. Finally, if f is independent of Y, then J (Re f ) = ∫︁
1444. u1444 Re f dR, which vanishes under the stated weighted integral condition.
1445. u1445 □
1446. u1446 The inverse is therefore exact up to a flat cutoff remainder whenever the weighted mean integral vanishes. This flatness uses estimates at every fixed derivative order. Frequencies comparable to M can be nearly resonant with vr; their small Fourier coefficients are controlled by the additional derivatives in the torus variables in (8.10). A fixed finite regularity bound would give only a fixed finite flatness order.
1447. u1447 8.3. Pressure reconstruction and divergence-free velocity corrections. In this subsection
1448. u1448 we will apply the radial inverse of Lemma 8.2 first to pressure pm, then to an azimuthal vector potential that produces a divergence-free velocity increment (Proposition 8.3). Its axial component will equal the prescribed source γd up to the cutoff remainder in (8.14). The pressure source may have a nonzero radial integral of its auxiliary mean. Subtracting a fixed bump times that integral isolates the obstruction and allows the remaining source to be integrated with compact support, as in (8.12).
1449. u1449 Let Imean be the reserved interval in X √ from Theorem 4.6(vi), and set The ∫︁ mean = patch is the region r/√q ∈Im. Choose a fixed bump ˆ︁ρ ∈C∞ c
1450. u1450 c (Im), normalized by ∫︁
1451. u1451 ˆ︁ρ(x) dx = 1. Define
1452. u1452 The It is bump fixed independently is independent of of the the velocity angular corrections. and auxiliary variables, and ∫︁
1453. u1453 ρ dR = 1. It is fixed independently of the velocity corrections.
1454. u1454 The operator T0 reconstructs pressure, and T1 reconstructs an azimuthal vector potential. The operators T1 and T2 will also recover radial fluxes of axial and azimuthal momentum. Write Ψ for the azimuthal coefficient of the vector potential. The physical normalizations, with a star denoting a normalized representative, are
1455. u1455 (8.11) In particular, Q2AIc,physgphys = Icg∗. Preservation of the ε exponent refers to these normalized representatives, with the integration length included.
1456. u1456 Proposition 8.3. Fix the radial operators of Lemma 8.2 and the bump ρ above.
1457. u1457 (i) Pressure. For a smooth shell-supported scalar source gr(R, Z, T, y), define the radial defect P and pressure pm by
1458. u1458 (8.12)
1459. u1459 Then
1460. u1460 (8.13) If gr ∈Mα, then P ∈Sα and pm ∈Mα. A change in gr of class Mα produces a pressure change of that same class. Under this class hypothesis, the cutoff remainder in (8.13) has zero auxiliary mean and is flat as q ↓0 at every fixed derivative order.
1461. u1461 (ii) Axial velocity. Let γd(R, Z, T, y) be a smooth shell-supported desired axial increment satisfying ∫︁
1462. u1462 R⟨γd⟩Y dR = 0 at every (Z, T). Define the physical azimuthal vector-potential coefficient Ψ = r−1Ic(rγd,phys), using (8.4), and define its normalized velocity increments by
1463. u1463 (8.14) The actual increment (∆β, 0, ∆γ) is divergence-free and has ⟨∆γ⟩Y = ⟨γd⟩Y. If γd ∈Mα, then Ψ∗, ∆γ ∈Mα and ∆β ∈Mα+1. If γd is independent of Y, its zero weighted integral implies ∆γ = γd pointwise. The pressure and vector potential are compactly supported in the active shell and retain the projection of the source support to (Z, T).
1464. u1464 Proof. Part (i): pressure. The source in (8.12) has zero radial auxiliary-mean integral:
1465. u1465 Apply Lemma 8.2 with e = 0, then take its auxiliary average. The moment P lies in Sα when gr ∈Mα: integrate its derivatives of any fixed order over the bounded shell, using the smooth zero extension and the boundedness of ζδ−B for each finite B. The interior bump ρP lies in Mα. Since ρ is fixed independently of the current velocity, the same argument proves the difference statement and completes part (i).
1466. u1466 Part (ii): axial velocity. The azimuthal potential generates the two physical velocity components −∂zΨ and (r + 1/r)Ψ. Commutation of the operators r and ∂z, which differentiate after evaluation at the phase map, proves that their divergence vanishes exactly:
1467. u1467 The weighted primitive identity gives the last equality in (8.14), including its sign. The zero weighted axial-flux integral gives zero auxiliary average of its cutoff remainder. Finally Q1/2−D = Qh = ε, which gives the extra factor of ε in ∆β in (8.14). The remaining claims follow from Lemma 8.2, completing part (ii). No slow derivative is commuted through Ic in
1468. u1468 □ this argument.
1469. u1469 8.4. Three compatibility defects in the integrated equations. In this subsection we identify
1470. u1470 the integral defects left by pressure reconstruction (8.13) and the two tangential equations in (8.3). We will then construct tangential stresses whose radial divergences cancel the auxiliaryaveraged residuals up to two explicit bump terms (Corollary 8.5). Pressure reconstruction leaves the integral P in (8.12) to be corrected. The auxiliary averages of the tangential residuals must likewise have zero weighted radial integrals before they can be represented by compactly supported stresses (Lemma 8.2). The moment constraints (8.2) remove the time and axial-viscosity terms from those integrals. The surviving axial fluxes give two further defects, as in Equations (8.15) and (8.16).
1471. u1471 Alongside P in (8.12), define the two flux defects and the second radial moment cρ of ρ by
1472. u1472 (8.15)
1473. u1473 Proposition 8.4. Let (β, v, γ) and w be the smooth shell-supported corrections of (8.1). Suppose that the two moments in (8.2) vanish at every (Z, T), and reconstruct pm from their radial source gr by (8.12). For the residual components Eθ, Ez in (8.3) and the defects Jθ, Jz, cρ in (8.15), one has
1474. u1474 (8.16) Proof. Auxiliary averaging turns Dr into ∂R and t∗into −ε∂T. Each radial divergence in (8.3), including that of the residual stress tensor, is an endpoint term after multiplication by its indicated weight. Those terms vanish by compact support. The radial viscous integrals vanish by two integrations by parts:
1475. u1475 R2(vRR + R−1vR −R−2v) dR = (2 −1 −1)
1476. u1476 These formulas are applied to the auxiliary averages. The time terms and axial viscosity terms are derivatives of the two zero integrals. The pressure moment is fixed by its compactly supported reconstruction:
1477. u1477 Substitution in the surviving axial fluxes yields (8.16). Since ρ is scaled by the moving q, cρ
1478. u1478 □ can depend on (Z, T). Its full product with P therefore remains inside ∂Z.
1479. u1479 The moments in physical variables and their normalized representatives are related by
1480. u1480 (8.17)
1481. u1481 The zero integral constraints are invariant under normalization. Each nonzero correction target uses the normalization of its corresponding row.
1482. u1482 The two tangential residual moments are therefore axial derivatives of Jθ and Jz + cρP. We subtract a fixed normalized bump times each weighted residual integral, then apply the radial inverse. The sources are independent of the auxiliary variable and have zero weighted integrals, so this inverse is exact. The resulting pair will be the target for an added covariance of the waves: its weighted radial divergence enters (8.3) with a positive sign through Wrθ and Wrz. Fix interior profiles ˆ︁σθ, ˆ︁σz with ∫︁
1483. u1483 Their supports lie in the mean patch. In each chart use σθ = Q3/2σθ,phys and σz = Qσz,phys, so ∫︁
1484. u1484 R2σθ dR = ∫︁
1485. u1485 Rσz dR = 1. This fixes the same physical bump functions on chart overlaps.
1486. u1486 Corollary 8.5. Under the hypotheses of Proposition 8.4, define the covariance targets
1487. u1487 (8.18) They are independent of the auxiliary variable, compactly supported in the active shell, and satisfy
1488. u1488 (Dr + 1/R)Hz = −⟨Ez⟩Y + σzε∂Z(Jz + cρP).
1489. u1489 The radial primitives act at fixed (Z, T). Thus Hθ and Hz vanish whenever their respective integrands in (8.18) vanish for every R; the construction does not enlarge support in the axial variable Z. Proof. The integrated identities and the normalization of the two bumps give
1490. u1490 Both sources are independent of the auxiliary variable. Their cutoff remainders vanish by
1491. u1491 □ Lemma 8.2, which gives the two identities and the support assertions.
1492. u1492 These radial divergence identities retain the factors ε∂ZJθ and ε∂Z(Jz + cρP) in the remaining bump terms. We can therefore correct the three defects at fixed (Z, T) without losing the displayed factors of ε∂Z.
1493. u1493 8.5. Inverting the auxiliary-time derivative on zero-mean functions. Corollary 8.5 reduces
1494. u1494 the auxiliary-averaged correction to the three integral defects and a compactly supported stress. The complementary part (E◦
1495. u1495 z) of the mean residual has zero auxiliary average at every slow point, by (3.9). In this subsection we construct its correction through the fast time derivative, whose Fourier multiplier is invertible on nonzero torus frequencies (Equation (6.7)). The axial velocity increment is then realized through its azimuthal potential in (8.14) to preserve incompressibility.
1496. u1496 Recall from Equations (6.4) and (6.6) that Nabs = vt · ∂Y acts in the original auxiliary coordinate. On a common torus y = Yi0, the normalized fast time derivative is ci0Ni0, where Ni0 = vt · ∂y and ci0 = Ti0g Q1+h. Here the normalized physical time derivative splits as t∗= −ε∂T + ci0Ni0. The slow term −ε∂T differentiates the coefficient’s explicit time dependence while holding y fixed. The fast term ci0Ni0 differentiates its auxiliary oscillations along the physical phase map. We invert the fast term here and retain the slow term in the residual.
1497. u1497 Lemma 8.6. Let N = vt · ∂y on T2, with vt as in (6.2), and let F be a smooth function of zero Haar mean. Then Nφ = F has a unique smooth zero-mean solution φ = N−1F. This solution is given by the following Fourier series and satisfies the displayed norm bound for every integer m ≥0:
1498. u1498 ∑
1499. u1499 (8.19) For a pair Eθ, Ez ∈Mα of normalized angularly invariant residual coefficients, use this inverse in the original auxiliary coordinate Y to define the physical tangential update −N−1
1500. u1500 z,phys). Here
1501. u1501 residuals and velocities have the normalizations in (8.11). This physical definition ensures agreement on chart overlaps. Its chart representatives are
1502. u1502 (8.20)
1503. u1503 Both desired increments ∆v, γd lie in Mα and have zero auxiliary mean at each slow and radial point. The realization of γd by (8.14) gives the actual velocity increment
1504. u1504 (8.21) This field is divergence-free and its addition preserves both integral constraints in (8.2). If a = −A1γd denotes the cutoff remainder in the axial increment, then its fast-time cancellation equations are
1505. u1505 (8.22) The angular cancellation is exact, and ci0Ni0a is flat at every fixed coefficient derivative order. Proof. The divisor bound (6.7) gives |vt · k|−1 ≤C(1 + |k|). Four more derivatives of F than the requested output leave a summable (1 + |k|)−3 bound on the Fourier series in two dimensions. This proves (8.19), smoothness, and uniqueness after fixing the zero Fourier coefficient. The multiplier commutes with every slow coefficient derivative and preserves reality. It acts at fixed (R, Z, T), so it preserves slow and radial support and their smooth zero extensions, although it need not preserve auxiliary-torus support.
1506. u1506 Let PiF(Y) = F(Ji
1507. u1507 gY). A mode k on the i-torus has frequency (Ji
1508. u1508 g)Tk in the original auxiliary coordinate Y. Since Jgvt = Tgvt,
1509. u1509 (8.23) The finite covering preserves Haar average, as follows also from its injective map of frequency lattices.
1510. u1510 Applying (8.19) in the common frequency coordinate k gives the same fixed derivative loss in every chart.
1511. u1511 In physical variables the update is −N−1
1512. u1512 Its chart velocity value has factor QA−(2A+1/2)T−i0 g
1513. u1513 i0 , which proves (8.20). This factor is bounded by CS∗by the choice of covering level and its bounded difference from a band level. For each output derivative, four additional derivatives in the torus variables suffice, so the class estimates at every fixed derivative order retain their ε exponent. Applying the slow-time term −ε∂T to these increments gives an additional factor of ε.
1514. u1514 The desired increments have zero auxiliary mean at each radial point. Hence both desired integral constraints hold, and the azimuthal-potential realization preserves them exactly. The bounds for ∆β and ∆γ in (8.21) follow from Proposition 8.3(ii). The fast derivative commutes with I, J , and Ic: the shifts are translations and χm is torus independent. Applying it to ∆γ = γd + a proves (8.22). Its last term is flat in every fixed coefficient derivative by (8.8). Full slow differentiation need not commute with Ic; those derivatives are estimated by
1515. u1515 □ Lemma 8.2 instead.
1516. u1516 8.6. Five-dimensional correction of the moment constraints. We have constructed inverses
1517. u1517 for pressure (Proposition 8.3), the auxiliary-averaged tangential residual (Corollary 8.5), and the part with zero auxiliary average (Lemma 8.6). The remaining quantities are the three integral defects P, Jθ, Jz in Equations (8.12) and (8.15). In this subsection we will correct their linear changes by a velocity increment that also preserves the two constraints (8.2). Three azimuthal coefficients and two axial coefficients will solve the five equations (8.25).
1518. u1518 On the reserved mean patch, the background retains the power law (4.30) by Proposition 5.5. We will use this form to reduce the equations to independent power moments of supported bumps. We first construct the fixed linear map for arbitrary targets (Lemma 8.7), then compute what remains when its targets are the current defects (Lemma 8.8).
1519. u1519 In x = what r/√q remains ∈Im, the when base its after targets velocity are normalization the current defects at q is
1520. u1520 (8.24) Here a(η) = 21/2+λcpatch(1 + η2)−1. The exact summed-base statement is (5.44). The tangential coefficients beyond the leading profile are supported elsewhere by (5.18). Multiplication of their azimuthal potentials by a cutoff in q does not create an axial component on this patch, because q is independent of radius. Thus (8.24) holds for the summed base used in the balances. The fixed parameter λ is positive.
1521. u1521 Lemma 8.7. Suppose that the fixed slow base satisfies (8.24) on the reserved interior interval Im, with λ > 0 and |a(η)| ≥a0 > 0. There are three fixed azimuthal bump profiles and two fixed axial bump profiles supported in this interval with the following property. At each (Z, T), for arbitrary scalar targets (P, Jθ, Jz), their physical rescalings have a unique linear combination (∆v, γd) satisfying
1522. u1522 (8.25)
1523. u1523 The target triple is normalized row by row as in (8.17). The resulting increments are independent of the angular and auxiliary variables and depend linearly on the targets. For every α, targets in Sα give increments in Mα, for every fixed coefficient derivative, with constants permitted to depend on the fixed λ. The azimuthal-potential realization (8.14) has ∆γ = γd exactly and ∆β ∈Mα+1. The two integral constraints are preserved exactly. When the targets are the current defects defined by Equations (8.12) and (8.15), the last three equations cancel the displayed contributions involving the fixed base.
1524. u1524 Proof. Step 1: choose the five profiles. Normalize every physical row and its target using q.
1525. u1525 On the patch, (8.24) splits the rows into an angular block whose powers of x are 2, −2 −2λ, −2λ, and an axial block whose powers are 1, 1 −2λ. Each list contains distinct powers for the fixed positive λ.
1526. u1526 To see invertibility explicitly, choose a nonnegative nonzero bump η0 supported near some x0 > 0 inside the patch. Choose a small fixed d > 0 and a sufficiently narrow support so that its geometrically scaled copies ηj(x) = a−1 j
1527. u1527 j η0(x/aj), aj = ejd, j = 0, 1, 2, lie on disjoint subintervals of the patch. Their power moments are
1528. u1528 For the three angular powers the matrix, after dividing its rows by µp, is the ordinary Vandermonde matrix in the distinct numbers edp. Its determinant is nonzero. Two copies give the same argument for the axial block. The remaining factors are nonzero multiples of a(η).
1529. u1529 Step 2: define the coefficients by matrix inversion. Use η0, η1, η2 as the azimuthal profiles and η0, η1 as the axial profiles. Let Pq, Jθ,q, Jz,q denote the physical targets multiplied by the powers in (8.17), with q in place of Q. Define the matrices
1530. u1530 Step 1 proves that both matrices are invertible. Define
1531. u1531 ∑
1532. u1532 ∑ The matrix Aθ enforces the zero angular-momentum constraint and the targets −Pq, −Jz,q; Az enforces the zero axial-flux constraint and the target −Jθ,q. Since Gq = 0, these two blocks exhaust the five equations. Their inverse defines one physical correction, linear in the target triple and smooth in the slow variables whenever the targets are smooth. The inverse may deteriorate as λ ↓0; no uniformity in that limit is required.
1533. u1533 Step 3: normalization, bounds, and axial realization. Passing from q-normalized rows to a fixed Q chart multiplies each row and its target by the same powers in (8.17) and by bounded smooth powers of q/Q. Those ratios have bounded derivatives of every fixed order. Matrix inversion and differentiation therefore cost fixed constants, and the radial weight is bounded above and below on the fixed interior support. This proves the Sα to Mα bound. The desired axial field is slow and its weighted axial-flux integral is zero; Lemma 8.2 gives an identically vanishing cutoff remainder. Its azimuthal potential and radial velocity component stay in
1534. u1534 □ the mean patch, including between the finitely many bump supports.
1535. u1535 The linear map has now been fixed independently of the target values. Its use in the nonlinear equations requires the defects of the updated field. The next statement records the exact difference and the gain available under the bounds used in the iteration. The base estimates for this gain are local to the support of the azimuthal potential, which includes the intervals between its bump supports.
1536. u1536 Lemma 8.8. Suppose that the slow base satisfies (8.24), with the fixed λ > 0 and lower bound |a(η)| ≥a0 > 0. Let (β, v, γ) be a smooth divergence-free mean correction supported in the active shell, and keep the base and wave field w fixed. Form gr and (P, Jθ, Jz) by Equations (8.3), (8.12) and (8.15). Apply Lemma 8.7 with these three defects as its targets, and realize the resulting γd by (8.14). For the updated mean velocity
1537. u1537 (βnew, vnew, γnew) = (β + ∆β, v + ∆v, γ + ∆γ),
1538. u1538 compute gr,new from (8.3) and define Rg = gr,new −gr −(2V/R)∆v. Then
1539. u1539 (8.26)
1540. u1540 + (2v∆v + (∆v)2)/R + ε(∆0 −R−2)∆β.
1541. u1541 The defects recomputed from the updated field satisfy
1542. u1542 (8.27)
1543. u1543 Suppose in addition that, on the support of the azimuthal potential for this correction, every fixed coefficient derivative of b is bounded by CIεSBI
1544. u1544 ∗. If the current corrections satisfy v, γ ∈M0.9, β ∈M1.9, and the three targets belong to Sα for α ≥0.9, then
1545. u1545 (Pnew, (Jθ)new, (Jz)new) ∈Sα+0.9−2κs.
1546. u1546 Proof. Subtract the two radial sources in (8.3). The covariance W is unchanged, and expanding each quadratic product gives (8.26). In the azimuthal flux defect, the terms linear in the base are ∫︁
1547. u1547 R2(G∆v + V∆γ) dR. In the axial flux defect they are ∫︁(2RG∆γ −1 ∫︁ R2(2V/R)∆v −RV∆v) dR, where the second term comes from the radial source contribution −1 2
1548. u1548 R2(2V/R)∆v dR. Since ∆γ = γd, the last three rows of (8.25) cancel the old defects and these linear terms. The remaining products give (8.27).
1549. u1549 For the estimates, Lemma 8.7 gives ∆v, ∆γ ∈Mα and ∆β ∈Mα+1. All three increments are independent of the auxiliary variable. In particular, t∗∆β = −ε∂T∆β. The terms in Rg satisfy
1550. u1550 The remaining transport products lie in Mα+2−κs. Since α ≥0.9, every exponent above is at least α + 0.9 −2κs. Radial integration preserves these estimates after normalizing each moment by the factors specified above. The radial weight is bounded below on the interior
1551. u1551 □ support of the azimuthal potential, so the stated local base bounds suffice.
1552. u1552 8.7. Recomputation and compatibility between charts. In this subsection we specify the
1553. u1553 order in which the residuals and defects in Equations (8.3), (8.12) and (8.15) are recomputed after a mean correction, and verify that the constructions agree between charts using Equations (8.4) and (8.23) and Lemma 6.2. Each correction changes the quadratic fluxes, so the next source must be computed from the updated divergence-free velocity. A cutoff remainder with zero auxiliary average can acquire a nonzero average when multiplied by another term. Its contributions to pressure and to the compatibility integrals in Equations (8.3) and (8.15) must therefore be retained at the next step.
1554. u1554 For fixed base coefficients and stress, an actual velocity tuple (β, v, γ, w) determines the pressure and mean residuals in the following order:
1555. u1555 (1) Form Wab = ⟨wawb⟩θ, using the full wave velocities, and define gr by the third row of
1556. u1556 (2) (3) With Define this P pressure, = ∫︁⟨gr⟩Y dR define and Eθ, pm Ez = by T0(gr the −ρP). first
1557. u1557 (3) With this pressure, define Eθ, Ez by the first two rows of (8.3).
1558. u1558 (4) Define Jθ, Jz by (8.15), using the same velocity and radial source.
1559. u1559 These definitions apply again after every change to the velocity. For either mean update, the new tuple is
1560. u1560 (β, v, γ, w)new = (β + ∆β, v + ∆v, γ + ∆γ, w).
1561. u1561 For the temporal update, use the current E◦
1562. u1562 z in (8.20); for the moment update, use the current (P, Jθ, Jz) in (8.25). In both cases ∆γ is the actual axial increment from (8.14). In particular, its cutoff remainder is included before the products in the four definitions above are evaluated. After the moment update, Equations (8.26) to (8.27) give the recomputed defects exactly.
1563. u1563 The correction cycle and its exponent estimates are given in Proposition 9.6 and its proof.
1564. u1564 The definitions in physical variables ensure consistency across charts. The radial integrals keep (z, t) fixed, and q = q(z, t) ensures that they introduce no new dyadic bands. One common index can therefore be fixed on a neighborhood containing all relevant support closures. The Fourier covariance (8.23) proves agreement on overlaps; the radial formulas agree by their physical definition. A change of representation uses only bounded differences of covering indices. The integer-valued covering indices remain fixed during differentiation, and the coordinate changes preserve the established derivative bounds. All estimates above hold on the same domain 0 < q < q∗fixed before the iteration.
1565. u1565 On a closed region q ≥c > 0, including η = ±1, only finitely many bands occur; the fixed-shift integrals, the Fourier inverse, and the finite-dimensional moment correction preserve all one-sided source derivatives on this closed region, including the endpoint derivatives.
1566. u1566 9. Residual improvement and the local field
1567. u1567 The wave and mean constructions in Sections 7 and 8 supply corrections for the different components of the momentum residual. We now combine them to prove Theorem 3.1. Starting from the fixed background (uB, pB) of Proposition 5.5 and the primary fields assembled in (7.35) and made divergence-free by Lemma 7.7, we must obtain a local velocity uloc whose residual and all its derivatives vanish to every order at the singularity, while preserving the inner velocity growth and the exterior heat flow. Each correction introduces new nonlinear errors, so we need a decay improvement after a complete cycle of corrections.
1568. u1568 A cycle first removes the nonzero angular harmonics by Proposition 7.2, then adjusts the averaged stress by changing the primary amplitudes through (7.31). The mean equations remove the part with zero auxiliary average through (8.20), and the five-equation system (8.25) corrects the three defects in compact support while preserving the two moment constraints. We prove in Proposition 9.6 that each cycle gains a fixed positive power of ε. All inverses retain the fixed background and primary amplitudes. This permits every finite stage to be defined on one domain (Lemma 9.7), where the final summation produces the local field of Proposition 9.9.
1569. u1569 The notation guide in Table 1 collects the recurring scales, operators, and field classes. Throughout this section κs = 10−5, and Wα, Mα, Sα have the meaning, at every derivative order, given in Section 6.4. An exponent is always a power of ε = Qh with the normalization
1570. u1570 used for these classes. Constants, powers of S∗, and inverse-distance powers may depend on a fixed correction stage and a fixed order of amplitude differentiation, but are uniform in band, label, copy, and point. The angular mean ⟨·⟩θ and auxiliary mean ⟨·⟩Y remain distinct; their definitions are (3.7). All class estimates refer to normalized chart coefficients. Physical velocity, pressure, and residual are recovered with the factors Q−A, Q−2A, and Q−2A−1/2, respectively. In particular, the residual of the physical fields is
1571. u1571 R(u, p) = ∂tu + (u · ∇)u −∆u + ∇p, We use physical fields in residual identities and normalized coefficients in the estimates below. The Cartesian space–time jet notation | · |m is defined in (5.23).
1572. u1572 9.1. Residual estimates for velocities constructed by curls. The amplitude equations (7.5)
1573. u1573 cancel the principal part of a prescribed source. The corresponding physical velocity also contains the terms introduced by taking a curl (Lemma 7.7), and its residual contains derivatives of the amplitudes and background; we will expand these terms in (9.2). We first estimate these remaining linear terms and the nonlinear interactions in Proposition 9.1 and Lemma 9.2, obtaining the gains needed when successive corrections are added in Proposition 9.6.
1574. u1574 We write an angular harmonic in a fixed label as a exp(ikmΦ), where m ∈Z \ {0} and a is its amplitude. The phase normal nΦ is defined in (7.4), and the chain-rule operators for evaluation at Y = Y(r, t) are (6.6). The background coefficients (b, V, G) are those of Sections 6 and 7.
1575. u1575 For a transverse amplitude tm, the potential in (7.38) gives
1576. u1576 Here rm is the curl correction. If tm ∈Wα, then Lemma 7.7 gives rm ∈Wα+1/2−κs. The longitudinal estimate (7.39) applies to the complete amplitude am. This distinction matters in quadratic transport: its advecting factor is the complete divergence-free velocity.
1577. u1577 We separate the principal linear operator, which is handled by the amplitude equation, from the remaining transport, pressure, and viscous terms. For later comparison with the residual, write the principal amplitude-pressure operator as
1578. u1578 Lm(a, π) = Q1+hNabsa + Ka + εk2m2|nΦ|2a + ikmnΦπ.
1579. u1579 (9.1) The inverse in Proposition 7.2 constructs (tm, πm) satisfying Lm(tm, πm) = −fm and nΦ · tm = 0, before the pulse cutoff is applied. The amplitude tm is not required to have temporal zero extension before cutoff.
1580. u1580 Proposition 9.1. Fix a label, m ∈Z \ {0}, and α ∈R. Let fm ∈Wα satisfy the support and smoothextension hypotheses of Proposition 7.2. Assume also that the source harmonic fmeikmΦ extends smoothly by zero outside the fixed local rectangle, including its pulse endpoints. Let (tm, πm) be the solution of Lm(tm, πm) = −fm. The same conclusions hold for a homogeneous pulse tm ∈Wα, with fm = 0. Multiply its potential and pressure by the pulse cutoff and take the curl as in (7.38). Then the normalized linear residual, after adding the prescribed source fmeikmΦ, belongs to Wα+1/2−3κs, apart from additive pulse-cutoff tails whose Cartesian space–time derivatives vanish to infinite order as q ↓0. The pressure amplitude πm belongs to Wα+1/2.
1581. u1581 Proof. We expand the linearized residual as its principal operator (9.1) plus a remainder. For this expansion, a denotes an arbitrary harmonic amplitude and π its pressure amplitude. Let
1582. u1582 Eik = (t∗+ bDr + (V/R)∂θ + GDz)Φ be the phase transport defect from (7.9). The remainder of the linearization about (b, V, G) is
1583. u1583 (a, π) = (−ε∂T + bDr + GDz)a + ikmEika
1584. u1584 + ar(∂Rb)er + azDz(b, V, G) + (b/R)aθeθ + (Drπ, 0, Dzπ)
1585. u1585 (9.2)
1586. u1586 Indeed, transport of a cylindrical vector v by u has the basis term (−uθvθ, uθvr, 0)/R. Linearizing it gives the connections in K and the displayed b/R term. The scalar Laplacian of the harmonic contributes the quadratic phase-gradient term to (9.1), its two mixed phase– amplitude derivative terms to (9.2), and the vector Laplacian contributes the last angular connection and the R−2 term. The amplitude coefficients are independent of θ.
1587. u1587 The transverse-amplitude equation of Proposition 7.2 gives Lm(tm, πm) = −fm, with πm ∈Wα+1/2. The respective gains above α in (9.2) are
1588. u1588 mixed derivatives and phase divergence
1589. u1589 The phase-defect bound permits polynomial factors of S∗. Applying the principal velocity operator to the curl correction preserves its exponent: fast time differentiates an amplitude coefficient, and εk2 = O(1). The potential formula and these chain-rule operators give, at every derivative order, the common lower bound α + 1/2 −3κs for the exponents of these terms.
1590. u1590 The pulse cutoff is applied to the potential and pressure before the curl. The uncancelled source (1 −ψ) f, and the principal pulse-cutoff derivative ψ′t with its fixed multipliers, are supported where Pv ≤e−cS∗. For every fixed physical derivative their estimates have the form CQ−MSP
1591. u1591 ∗e−cS∗, hence are flat because S∗= ℓ2 and Q = 2−ℓ. Retain these terms in the additive residual throughout the iteration; apply subsequent forward solves only to the
1592. u1592 □ supported source terms.
1593. u1593 The nonlinear terms require separate estimates for interactions between waves and means and between two waves. For two waves, incompressibility controls the contraction of the advecting amplitude with the phase gradient.
1594. u1594 For the next lemma, (w · ∇∗)v denotes cylindrical transport with normalized derivatives (Dr, R−1∂θ, Dz), including differentiation of the cylindrical frame. Thus the lemma estimates the normalized nonlinear residual.
1595. u1595 Lemma 9.2. Let w be a wave in Wα, and let v be a mean field with tangential components in Mµ and radial component in Mµ+1. The nonzero harmonics of (w · ∇∗)v + (v · ∇∗)w belong to Wα+µ−1/2.
1596. u1596 For two curl-generated waves w, w′ with the same label and exponents α, α′, the nonzero harmonics of (w · ∇∗)w′ + (w′ · ∇∗)w belong to Wα+α′−κs; the zero harmonic belongs to the mean class Mα+α′−κs. Distinct labels have zero products on their closed supports.
1597. u1597 Proof. Mean advection of a wave phase costs k = O(ε−1/2). Its radial transport of the amplitude has a radial mean factor of order µ + 1 and one Dr; axial transport gains one through Dz. Wave transport of a mean costs at most κs radially, and the cylindrical connections contain no derivatives. These terms are all bounded by the asserted mean–wave exponent.
1598. u1598 For two harmonics of one label, phase differentiation in the transport of a′ exp(ikm′Φ) by a exp(ikmΦ) gives ikm′(a · nΦ)a′. Equation (7.39) cancels the apparent half-power loss. Transport terms in which the derivative acts on the amplitude cost at most κs, and the connections cost none. Products of the weights obey P2 v
1599. u1599 v ≤Pv and have radial weight ζ; this connections is also stronger cost than none. the Products required of √ζ the weight weights for obey a nonzero P2 v ≤Pv output. and have The radial support weight separation ζ; this in Lemma 6.1 proves the last assertion. Leibniz’ rule proves each statement for amplitude
1600. u1600 □ derivatives through every fixed order.
1601. u1601 9.2. Sources and errors retained during iteration. To use the estimates in Proposition 9.1 and Lemma 9.2 at successive stages, each new source must satisfy the support and derivative hypotheses of the pulse inverse in Proposition 7.2. Mean operations can enlarge auxiliary support, so we must check these hypotheses after a complete cycle. We keep the errors already flat at the singularity separate from the sources to be corrected, as in (9.3), while using the full velocity fields in every nonlinear product. The supported sources must remain inside the fixed enlarged rectangle; Proposition 9.3 establishes these properties.
1602. u1602 We use the source supports Ωγ and enlarged domains E+ γ
1603. u1603 The flat term below collects the base error EB from (5.41), pulse-cutoff tails, and uncompensated cutoff remainders. The exact mean balances can contain such flat terms; their definition always uses the complete current velocity. Angular means may occupy the full auxiliary torus; the rectangle and envelope conditions below concern the nonzero harmonics.
1604. u1604 Proposition 9.3. Fix j ∈N0. Let (u[j], p[j]) be obtained from the fixed slow base and primary waves by a finite sequence of the pulse inverse and curl construction, the signed amplitude map for auxiliary-independent stresses, and the mean maps of Section 8. Suppose that pressure is reconstructed by (8.12) after each update and all products use the complete velocity fields. There is a physical residual decomposition
1605. u1605 (9.3) In a common chart, put G[j]
1606. u1606 (i) Supported sources. On the coarsest common auxiliary torus, the nonzero angular harmonics
1607. u1607 (9.4)
1608. u1608 The sum is locally finite, and Hj depends on the finite construction but not on the band. The
1609. u1609 with smooth zero extension across the fixed slow and transverse supports and the enlarged local
1610. u1610 torus rectangle. Let α, µ be any lower bounds for the wave and mean exponents furnished by
1611. u1611 applying Propositions 9.1, 7.2, 6.6 and 7.6, Section 8, and Lemma 9.2 along this finite sequence.
1612. u1612 The mean coefficients have smooth zero extension outside the active shell. The next pulse inverse
1613. u1613 is applied precisely to the coefficients f [j]
1614. u1614 (ii) Flat remainder. For every fixed j, m, N,
1615. u1615 (9.5)
1616. u1616 This physical bound is uniform over bands, labels, and points.
1617. u1617 (iii) Exact mean equations. Let (β[j], v[j], γ[j], w[j]) be the complete current velocity correction as
1618. u1618 in (8.1), and set EB,∗= Q2A+1/2EB. Then
1619. u1619 (9.6)
1620. u1620 Proof. We first prove (i), then track the additive errors in (ii) while preserving the exact mean equations in (iii).
1621. u1621 Part (i): support and coefficient bounds. The primary pulses have the claimed supports and weighted bounds at every derivative order. Differentiation preserves closed support and the amplitude class bounds. An explicit Dr costs κs each time it occurs; taking further amplitude derivatives of that estimate enlarges its constants and polynomial degrees, without repeating that exponent loss.
1622. u1622 For products, a mean times a wave inherits the wave’s slow, transverse, and local torus rectangle support, and its Pv factor.
1623. u1623 Two waves of one label add their integer phase multipliers. Their zero harmonic is exactly their angular mean, since the angular frequency kp is a nonzero integer; every other output retains the wave weight by P2 v
1624. u1624 v ≤Pv. Different labels do not interact. Curls, pressure coefficients, and particular pulse propagation preserve the phase integer. Mean operations cannot introduce a nonzero angular harmonic. Any subsequent nonzero harmonic product contains a wave factor and therefore inherits that factor’s support and envelope.
1625. u1625 The radial integration, pressure reconstruction, mean-vector-potential construction, and auxiliary-time inversion in Section 8 preserve all mean-amplitude bounds. For the stress correction entering Proposition 7.6, subtract a supported interior profile with the same weighted radial moment as the source. The resulting weighted radial integral can be taken forward from the left support boundary or backward from the right support boundary, because the corrected source has zero total moment. Near either boundary, let s be the corresponding logarithmic distance in (6.23). The weight ζ is e−a/s2 times a smooth positive factor bounded above and below there, with a > 0 fixed. For each fixed M ≥0 and integer k ≥0, the following bounds hold for sufficiently small positive δ, s:
1626. u1626 The first bound follows by the substitution v = a/s2; the second follows by differentiation. Radial weights and the radial Jacobian are bounded on the normalized shell. Differentiating the radial integral therefore retains ζδ−M′, with M′ allowed to depend on the derivative; in the interior ζ is bounded below. Thus a source of order α gives a stress correction Σ ∈Mα.
1627. u1627 For the amplitude division, use the same normalization as (7.31): dΣ = H−1(Σ/ε) and δaσ = (dΣ)σ/(2aσ). The inverse-matrix bounds and (7.25) give, for every fixed coefficient multi-index I,
1628. u1628 This recovers (7.33). The last estimate uses the fundamental pulse bounds; after the slow cutoff it is the Wα−1/2 estimate of Proposition 7.6. The remaining square-root edge weight gives smooth extension by zero.
1629. u1629 The hypotheses of Proposition 7.2 now hold for each new source. Its prescribed path holds the slow and transverse variables fixed; zero data along an entire such path give zero solution. The final cutoff restores the fixed enlarged local torus rectangle support. Smooth source extensions give smooth extensions of the solution. This proves the support-containment hypothesis of the inverse; divisibility by the original cutoff is unnecessary.
1630. u1630 The local band set remains fixed: radial paths hold (z, t) fixed, and q depends only on (z, t); temporal inversion and pulse propagation hold all slow variables fixed. Hence the common-torus construction of Lemma 6.2 applies at every step of the finite construction. A radial mean can aggregate polynomially many labels in S∗, whereas only boundedly many wave labels overlap at a point. Radial aggregation changes only the polynomial factors in S∗, while preserving the common chart and the ε exponent.
1631. u1631 Parts (ii) and (iii): flat errors and exact means. For an actual increment (w, π), the residual changes by
1632. u1632 R(u + w, p + π) = R(u, p) + ∂tw −∆w + ∇π
1633. u1633 The old F enters additively, while all displayed products use actual fields, including their cutoff curls and remainders from the compactly supported modified radial integral. Add newly created flat terms to F. At a fixed stage there are finitely many kinds of such terms. The Gaussian bound in Proposition 9.1 is uniform over labels. The estimate in Lemma 8.2 bounds each cutoff remainder by any prescribed power of q, using finitely many additional source derivatives. Summing polynomially many labels preserves these bounds. Keep Gaussian nonzero-harmonic cutoff tails in the additive residual and exclude them from subsequent forward pulse inputs.
1634. u1634 The mean residuals Eθ, Ez are the exact conservative balances and may contain flat cutoff remainders, which a fixed mean inverse may cancel. Remove any canceled contribution from F. The cutoff-remainder bounds place the remainder in Mα for every α, so this update preserves the stated classes and the exact residual decomposition. This proves (9.5) at each finite stage, hence (ii). Applying
1635. u1635 □ Proposition 8.1 to the complete current tuple, and using (8.12), gives (9.6) and proves (iii).
1636. u1636 The realization step will use a comparison with one fixed finite stage; it will not sum these flat residuals directly.
1637. u1637 9.3. Initialization and a full correction cycle. We now put the wave and mean corrections together. The primary pulses, followed by the initial mean corrections, will give the first set of residual bounds in Proposition 9.5. We then show in Proposition 9.6 that a complete cycle improves all these bounds while preserving the support and moment conditions. The induction statement below combines the residual orders in (9.8) with the bounds (9.9) on the total velocity correction that control its interactions with later increments.
1638. u1638 We keep the base (b, V, G), the primary amplitudes, and all inverse operators fixed throughout the construction.
1639. u1639 be the normalized representative of the assembled primary transverse field in (7.35). For each finite construction write its normalized velocity and pressure as
1640. u1640 (9.7) Here pB,∗is the base pressure, pw consists of the pressure harmonics supplied by the amplitude equations, and pm is reconstructed from the current gr by (8.12). The quantities Eθ, Ez, P, Jθ, Jz are then determined by (8.3), (8.12), and (8.15).
1641. u1641 Definition 9.4 (Finite correction state). For j ∈N ∪{0}, a state at stage j consists of real fields of the form (9.7) with the following properties. The wave w is a sum of curls of supported wave potentials with the fixed label phases, and the mean correction (β, v, γ) is the sum of a curl of an azimuthal potential and a direct azimuthal field. In particular, both velocity corrections are exactly divergence-free. The source supports, finite harmonic sets, smooth zero extensions, and physical residual decomposition are as in Proposition 9.3.
1642. u1642 The normalized nonzero harmonics G[j]
1643. u1643 ∗, the exact tangential mean residuals, and the three integral defects obey
1644. u1644 (9.8)
1645. u1645 with the following bounds for the total velocity and pressure corrections
1646. u1646 (9.9) Wave assertions refer to each grouped nonzero harmonic per label. The two normalized angular-momentum and axial-flux moments are maintained exactly:
1647. u1647 (9.10) The pressure reconstruction is part of the state.
1648. u1648 Consequently, the radial residual of every such state is −ρP plus the flat cutoff remainder of Proposition 8.3.
1649. u1649 Proposition 9.5. Starting with the fixed base and the curls of the primary potentials, perform the temporal mean update (8.20) and the five-equation correction (8.25), reconstructing pressure by (8.12) before and after each update. The resulting fields form a state at stage 0 in the sense of Definition 9.4; in particular, their residual orders are B0 = 0.7 and C∗ 0 Proof. We first estimate the primary waves, then remove the torus-dependent means and the three moment defects. The initialization uses the higher decay order of the auxiliary-averaged primary residual. The primary transverse amplitudes have exponent 1/2. Their supported linear residual components have exponent 1 −3κs, and their nonlinear wave residuals have
1650. u1650 exponent 1 −κs. The exact mean balances (8.3) give gr, pm, Eθ, Ez ∈M1−κs and defects in S1−κs before any mean update.
1651. u1651 For the auxiliary means, the primary covariance of Proposition 7.5, assembled by (7.35), cancels the order-zero auxiliary radial–tangential stress tensor exactly, including derivatives of the slow partition because the squared partition functions sum to one. To display this cancellation, write Σ(0) a
1652. u1652 for that leading tensor and Σa for the full tensor in (8.3). Before the mean velocities are added, those equations read
1653. u1653 (9.11)
1654. u1654 The first differences contain at least one curl correction. The tensor cross interaction between the curl correction and a primary transverse amplitude belongs to M3/2−κs before applying the divergence. Axial fluxes, the axial pressure term, and the higher-order auxiliary stress tensor have residual exponent at least 2 −κs. Thus ⟨Eθ⟩Y, ⟨Ez⟩Y ∈M1.49.
1655. u1655 Set H0 = 1 −κs and apply the temporal update (8.20). The tangential increment is MH0 and its induced radial velocity component is MH0+1. The pressure change has exponent H0, since 2V∆v/R appears in ∆gr. Its contribution to each tangential residual includes an axial derivative and therefore has the corresponding additional gain. Beyond the cancelled fast-time term, slow-time differentiation supplies one additional power of ε, radial linear fluxes contain b = O(ε) in every fixed-order amplitude derivative on the shell or the induced radial velocity increment, axial fluxes gain one, and viscosity gains 1 −2κs. Radial quadratic mean fluxes also contain a radial mean. All these residual changes have exponent at least H0 + 1 −2κs. Mean interaction with w ∈W1/2 has wave exponent at least H0. Defects remain in SH0 after pressure is recomputed.
1656. u1656 Now use the five-equation correction map (8.25) at order H0 and recompute pressure. Its slow increments have no fast-time term. The preceding tangential and wave estimates still hold. In the defect changes, the term 2V∆v/R and its two contributions to the weighted flux moments are the linear contributions to ∆P, ∆Jθ, ∆Jz canceled by this system. Every other term contains slow-time differentiation of the induced radial velocity, a radial flux with b or another radial mean, an axial derivative, radial viscosity, or an additional tangential mean in v2, γv, γ2. These have gain greater than 0.8 above H0. Hence the remaining defects have exponent greater than 1.8 −κs, and both tangential means retain exponent 1.49. These estimates imply (9.8) with B0 = 0.7 and C∗ 0
1657. u1657 The primary curl correction has exponent 1 −κs > 0.68. All mean increments and pressure changes have exponent H0 > 0.9, with radial exponent H0 + 1 > 1.9. This proves (9.9). The prescribed temporal increments have zero auxiliary mean. The first two homogeneous equations of the correction system enforce (9.10), and the mean-field reconstruction preserves
1658. u1658 □ the prescribed auxiliary mean exactly.
1659. u1659 The initialization provides the bounds at stage zero. We now repeat the four corrections and compare every new error with the bounds required at the next stage. The total corrections in (9.9) remain available when estimating interactions with the new increments. Proposition 9.6. Let (u[j], p[j]) be a state at stage j as in Definition 9.4. There is a state (u[j+1], p[j+1]), constructed with the same base, primary amplitudes, supports, and inverse operators, whose residual orders are
1660. u1660 It is obtained by the following ordered corrections:
1661. u1661 (i) solve the inhomogeneous amplitude equation for each supported nonzero harmonic;
1662. u1662 (ii) change the wave amplitudes to correct the auxiliary-averaged tangential residual;
1663. u1663 (iii) apply the temporal mean inverse to the part with zero auxiliary average;
1664. u1664 (iv) apply the five-equation correction to the three current defects.
1665. u1665 Pressure is reconstructed after each correction. Wave and meridional velocity increments are realized by their potentials; azimuthal mean increments are added directly. The cumulative bounds (9.9) and the exact moments (9.10) are preserved.
1666. u1666 Proof. The four steps below implement (i)–(iv) in order. For this cycle abbreviate B = 1 2
1667. u1667 1 2 + σj and C∗= B + 1 2
1668. u1668 1 2 = 1 + σj. At every occurrence of gr and pm below, an asserted improved order concerns their changes; the total corrections continue to satisfy (9.9). Within each step, w, β, v, γ denote the fields before that step. An increment replaces them by
1669. u1669 Compute gr,new from these complete fields using (8.3), and set
1670. u1670 All residuals and defects in the next step are computed from this updated state. Step 1: Cancel the supported harmonics. We apply Proposition 7.2 to each grouped supported source, with zero initial data at the entrance to the prescribed path in the common auxiliary torus. For source coefficient fm, the normalized harmonic increments are
1671. u1671 Assemble these fields with their physical rescalings over all labels and harmonics, retaining conjugate pairs. The velocity increment has class WB, and its pressure has class WB+1/2. The new nonzero harmonic errors have the following exponents:
1672. u1672 The first row follows from Proposition 9.1; the other three follow from Lemma 9.2. The mean covariance changes caused by these increments have exponent at least B + 1/2 = C∗; a radial divergence costs κs. The exact balances and linear pressure map therefore give
1673. u1673 The preserved angular-momentum and axial-flux constraints and the exact integrated identities (8.16) improve the weighted moments by one:
1674. u1674 (9.12) The axial identity includes cρP inside its axial derivative.
1675. u1675 This term accounts for the normalized correction −ρP left in the radial equation after pressure reconstruction. Step 2: Correct the auxiliary-averaged residual by adjusting the stress. A compactly supported stress can cancel the auxiliary-averaged tangential residual once its total weighted radial moments have been removed. We make this correction in physical coordinates so that the stress agrees across charts. Use the residual components Eθ, Ez of (8.3), supported in the prescribed annulus, and retain the background residual separately with its flatness
1676. u1676 Their auxiliary averages in physical coordinates are F2 = Q−2A−1/2⟨Eθ⟩Y and bound.
1677. u1677 F1 = Q−2A−1/2⟨Ez⟩Y; these definitions agree between charts. For e ∈{1, 2}, choose a fixed
1678. u1678 xeˆ︁be(x) dx = 1, put be(r, z, t) = q−(e+1)/2ˆ︁be(r/√q), and set interior profile ˆ︁be satisfying ∫︁
1679. u1679 The radial integrals are taken with (z, t) fixed. The integral has zero total moment, since ∫︁
1680. u1680 re(Fe −beMe) dr = Me −Me = 0. Its lower endpoint fixes the integration constant, and the zero weighted moment makes σe vanish beyond the annulus supporting the source. Thus σe is compactly supported and (∂r + e/r)σe = −Fe + beMe. Define the normalized tensor pair Σ = Q2A(σ2, σ1). The weighted radial-integral estimate in the proof of Proposition 9.3 gives Σ ∈MC∗−κs. Since the construction uses the auxiliary averages Fe, the result is independent of every auxiliary torus coordinate, including when the original fields have nonzero Fourier modes on that torus.
1681. u1681 Apply the amplitude-correction operator Las
1682. u1682 phys from (7.35), which sums the localized operators in physical coordinates, to the pair σ = (σ2, σ1). Its normalized representative LasΣ belongs to the supported class WB−κs and has the prescribed symmetrized cross covariance Σ with wtan 0
1683. u1683 0 , by (7.36). Each local summand uses (7.31) before its slow cutoff and physical rescaling. For each sign σ ∈{+, −}, the formula divides the known numerator by physical the fixed rescaling. positive primary For each amplitude sign σ ∈{+, 2√yσ, −}, so the it permits formula increments divides the of known either numerator sign without by requiring a square root of the current covariance or y + dΣ. We use the pressure coefficient for the homogeneous amplitude equation (with source zero) and construct the velocity by taking the curl, as in Lemma 7.7. The covariance identity is for the paired signs of one slow label; summing the resulting covariances uses each slow label once and the squared slow partition. The quadratic self-interaction of the correction remains in the residual.
1684. u1684 The nonzero harmonic errors now have exponents
1685. u1685 For the auxiliary mean, split the old exact wave as its fixed primary transverse amplitude plus a remainder in W0.68. More explicitly, write the signed increment as s = ts + rs, with transverse amplitude ts and curl remainder rs, and set B(a, b) = ⟨⟨a ⊗b + b ⊗a⟩θ⟩Y. Then its exact covariance change is
1686. u1686 (9.13) The radial–tangential components of the first term equal Σ, whose divergence cancels Fe −beMe. The remaining tensor terms, before divergence, have exponents
1687. u1687 transverse correction with old-wave remainder
1688. u1688 signed curl correction times old exact wave
1689. u1689 These are the remainders in the exact covariance expansion of Corollary 7.8. Their radial divergence incurs one loss of κs. Axial fluxes gain one on tensor changes of order at least C∗−κs. The changes in gr, pm are of order at least C∗−2κs, and the pressure effect on Ez gains one. The retained moment-correction term beMe is of order C∗+ 1 −κs by (9.12). Since
1690. u1690 0.18 −2κs > 0.17 and σj −3κs > 0.17, the state after pressure reconstruction satisfies
1691. u1691 (9.14) Step 3: Remove the nonconstant auxiliary means. The auxiliary averages now have the improved order in (9.14). The complementary parts still have order H and can be canceled by the temporal mean inverse. For the current exact mean residuals, set E◦ a
1692. u1692 a = Ea −⟨Ea⟩Y, a = θ, z. The temporal inverse (8.20) prescribes
1693. u1693 The inverse N−1 i0
1694. u1694 is the zero-average inverse from Lemma 8.6. The prescribed tangential increments have class MH and the induced radial velocity increment has class MH+1. Recompute pressure. The term 2V∆v/R again has order H in ∆gr, so ∆pm ∈MH. Every other part of ∆gr has the following orders: the fast-time derivative of ∆β has order H + 1; radial fluxes contain b or an old or new radial mean; axial fluxes gain one; the other quadratic azimuthal-velocity terms have order at least H + 0.9; and the viscous term contains a factor ε and at most two radial derivative losses.
1695. u1695 Write ∆γ for the actual axial increment and γd for the prescribed increment in (8.20). Since t∗= −ε∂T + ci0Ni0, the fast-time identities are
1696. u1696 The last term is the flat axial reconstruction remainder from Lemma 8.6. The wave covariance is unchanged during this mean update. Subtracting the two versions of (8.3) gives
1697. u1697 Eθ,new −⟨Eθ⟩Y = −ε∂T∆v −ε(∆0 −R−2)∆v
1698. u1698 Ez,new −⟨Ez⟩Y = −ε∂T∆γ −ε∆0∆γ + Fax
1699. u1699 Using (9.9), the bounds |DIb| ≤CIεSbI∗on the shell, and the increment orders above, the four types of terms have the following lower bounds for their class exponents:
1700. u1700 Thus the displayed residual changes, after subtracting Fax in the axial component, belong to MH+1−2κs. Retain Fax in the additive flat term. The wave change is in WH by interaction with the waves already added, whose total satisfies the W1/2 bound. Remainders from the compactly supported modified radial integral enter the additive flat term while their actual velocities continue to be used. We have therefore reached
1701. u1701 Step 4: Enforce the three compatibility conditions for compact support. The tangential residuals have improved, while the compatibility defects still have order H. Apply the fiveequation correction map (8.25) to these defects at order H. The azimuthal and prescribed axial increments are slow, supported on the fixed support of the test-function profiles in the five-equation correction system, and in MH. Their induced radial velocity increment is in MH+1. The first two rows preserve (9.10); the last three remove the linear contributions to the changes in (P, Jθ, Jz). Recompute pressure from the new actual fields.
1702. u1702 The preceding bounds for changes in the nonzero harmonics and in the tangential mean residuals still apply; a slow correction from this system has no fast-time term to cancel. To estimate the remaining defects, subtract 2V∆v/R from ∆gr. The remainder contains slow-time differentiation of the induced radial velocity, radial derivatives of b∆β and of products involving old or new radial means, axial flux derivatives, products of ∆v with old or new means, and the radial viscous term. They have exponent at least H + 0.9 −2κs. In ∆Jθ and ∆Jz, the terms beyond the linear contributions canceled by this system are products of tangential means or moments of this same radial remainder; see the exact formulas (8.27). Hence
1703. u1703 (P, Jθ, Jz) ∈SH+0.9−2κs = SC∗+0.9−4κs.
1704. u1704 The cycle closes with the inequalities
1705. u1705 We used B ≥0.7. Thus the wave, full tangential mean, and defect orders all improve by 0.1. The smallest signed increment order is B −κs ≥0.69999 > 0.68; particular increments start at B ≥0.7. Every new mean-velocity and mean-pressure increment has decay exponent above 0.9, and every radial mean-velocity increment has exponent above 1.9. Finite summation therefore preserves (9.9). The pressure reconstruction gives the asserted radial residual. Proposition 9.3 supplies the source hypotheses for the next cycle, completing the induction.
1706. u1706 □
1707. u1707 9.4. A common domain and physical derivative estimates. To apply the summation lemma
1708. u1708 Lemma 5.4, all finite states (u[j], p[j]) must exist on one domain (Lemma 9.7). We also need their gains in decay to survive differentiation in the physical variables: for each fixed derivative order, the loss in powers of q must be independent of the correction stage. Constants in the bounds may still depend on the stage, as in Lemma 9.8.
1709. u1709 Lemma 9.7. There exists qbig > 0, independent of the correction stage and derivative order, such that every finite partial sum above, before applying summation cutoffs, is well defined on 0 < q < qbig. For every fixed stage and output amplitude derivative, its construction and estimate require finitely many input amplitude derivatives. The constants may depend on the correction stage. Proof. Choose 0 < qbig ≤q∗once, so that the pulse estimates hold for q < qbig, with the fixed background, the stated positive lower bound for |nΦ|, the primary covariance, and the fixed supports of the normalized mean-correction profiles. All later inhomogeneous amplitude problems are linear equations with these fixed coefficients and known sources. The same threshold applies to every harmonic: (7.18) expresses its propagator as the fundamental propagator multiplied by a forward damping factor of magnitude at most one. At each
1710. u1710 fixed derivative order, differentiation introduces powers of the harmonic integer into the constants and permits polynomial growth in S∗. It does not require a smaller q-threshold; see Corollary 7.3.
1711. u1711 Signed Corollary updates 7.3. divide by the original 2√yσ in (7.31); the finite-dimensional correction system uses the fixed linear map of Lemma 8.7. Pressure, temporal inverses, and the modified radial integrals defining the mean potentials are fixed linear maps. All inverse coefficients and denominators are fixed by the primary construction. Consequently, these finite operations are defined on the same domain for sources of arbitrary size, including near qbig, and each cycle improves the decay exponent on that domain.
1712. u1712 To count the input derivatives required at each stage, regard the finite construction as a directed acyclic graph of sums, products, derivatives, and the proved inverses. Its vertices are intermediate expressions and its source vertices are the input fields. For a source vertex a, let Dv,a(m) be a sufficient number of derivatives of input a to bound derivatives through order m of the expression at vertex v. If the operation at v requires dv,w additional derivatives of its input at w, define
1713. u1713 with initial values Db,a(m) = m when the source vertices b = a, and Db,a(m) = 0 when b̸ = a. The graph has finitely many vertices at each fixed correction stage, so all resulting derivative counts are finite. Differentiating a pulse equation to order m gives an equation involving amplitude derivatives of order at most m; lower-order derivatives are already controlled. Torus inverses require finitely many additional amplitude derivatives, while the modified radial integrals preserve the stated derivative bounds. Flatness of the cutoff remainder at a chosen power uses finitely many integrations by parts of the same exact integral. These counts specify how many source derivatives must be estimated. The explicit ε losses instead specify reductions in the decay exponent. For example, estimating additional amplitude derivatives of N−1Dr f retains the single explicit radial loss associated with Dr.
1714. u1714 After constructing and estimating each finite-stage coefficient as a function of independent slow and auxiliary variables, evaluate it at the auxiliary phase map Y = Y(r, t) and differentiate
1715. u1715 □ with respect to (x, t).
1716. u1716 We group the base and finite initialization into U0 = (u[0], p[0]). For j ≥1, let Aw j
1717. u1717 j be the sum of the physical wave-potential increments in the cycle from stage j −1 to stage j. Let Ψj be the sum of its physical azimuthal-potential coefficients, and let bj be the sum of its direct azimuthal velocity coefficients in physical variables. For the remaining summation argument, we use Bj for the azimuthal vector representative in (5.29) and define
1718. u1718 (9.15)
1719. u1719 ∑
1720. u1720 Thus Zj is the tuple in (5.29), with the optional stress entry omitted. In a fixed Q chart, linearity of T0 and the fixed choice of ρ give the pressure increment explicitly:
1721. u1721 (9.16)
1722. u1722 Here p[j] w
1723. u1723 r , P[j] are the normalized quantities in (9.7) and (9.6). The pressure differences from intermediate steps telescope to (9.16).
1724. u1724 Lemma 9.8. For every Cartesian space–time derivative order m,
1725. u1725 |Zj|m + |∆uj|m ≤Cj,mqgj−ℓm(1 + | log q|)Pj,m,
1726. u1726 (9.17) where ℓm is independent of j and includes the additional spatial differentiation recovering velocities from wave and mean vector potentials. The finite partial sums satisfy
1727. u1727 |(u[j], p[j])|m ≤Cj,mq−Km(1 + | log q|)Pj,m, with Km independent of j. Their residuals satisfy
1728. u1728 |R(u[j], p[j])|m ≤Cj,mqhσj−Km(1 + | log q|)Pj,m + Ej,m,
1729. u1729 for every N, (9.18) with Km independent of j.
1730. u1730 Proof. On perturbation supports r ≍√Q and q ≍Q. In a fixed chart Q is held fixed while differentiating. The graph operators of (6.6) give the following losses in powers of Q for an amplitude coefficient:
1731. u1731 (9.19)
1732. u1732 For instance, the coefficient multiplying an auxiliary-coordinate derivative in the physical radial derivative is rdr−1Λi0g = O(Q−1/2ε−κsSP
1733. u1733 ∗). Each additional fixed derivative contributes a fixed reduction in the power of Q. These exponent bounds remain uniform when the relevant chart indices differ by a bounded amount.
1734. u1734 The label phase satisfies |∇xΦ| ≤CQ−1/2SP
1735. u1735 ∗; the axial term pzZ/ε costs Q−Dε−1 = Q−1/2. More generally, (7.3) and (6.12) give, for every fixed pair of nonnegative integers a, b with a + b ≥1,
1736. u1736 uniformly over bands and labels. In each chart, the pulse coordinate has zero physical spatial derivatives and a fixed first physical time derivative; the remaining factors have the stated slow derivative bounds. Set
1737. u1737 Since k ≤2Q−h/2, the chain rule gives
1738. u1738 These costs dominate (9.19). The finitely many harmonic integers at a fixed stage change constants only: a quadratic residual can at most double the current harmonic range, and inverse and curl operations preserve each integer. Thus the range is finite independently of band.
1739. u1739 For a class exponent α, Leibniz’ rule now gives Cartesian space–time derivatives of
1740. u1740 The factors space–time √ζδ−M derivatives and ζδ−M of the normalized coefficient bounded by CQhα−asx−bstSP are bounded on the closed shell for every fixed M and cause no additional power loss. Physical velocity, pressure, wave-potential, and mean-potential rescalings are respectively Q−A, Q−2A, Q1/2−A, Q1/2−A, all bounded by Q−2A. The factor (km)−1 is already included in the normalized chart wave potential, before physical rescaling. An additional spatial derivative covers recovery of a velocity from a wave or mean vector potential. A sufficient common choice is Every retained stage-j increment has normalized exponent at least j/10, and its physical prefactor is bounded by Q−2A. This proves (9.17).
1741. u1741 Apply the same calculation to (9.9), to the fixed background, and to (9.8). Converting the residual from normalized to physical coordinates adds a fixed power of q. The remaining radial residual consists of −ρP, expressed in physical coordinates, and its cutoff remainder. Equation (9.5) supplies Ej,m uniformly over all labels at each fixed stage. This proves (9.18) and the bound for the total correction. On overlapping coordinate representations, the fields agree under the prescribed torus coordinate changes and relabeling. Together with their
1742. u1742 □ smooth zero extensions, this gives the estimates across support and band boundaries. 9.5. Realization of the local field. The finite partial sums (u[j], p[j]) now have arbitrarily high residual decay on a common domain (Lemmas 9.7 and 9.8), with the physical derivative bounds required by Lemma 5.4. In this subsection we sum their potentials with shrinking cutoffs to obtain the local velocity uloc of Proposition 9.9. It remains to verify that this summation preserves the endpoint regularity, exterior heat flow, and inner growth asserted in Theorem 3.1(ii)–(iv).
1743. u1743 Proposition 9.9. There are smooth fields (uloc, ploc) on Ω∗, obtained by summing the finite corrections above, such that div uloc = 0 and all conclusions of Theorem 3.1 hold.
1744. u1744 Proof. We apply the summation lemma with F(u, p) = R(u, p), using q∗= qbig from Lemma 9.7 and shrinking the earlier local thresholds to this common value. The inputs are the velocity and pressure fields. Step 1 verifies the hypotheses of Lemma 5.4. Steps 2–5 establish Theorem 3.1(i)–(iv), respectively: Cartesian regularity, endpoint regularity, the exterior formula and flat residual, and the inner swirl asymptotic.
1745. u1745 Step 1: Representatives and the summation hypotheses. Take U0 to consist of the fixed slow base and the finite initialization block. If necessary cut that block once inside q < qbig, on its wave and mean potentials before differentiation, and on its direct angular means and pressures. The cutoff equals one near q = 0, so each finite state agrees there with its original expression.
1746. u1746 For j ≥1, use the representatives Zj = (Aj, Bj, pj) from (9.15). Their mean-potential part gives
1747. u1747 curl(Ψjeθ) = (−∂zΨj, 0, (∂r + r−1)Ψj).
1748. u1748 Since Bj = bjeθ and ∂θbj = 0, multiplying Bj by a function of q(z, t) preserves its zero divergence. All these representatives have smooth zero extensions and vanish near the axis for each fixed t < 1; their vanishing near the axis gives a smooth Cartesian extension there.
1749. u1749 Lemma 9.7 supplies one domain for every finite partial sum before summation cutoffs. The similarity identities give (5.28), and q ≥1 −t gives the required local lower bound. Lemma 9.8 supplies (5.30) with gj = hj/10 and exponent reductions independent of j.
1750. u1750 For the fixed background and initialization, derivatives through order m are bounded by Cmq−Km(1 + | log q|)Pm. Finally, (9.18) is (5.33) for the fixed differential polynomial
1751. u1751 F(u, p) = R(u, p) = ∂tu + (u · ∇)u −∆u + ∇p, with ρj = hσj →∞. The estimate (9.18) concerns the actual pointwise residual, after evaluating the auxiliary variables at Y(r, t). The finite-stage flat term is uniform over its infinitely many dyadic labels. The finite initialization cutoff only adds errors supported away from q = 0, which also meet that fixed-stage flatness condition. These residual estimates are uniform on the whole local domain: choose a fixed bounded interval in X containing all slow and annular correction supports. The estimates above and (5.41) apply there. Outside it, every finite state agrees with the exact exterior heat field of (4.29), with its centrifugal pressure, so its residual is identically zero.
1752. u1752 Step 2: The summed fields and their Cartesian regularity. Lemma 5.4 therefore gives potentials whose locally finite sum defines a velocity satisfying div uloc = 0, and
1753. u1753 (9.20) The tail estimate compares the constructed field with one fixed finite state and its flat remainder. The flat residuals separated at each finite stage are estimated through this comparison and are not summed. To identify the potentials and pressure of the summed fields, let A0, B0eθ, p0 be the representatives of U0: the realized slow base from (5.27), together with the finite initialization and its pressure. Define
1754. u1754 (9.21)
1755. u1755 This gives the required vector potential, azimuthal field, and pressure. At the axis the base Stokes streamfunctions have the form r2 times a smooth function of (r2, z, t), so the base vector potential is smooth in Cartesian coordinates. The base angular field is r times such a scalar times eθ. All annular representatives vanish in a neighborhood of the axis at each t < 1. Thus (9.21) proves Theorem 3.1(i).
1756. u1756 Step 3: One-sided regularity away from the singular point. It remains to verify the endpoint regularity away from q = 0. Fix a compact spatial set and 0 < c < c′ < q∗. Only finitely many terms in the expansion in powers of q2h and correction stages have nonzero cutoff factors on a neighborhood of c ≤q ≤c′, because both cutoff sequences tend to infinity. There are also finitely many dyadic bands and slow labels on this compact set. The base profiles and their Stokes streamfunctions have bounds for derivatives of every fixed order on the closed parameter range −1 ≤η ≤1, by Theorem 4.6 and Proposition 5.5. The geometry chooses its labels, representatives, rounded frequencies, and enlarged local torus rectangles from the closed τ ≥0 range and keeps these discrete choices fixed as τ →0. The pulse inverse acts along a path with the physical slow variables fixed, within τ ≥0. Differentiating its fixed linear ODE gives the bounds at every derivative order. These bounds, the quotient bound (7.33) using the fixed positive primary amplitudes, and the radial and temporal mean maps give finite one-sided bounds for every fixed derivative of each wave potential, mean
1757. u1757 vector potential, azimuthal mean component, and pressure whose cutoff is nonzero on the chosen neighborhood. Their smooth zero extensions cover every support boundary.
1758. u1758 Coordinate conversion has denominator L ≥1 −2h > 0, so on q ≥c it preserves these bounds for derivatives of each fixed order. On annular supports r is bounded away from zero; at the axis use the preceding Cartesian factors. The finite sum therefore has uniformly bounded Cartesian space–time derivatives of every order up to τ = 0, also inside the active shell. Bounding one additional time derivative and applying the fundamental theorem of calculus gives the uniform one-sided limit of each derivative. The same theorem on compact spatial subsets proves compatibility with spatial derivatives; compatibility between limits of consecutive time derivatives follows by integration in time. This proves Theorem 3.1(ii) for A, Beθ, ploc themselves.
1759. u1759 Step 4: The exterior heat field and flat residual. All annular wave and mean correction potentials, azimuthal mean components, and pressures vanish beyond Xb. The zero total axial moments of the leading profile and of each coefficient in the expansion in powers of q2h also make their Stokes streamfunctions vanish in the common exterior. Consequently, for a fixed Xext beyond all slow supports, A = 0 and B = K. The leading-profile pressure, normalized to vanish at radial infinity, and the compactly supported pressure corrections of positive order give precisely −∫︁∞ r
1760. u1760 r K(ρ, τ)2 dρ/ρ there. Using (4.29), define
1761. u1761 This is (3.5). The integral formula (A.34) gives, for every integer m ≥0,
1762. u1762 where (b)m is the rising factorial and (b)0 = 1. Indeed, the factor (1 + Zv)−h−m in that integral is at most one for every Z, v ≥0. Thus the required derivative bounds hold on the full nonnegative half-line. By Lemma A.6, K solves the radial heat equation; its centrifugal pressure makes the Navier–Stokes residual identically zero in the exterior. Together with the background residual estimates on compact sets of profile variables in (5.41) and (9.20), this proves all of Theorem 3.1(iii).
1763. u1763 Step 5: The inner velocity growth. Finally choose any fixed Xin ∈(0, Xa) where the positive leading azimuthal velocity is e0 = E0(Xin, 0) > 0. All annular corrections vanish there. The realized expansion in powers of q2h in (5.1) has its leading value plus a remainder O(q2h) after removing the velocity factor q−A, at the fixed point (X, η) = (Xin, 0) in profile coordinates. At z = 0 one has η = 0 and q = τ, hence
1764. u1764 □ This is Theorem 3.1(iv), and completes its proof.
1765. u1765 10. Compact forcing and whole-space breakdown
1766. u1766 In this section we turn the fields supplied by Theorem 3.1 into the whole-space solution and force in Theorem 1.1. We write the local fields as
1767. u1767 Here A and B are the representatives constructed in (9.21), and B is independent of θ. We first work at viscosity one. The local velocity has the growth (3.6) required for Theorem 1.1;
1768. u1768 we must give it zero initial datum and compact spatial support, while making its momentum
1769. u1769 c (R3 × (0, ∞); R3) (Proposition 10.1 and Lemma 10.3). residual the restriction of a force in C∞ c
1770. u1770 We multiply the vector potential, azimuthal coefficient, and pressure by cutoffs equal to one near (0, 1). We then prove that every derivative of the resulting residual has a uniform limit as t ↑1 (Lemma 10.2), and construct a force attaining those limits from t > 1 (Lemma 10.3). The equation gives a uniform energy bound before time one (Lemma 10.4). Finally, comparison with any smooth solution of bounded kinetic energy (Lemma 10.5) transfers the local velocity growth to that solution, giving the contradiction needed for Theorem 1.1. Rescaling gives arbitrary positive viscosity and the periodic construction (Corollary 10.6).
1771. u1771 10.1. Localization within the local domain. In this subsection we localize the velocity and pressure inside the domain of Theorem 3.1. We first recall why the chosen vector potential A is smooth at the axis (Theorem 3.1(i)) and vanishes in the heat exterior (3.5). These properties will allow us to take its curl after localization and to estimate the force at time one (Lemma 10.2). For the background coefficients Un in (5.1), the Stokes streamfunctions and their sum are
1772. u1772 (10.1)
1773. u1773 The for every cutoffs n χ(cnq) ≥0. Thus are those each radial used in integral Proposition in (10.1) 5.5. vanishes By (4.28) beyond and (5.10), the ∫︁∞ 0
1774. u1774 0 Un(X, η) dX = 0 for every n ≥0. Thus each radial integral in (10.1) vanishes beyond the common support of the axial coefficients. Near the axis, S = r2a(r2, z, t) for a smooth scalar a, and hence
1775. u1775 This formula gives the smooth Cartesian representative of the background potential.
1776. u1776 Every wave velocity is the curl of its supported physical potential. For a prescribed axial mean increment γd,phys, the mean construction uses the azimuthal potential coefficient
1777. u1777 with the radial operations defined in (8.4). Beyond the source and the transition of χm, we have χm = 1 and Ig = J g, so Ψ = 0. These wave and mean potentials, with their summation cutoffs, are the terms added to Abase in (9.21). The direct azimuthal mean increments are included in B. Evaluation at Y(r, t) preserves their independence of θ, and all the annular corrections vanish near the axis for each fixed t < 1. In particular, for the fixed Xext in Theorem 3.1, the chosen potential satisfies A = 0 on X ≥Xext.
1778. u1778 We now choose cutoffs supported inside the local domain Ω∗= {τ > 0, q < q∗}, where τ = 1 −t. Applying the spatial cutoff before taking the curl preserves incompressibility, and the temporal cutoff gives zero initial datum.
1779. u1779 Proposition 10.1. There are a compact set K ⊂R3, a smooth vector field u and a smooth scalar field p on R3 × [0, 1), with supp u(·, t) ∪supp p(·, t) ⊂K for every 0 ≤t < 1, satisfying
1780. u1780 (10.2) They agree with uloc, ploc on a spatial neighborhood of the origin for all times sufficiently close to 1.
1781. u1781 Proof. To keep a fixed spatial cutoff inside the local domain, we need an upper bound for q in terms of z and τ. The coordinates in (3.2) give √ such a bound uniformly in r. If 1 −η2 ≥1/2, then q ≤2τ; otherwise |η| > 2−1/2 and q ≤(
1782. u1782 (10.3) We choose 0 < τ0 < 1/2 and z0 > 0 with C0(τ0 + z1/D 0
1783. u1783 ) < q∗/2. For any fixed r0 > 0, we choose a smooth axisymmetric cutoff 0 ≤χx ≤1, smooth as a function of r2, z, such that
1784. u1784 We also choose a smooth function 0 ≤χt ≤1 of t with
1785. u1785 By (10.3), the support of c for t < 1 lies in q < q∗/2. In particular it stays a fixed distance in the q coordinate from the outer boundary of Ω∗. On that domain we define
1786. u1786 (10.4) and extend by zero outside the cutoff support. The product rule gives
1787. u1787 div u = div curl(cA) + r−1∂θ(cB) = 0. The Cartesian representatives recalled above make the fields smooth at the axis. The cutoffs and the margin from q = q∗give smooth zero extension elsewhere. The support is contained in K = supp χx, and the temporal cutoff gives (10.2). Both cutoffs are one near (0, 1), so the
1788. u1788 □ local fields are unchanged there.
1789. u1789 For 0 ≤t < 1 we define
1790. u1790 f = R(u, p) = ∂tu + (u · ∇)u −∆u + ∇p.
1791. u1791 (10.5) This definition includes every derivative of the cutoffs. Taking its divergence gives
1792. u1792 ∑ Thus the localized pressure satisfies the Poisson equation with the corresponding forcedivergence term; the force may have nonzero divergence.
1793. u1793 10.2. Limits of force derivatives at the terminal time. The force in (10.5) is so far defined only for t < 1. A smooth extension requires compatible limits of all its derivatives as t ↑1. Flatness of the local residual in (3.4) controls these limits at the origin; away from the origin, the endpoint bounds in Theorem 3.1(ii) and the exact exterior heat flow (3.5) control the terms introduced by localization. In this subsection we first establish the derivative limits (Lemma 10.2), then construct an extension with compact time support (Lemma 10.3).
1794. u1794 t f converges uniformly on R3 Lemma 10.2. For every spatial multi-index α and integer j ≥0, ∂α as t ↑1. There are functions Fj ∈C∞ c
1795. u1795 c (R3; R3), all supported in K, such that
1796. u1796 (10.6) These are the derivative limits of f from t < 1, compatible under spatial and time differentiation.
1797. u1797 Proof. We first establish endpoint bounds away from the origin. On compact spatial sets with q bounded below and bounded away from q∗, Theorem 3.1(ii) bounds every physical derivative of the actual potentials and the scalar fields. More precisely, on such a compact set K0, for G = A, Beθ, ploc and t < t′ < 1, the fundamental theorem of calculus gives
1798. u1798 The last supremum is bounded independently of t, t′ near one. Thus each fixed derivative is uniformly Cauchy as t ↑1. All constants here may depend on the positive lower bound for q.
1799. u1799 The argument using finitely many nonzero cutoff terms in Proposition 9.9 supplies these bounds on the closed parameter range −1 ≤η ≤1, for the actual representatives and the smooth Cartesian factors in their near-axis representations, with the labels and integer frequencies fixed. Multiplication by the fixed cutoffs and the finite differential expression (10.5) preserve the bounds.
1800. u1800 The remaining region has r bounded below and q sufficiently small, so X ≥Xext and the velocity is the exterior azimuthal field K(r, τ)eθ of (3.5). In terms of the heat profile H defined by (A.32),
1801. u1801 (10.7) Thus the function in Theorem 3.1 is Hext(Z) = 2Ac∞H(4Z). For each j, differentiating the integral in (A.32) is justified uniformly for Z ≥0 by a constant times e−vvh+j. It follows that
1802. u1802 (10.8) The pressure is normalized to vanish at radial infinity.
1803. u1803 On a compact positive-radius interval r ∈[r−, r+], with r−> 0, the second majorant is integrable for ρ ≥r−. Dominated convergence therefore justifies
1804. u1804 It also gives a uniform bound and a one-sided limit for each derivative on compact positiveradius intervals. Here ∂t = −∂τ. Further radial derivatives of the pressure follow by differentiating the lower endpoint; those of K follow directly from (10.7). This proves the required smooth one-sided extension to τ = 0 using the integral on τ ≥0. In this region A = 0, so the localized fields involve only these smooth exterior quantities and fixed cutoffs. The heat equation for K and ∂rploc = K2/r also show directly that the uncut exterior residual is exactly zero.
1805. u1805 These two regions cover the terminal slice away from the origin: if z̸ = 0, then q ≥ |z|1/D > 0; if z = 0 and x̸ = 0, then r > 0 and the preceding exterior argument applies on a small neighborhood. Hence every Cartesian space–time derivative of the force converges uniformly on compact spatial sets avoiding the origin.
1806. u1806 Near the origin the cutoffs equal one. We use (3.4) on X ≤Xext, and the exact zero residual for larger X. For every α, j, N this gives, on a fixed neighborhood of (0, 1),
1807. u1807 (10.9) For a prescribed tolerance, we first choose a small ball and then a late enough time so that this bound is below the tolerance there. A finite cover of the rest of K gives uniform convergence outside that ball. All these derivatives vanish off K. They are therefore uniformly Cauchy on R3, and their limits vanish at the origin.
1808. u1808 We define Fj to be the uniform limit for α = 0. Passing to the limit in the fundamental theorem of calculus along spatial coordinate segments identifies every other limit as ∂α
1809. u1809 xFj. Thus Fj is smooth and has the stated support. Passing to the limit in the corresponding identity in time gives
1810. u1810 (10.10) This proves compatibility of the derivative limits under spatial and time differentiation and
1811. u1811 □ completes (10.6).
1812. u1812 The limits Fj give the Taylor coefficients needed to continue the force through time one. We realize them with time cutoffs whose supports shrink with the derivative order, so that the extension is also compactly supported in time. Such a smooth compactly supported force satisfies the decay conditions in [13, (5)]; the zero datum satisfies its initial-data conditions as well.
1813. u1813 c (R3 × (0, ∞); R3), with support contained in Lemma 10.3. The force in (10.5) extends to f ∈C∞ c K × [0, 2].
1814. u1814 Proof. We choose χ0 ∈C∞ c
1815. u1815 c (R), equal to one near zero and zero for arguments at least one. For σ = t −1 ≥0, we set
1816. u1816 ∑
1817. u1817 (10.11) where the increasing integers bj ≥1 are chosen below. The product rule and 0 ≤σ ≤b−1 j
1818. u1818 on the support of each summand give⃦⃦⃦⃦
1819. u1819 (10.12) Indeed, when a derivatives fall on the cutoff, their factor ba j
1820. u1820 j is multiplied by a monomial of degree j −m + a; its bound is b−(j−m+a) j
1821. u1821 . Terms whose monomial has been differentiated to zero are absent. We set b0 = 1, and for j ≥1 define
1822. u1822 where the maximum is over nonnegative integers a, m. This is a finite set of constraints, and m −j < 0 in each of them; thus the defining set of integers is nonempty. With this choice, (10.12) is at most 2−j whenever |α| + m ≤⌊j/2⌋.
1823. u1823 Every fixed differentiated tail now converges uniformly. At σ = 0 the cutoff in each fixed summand is constant near zero, so its m-th derivative contributes only when j = m, giving Fm. Uniform differentiated convergence and (10.10) show that (10.11) matches all mixed space–time derivative limits from t < 1. Each summand retains support in K and is zero for σ ≥1. The same uniform convergence proves smoothness at those support boundaries. Finally, the original force is zero near t = 0, by (10.2) and the construction of p.
1824. u1824 To state the decay bounds, we choose R with K ⊂B(0, R) and write Mα,m = ∥∂α
1825. u1825 t f ∥∞. Compact support gives, for every integer k ≥0,
1826. u1826 t f (x, t)| ≤Mα,m(3 + R)k(1 + |x| + t)−k.
1827. u1827 □
1828. u1828 10.3. Energy and comparison. In this subsection we first prove that the smooth compactly supported force f of Lemma 10.3 gives a uniform energy bound for the constructed velocity u up to time one (Lemma 10.4). We then show that any smooth solution with the same force, zero datum, and bounded kinetic energy agrees with it on every shorter time interval (Lemma 10.5). This comparison will transfer the local velocity growth (3.6) to a hypothetical global smooth solution. In these whole-space energy estimates, ∥· ∥p denotes the spatial Lp(R3) norm, and unmarked spatial integrals are over R3.
1829. u1829 Lemma 10.4. Let F(t) = ∫︁t 0
1830. u1830 0 ∥f (s)∥2 ds for 0 ≤t ≤1. Then F(1) < ∞, and
1831. u1831 (10.13) In particular the kinetic energy is uniformly bounded, and the total dissipation on [0, 1) is finite. Proof. On each [0, T], T < 1, the localized velocity and pressure are smooth with fixed compact spatial support. Integrating the equation against u therefore gives
1832. u1832 (10.14) For δ > 0, we divide this identity by (∥u(t)∥2 2
1833. u1833 2 + δ2)1/2, discard dissipation, and use Cauchy– Schwarz. Integration from the zero datum and δ ↓0 yield ∥u(t)∥2 ≤F(t). Returning to (10.14) and integrating gives
1834. u1834 The compact smooth force has F(1) < ∞. Monotone convergence gives finite integrated dissipation on [0, 1), using the bounds for t < 1 alone.
1835. u1835 □
1836. u1836 The comparison argument uses smoothness and a uniform spatial L2 bound on [0, T], where T < 1. These hypotheses leave the growth of spatial derivatives at infinity unrestricted. We recover the pressure gradient from the equation before passing to the limit in the localized energy identity.
1837. u1837 Lemma 10.5. Fix T < 1. If v, P is a smooth solution of (1.1) at viscosity one on R3 × [0, T], with the force f of Lemma 10.3, zero initial velocity, and v ∈L∞([0, T]; L2(R3)), then v = u on that interval, where u is the localized velocity of Proposition 10.1.
1838. u1838 Proof. We estimate the difference of the two solutions on expanding balls. The pressure term requires separate control because no spatial growth condition has been imposed on P. Constants denoted by CT below may depend on T and the stated bounds for the fields, but not on the cutoff radius. We set
1839. u1839 gij = vivj −uiuj = wiwj + wiuj + uiwj. We use the convention (div g)i = ∑j ∂jgij. On [0, T], ∥w(t)∥2 ≤CT and ∑i,j ∥gij(t)∥1 ≤CT. The common force cancels, so
1840. u1840 ∂tw + (v · ∇)w + (w · ∇)u = ∆w −∇π,
1841. u1841 (10.15)
1842. u1842 Pressure gradient. Let Ri be the Riesz transform with Fourier multiplier iξi/|ξ|, and define the distribution
1843. u1843 ∑
1844. u1844 (10.16)
1845. u1845 Its multiplier is −ξiξj/|ξ|2, with the value at zero irrelevant. Since the Fourier transform of an L1 function is bounded, π∗lies uniformly in H−s(R3) for each s > 3/2: its squared norm
1846. u1846 R3(1 + |ξ|2)−s dξ. The multiplier gives is bounded by a constant times ∥g∥2 1
1847. u1847 We claim that ∇π = ∇π∗in the space–time interior. For a ∈C∞ c
1848. u1848 c (0, T), the conservative form of (10.15) gives
1849. u1849 The three terms on the right belong respectively to H−2, L2, and H−3: the last assertion follows from g ∈L∞ t
1850. u1850 x and the Fourier estimate above with s = 2. Also π∗∈L∞ t
1851. u1851 Taking divergence of the difference equation gives ∆π = −∑i,j ∂i∂jgij = ∆π∗.
1852. u1852 Hence ∆Ha = 0. The Fourier transform of Ha is a weighted L2 function supported at {0}, and is therefore zero. Testing also in space proves the claim. This determines the pressure gradient under the stated hypotheses, with the spatial growth of P unrestricted.
1853. u1853 Pressure flux. We now bound the pressure contribution at the boundary of an expanding a ball weighted in terms L6 of norm the weighted of w that dissipation can be absorbed inside by it. We dissipation. first estimate We ∫︁
1854. u1854 πw · ∇χR by powers of a weighted L6 norm of w that can be absorbed by dissipation. We choose 0 ≤φ ≤1 smooth, compactly supported and equal to one on the unit ball. For R ≥1, we write φR(x) = φ(x/R), χR = φ8
1855. u1855 The Sobolev inequality and the product rule imply
1856. u1856 (10.17) We use the standard Lp boundedness of Riesz transforms for 1 < p < ∞; see [20]. For multiplication by a scalar function a and a linear operator T , we write [a, T ]g = a(T g) − T (ag). Multiplication of (10.16) by φ4
1857. u1857 (10.18) The first sum has L3/2 norm at most CT(BR + 1), since
1858. u1858 The nonlocal kernel of RiRj is bounded by C|x −y|−3. The possible multiple of the identity cancels in the commutator, whose kernel is therefore bounded in absolute value by
1859. u1859 KR(x −y) = C|x −y|−3 min{|x −y|/R, 1}.
1860. u1860 Direct radial integration gives
1861. u1861 Young’s convolution inequality bounds the second sum in (10.18) in L4/3 by CTR−3/4. These identities hold first for compact smooth cutoffs of g, then for g by convergence in L1, the uniform commutator bound, and convergence of its Riesz transforms in H−s. In particular π∗is locally integrable in space and time.
1862. u1862 The equality of pressure gradients, tested against the compact vector field a(t)χRw, now gives
1863. u1863 Since |∇χR| ≤CR−1φ7
1864. u1864 R, the two terms in (10.18) pair with CR−1φ3
1865. u1865 R|w|. Interpolation yields
1866. u1866 Consequently⃓⃓⃓⃓
1867. u1867 (10.19)
1868. u1868 Difference energy. The pressure flux can now be included in the energy estimate for the difference. All integrations have compact support, so smoothness suffices. Pairing (10.15) with χRw gives
1869. u1869 We take R large enough that χR = 1 on a neighborhood of supp u. Then v = w on supp ∇χR, and the transport flux is bounded by
1870. u1870 R|w|)1/2|w|1/2 and Hölder’s inequality give the last bound. The Laplacian Here φ2 term is at most CTR−2. Using (10.17) in (10.19), every power of AR in these flux bounds is at most 3/2 < 2. Young’s inequality therefore gives
1871. u1871 For example, R−1A3/2 R
1872. u1872 R + CδR−4; the terms of powers 1/2 and 3/4 have the same required bound. The coefficient ∥∇u∥∞is bounded on [0, T], and ER(0) = 0. Gronwall gives
1873. u1873 Since χR = 1 on each fixed ball for all sufficiently large R, letting R →∞gives w = 0 throughout [0, T].
1874. u1874 □ 10.4. Growth, lifespan, and viscosity. The force extension and energy estimate are now established for the localized fields (u, p) (Lemma 10.3, (10.13)). In this subsection we use the inner growth asymptotic (3.6) from Theorem 3.1 to identify the maximal classical lifespan and, with the comparison lemma (Lemma 10.5), exclude a global smooth solution of bounded energy. A spatial rescaling then gives the result for every fixed positive viscosity without changing the singular time.
1875. u1875 Proof of Theorem 1.1. The field constructed above solves the equation exactly on [0, 1), by (10.5). Smoothness and fixed compact support give u ∈C([0, T]; H3) for each T < 1. The force has the required smoothness and support by Lemma 10.3, and Lemma 10.4 gives bounded energy.
1876. u1876 For the fixed Xin in Theorem 3.1, we take the Cartesian points
1877. u1877 (10.20) Along this path z = 0, q = τ, and the spatial and time cutoffs in (10.4) equal one for all sufficiently small τ > 0. Hence (3.6) gives
1878. u1878 (10.21) Here the annular corrections vanish in the fixed inner similarity region, and the cutoff sum of terms with n ≥1 in the expansion in powers of q2h satisfies the stated relative-error estimate. The summation cutoffs remain in this estimate. The points xτ along which the velocity diverges converge to the origin. The embedding H3(R3) ↪→L∞(R3) excludes a classical H3 continuation through time one. In the classical H3 class, the same difference energy calculation is justified by Sobolev approximation and H3 ↪→W1,∞; Gronwall gives uniqueness on every shorter interval. Thus the maximal classical existence interval is [0, 1).
1879. u1879 Suppose that a global smooth solution v, P with the same data had uniformly bounded kinetic energy. For every T < 1, Lemma 10.5 applies on the closed interval [0, T]. It follows that v = u throughout [0, 1). Equation (10.21) then contradicts the boundedness of the smooth v on a compact neighborhood of (0, 1). The force is nonzero, since (10.13) would otherwise give u = 0.
1880. u1880 Finally, for fixed ν > 0, we define
1881. u1881 (10.22) With y = x/√ν, the chain rule gives
1882. u1882 The datum remains zero and the common spatial support becomes Kν = √ν K. Time is unchanged, and
1883. u1883 Thus the force remains smooth and compactly supported in Kν × [0, 2], and satisfies all required decay bounds for each fixed ν > 0. The energy and dissipation scale by
1884. u1884 (10.23) Growth occurs at √ν xτ at the same time one. 0 Any smooth bounded-energy 0 competitor vν, Pν for the rescaled data would give the viscosity-one competitor
1885. u1885 This competitor has already been excluded. The theorem follows for every fixed positive
1886. u1886 □ viscosity.
1887. u1887 10.5. The periodic equations. The fixed compact support of the velocity, pressure, and force (Proposition 10.1 and Lemma 10.3) also allows a periodic construction. In this subsection we rescale their supports into the interior of a fundamental cube, then sum their integer translates. The translates have disjoint supports, so the momentum equation is preserved, including its nonlinear term. The pressure is periodic as well, as required in the erratum to the problem statement [13].
1888. u1888 The periodic domain and the interior of its fundamental cube are
1889. u1889 The following corollary gives the periodic breakdown alternative (D) in [13]. Corollary 10.6. For every ν > 0, there is a smooth force on T3 × [0, ∞), compactly supported in time, for which the solution (U, Pper) of the periodic Navier–Stokes equations with viscosity ν and zero initial velocity is smooth on [0, 1) and satisfies
1890. u1890 Its velocity and pressure have support in a fixed compact subset of the interior of Q0 at every time before one. There is no global smooth periodic solution for the same datum and force. Proof. At a fixed viscosity ν, we take the whole-space fields and force already constructed Proof. and denote At a fixed them viscosity here by u, ν, p, we f. take Their the common whole-space spatial fields support and is force contained already in constructed Kν = √ν K, as in the viscosity rescaling (10.22). We choose λ > 1 so large that λ−1Kν ⋐Q0, and put t0 = 1 −λ−2. For t ≥t0, we set
1891. u1891 The first two formulas are used for t < 1, and the force formula is used for all t ≥t0. We extend all three by zero for t < t0. They are smooth across t = t0, since the original fields and force vanish on an interval after their initial time. Each term of the momentum equation is λ3 times its original value, so the viscosity stays equal to ν. The original time one is reached precisely when λ2(t −t0) = 1, namely at t = 1.
1892. u1892 The periodic fields are defined by
1893. u1893 Distinct translates have disjoint support, with a positive gap between them. Derivatives commute with the locally finite sums, and at every point at most one translate is nonzero. In particular,
1894. u1894 Thus the periodized velocity, pressure and force solve the equation exactly:
1895. u1895 ∂tU + (U · ∇)U −ν∆U + ∇Pper = Fper,
1896. u1896 The force is smooth, with time support contained in [t0, 1 + λ−2]. On the torus this implies every decay estimate ∥∂α
1897. u1897 t Fper(t)∥∞≤Cα,m,N(1 + t)−N. For the path xτ defined in (10.20), the rescaled growth path is
1898. u1898 It remains inside Q0 near (0, 1), where the zeroth translate is the only nonzero term. By (10.21) and (10.22),
1899. u1899 Finally, on each [0, T] with T < 1, two smooth periodic solutions with the same force and datum have difference w obeying
1900. u1900 Periodic integration removes both the transport and pressure terms. Gronwall gives equality of the solutions through T. A global smooth competitor would therefore coincide with U
1901. u1901 □ before time one and contradict its velocity blowup at t = 1.
1902. u1902 Appendix A. Matching radial moments and constructing the heat exterior
1903. u1903 In this appendix we construct the outer part of the leading profile (U, E, Π) in Theorem 4.6. Its pressure integral supplies the datum Π0 for the regular axis problem (Lemma A.5). Its moment conditions, including (A.8), ensure that the radially integrated leading residual stress T0 vanishes in the exterior after the heat replacement (Lemma A.8). The construction must satisfy both requirements while retaining the strict stress cone inequalities on the designated radial intervals (Propositions A.4, A.7 and A.10).
1904. u1904 We begin with finite-dimensional moment corrections (Lemmas A.1 and A.2) that will also be used to attach the axis profile (Proposition B.8) and modify the shear (Proposition C.2). We then build the outer profile from a prescribed inner reference profile (Section A.2). Its terminal power law Epow is finally replaced by an exact heat flow, with compensating moment corrections that preserve the pressure datum and exterior stress conditions (Lemmas A.6 and A.8 and Proposition A.7).
1905. u1905 A.1. Adjusting finitely many moments. Changes to a profile on a compact radial interval alter its cumulative moments (4.15). In this subsection we correct these changes by adding fixed smooth bumps with adjustable coefficients. The following results show that distinct power weights give independent moment changes (Lemma A.1) and that the resulting quadratic moment equations can be solved for small discrepancies (Lemma A.2). They will allow each radial modification to preserve the data prescribed farther out, by Lemma 4.4. Lemma A.1. Let α1, . . . , αm be distinct real numbers. For each j let βj be a nonnegative, nonzero smooth function supported in the interior of a compact interval Ij ⊂(0, ∞), with I1 < · · · < Im. The matrix
1906. u1906 is invertible. The assertion also holds for exponential weights in a logarithmic coordinate. If these data vary smoothly over a compact parameter set and the stated separations persist, the inverse and each fixed parameter derivative of it are bounded.
1907. u1907 Proof. A nonzero linear combination of m distinct powers has at most m −1 distinct positive zeros. Indeed, divide by the least power; if the quotient had m zeros, Rolle’s theorem would give m −1 zeros of its derivative, contrary to the induction hypothesis for the remaining m −1 powers. Consequently det[xαi j
1908. u1908 j ] is nonzero whenever 0 < x1 < · · · < xm, and has constant sign on that connected set. Multilinearity gives
1909. u1909 The integrand has one sign and is nonzero on a set of positive measure. The substitution x = ey proves the logarithmic version. Compactness and differentiation of B−1B = Id prove the last assertion. This assertion does not give a uniform inverse when two exponents
1910. u1910 □ approach one another.
1911. u1911 Some of the moment weights approach one another as λ tends to zero.
1912. u1912 To choose corrections small enough to preserve the cone, we need the resulting loss in the inverse bound. For identical translated bumps β(y −c1) and β(y −c2), put ∆= c2 −c1 > 0 and b(s) = ∫︁
1913. u1913 estβ(t) dt. Dividing row i by esic1 gives
1914. u1914 (A.1) For a fixed bump and fixed ∆> 0, one has ∥B−1∥= O(λ−1) when s1 −s2 = λ and the slopes remain in a fixed compact interval; it is bounded when their limiting separation is nonzero. The same O(λ−1) bound for moment matrices with weights 1, x−λ and bumps supported on ordered disjoint intervals follows by expanding x−λ = 1 −λ log x + O(λ2): the two bump averages of log x have strictly separated values.
1915. u1915 The rank calculation controls the linearized moment map. The actual moments are quadratic in the velocity profiles, so the correction coefficients must solve a finite system with a quadratic remainder. The following estimate gives a small solution and controls its parameter derivatives. No simultaneous smallness of all derivative orders is required. Lemma A.2. Let K = [−1, 1] and suppose a finite family of correction coefficients c ∈Rm changes the required moments, after taking specified linear combinations of the moments and dividing each resulting component by a specified nonzero factor, by
1916. u1916 (A.2) Here B and the bilinear map Q are smooth, and B is invertible on K, and the prescribed moment discrepancy is d ∈C∞(K; Rm). Write β0 = supK ∥B−1∥, κ0 = supK ∥Q∥, and d0 = ∥d∥C0(K). If
1917. u1917 there is a unique solution of Fη(c) = d(η) in the ball ∥c∥C0 ≤2β0d0. It is smooth in η and is selected by iteration from zero of c ↦→B(η)−1(d(η) −Qη(c, c)). If Q = 0, there is no smallness requirement.
1918. u1918 For any preselected finite k, let βk be the operator norm of multiplication by B−1 on Ck(K), and let κk be the corresponding bilinear norm of Q. If also 8β2
1919. u1919 (A.3) Proof. For r = 2β0d0 the map c ↦→B−1(d −Q(c, c)) preserves the closed C0 ball of radius r. Its Lipschitz constant there is at most 2β0κ0r ≤1/2. The same inequality makes B + DcQ(c, c) invertible, so the pointwise implicit function theorem gives smoothness, including one-sided endpoint derivatives. Applying the identical contraction argument in the Banach algebra
1920. u1920 Ck(K) gives (A.3). The iterations are the same as in C0, so their limits agree. All other fixed derivatives are finite by implicit differentiation, with constants depending on the corresponding data. An additional parameter in a fixed compact set, such as the pulse
1921. u1921 □ amplitude below, is treated in exactly the same way.
1922. u1922 The correction estimates depend on the radial support scale. For example, if δV = v∗∑cj(η)βj(X/ρ), then
1923. u1923 For bumps of fixed width in log X, the analogous estimate uses (X∂X)r. Thus small coefficients preserve any fixed collection of strict inequalities involving finitely many field derivatives on that patch. This estimate controls the moment-correction bumps. The rapid modulation preceding a correction has separate radial-derivative estimates.
1924. u1924 For the five cumulative integrals of (4.15), the linearized system separates into two axial and three azimuthal corrections on a power-law interval. We now identify their weights so that the preceding rank and small-solution results can be applied.
1925. u1925 Corollary A.3. Use the five radial moment functions (M, I, J, S, Cp) of (4.15), where Cp is the pressure increment. Under X = ρx their normalizing factors are
1926. u1926 (A.4) On a fixed correction interval suppose U0 = uc(η) and E0 = e∗f∗(η)xα, with e∗> 0 and smooth f∗> 0. Two additive U bumps and three additive E bumps give an invertible Jacobian of the normalized five-moment map provided α /∈{−1/2, 1/2, 3/2}. The moment changes have the exact quadratic form (A.2). At U0 = 0, the three corrections to E can prescribe changes in (I, S, Cp) while preserving M, J.
1927. u1927 Proof. In the normalized rows, replace J by J −ucI and S by S −2ucM, with uc fixed at
1928. u1928 2xE0 δU dx, while that of S −2ucM is its base value.
1929. u1929 −∫︁
1930. u1930 E0 δE dx. Along with the other three rows, the resulting matrix has two blocks whose weights, up to nonzero normalizing factors, are
1931. u1931 The factors e∗, f∗and
1932. u1932 2 are retained in the normalization of the moment components; their inverses and fixed parameter derivatives are bounded. The hypotheses on α say precisely that the powers in each block are distinct. Lemma A.1 therefore applies. The original functionals have degree at most two in U, E, so all remaining terms are quadratic. In particular, the row
1933. u1933 □ operations do not assume that the corrected U remains constant.
1934. u1934 For the axis moment correction α = 1/10, the two blocks have powers (0, 3/5) and (1/2, 1/10, −9/10). On the intermediate power-law interval, α = −1/2 −λ gives (0, −λ) and (1/2, −1/2 −λ, −3/2 −λ). The first of the latter blocks can introduce an inverse bound of λ−1; λ is fixed before the later radial frequency is chosen. These normalized inverses, together with the exact restoration of matching data Lemma 4.4, are the tools used for both corrections.
1935. u1935 A.2. Constructing and matching the outer radial profile. We now specify a profile connecting an inner reference power law to a purely azimuthal exterior power law Epow. We will prescribe the inner data in (A.7). The intermediate intervals allow us to set the required total moments (A.8) and retain the stress cone inequalities. Four subintervals will remain available for the later profile, heat, background, and mean corrections; we will list them after (A.9). We give the radial pieces first, then choose their free amplitudes and verify the stated properties (Sections A.3 and A.5).
1936. u1936 We fix once and for all the smooth step
1937. u1937 (A.5) We take the positive moment-correction bumps to be rescaled copies of σ′ on ordered disjoint subintervals of the designated patch. Each stage uses a local logarithmic radial coordinate y, with d/dy = DX = X∂X and origin at the start of the stage.
1938. u1938 The following profile depends on parameters to be fixed in the order
1939. u1939 (A.6) Here ≪has the smallness meaning specified in Definition 3.3: choose λ sufficiently small after fixing Md, Td, P∗, then choose h sufficiently small relative to both λ and e−Td. The large radius XR will be fixed afterward. On the reference inner interval, writing x = X/XR and y = log x, we prescribe
1940. u1940 U = 4η,
1941. u1941 (A.7) This ideal region is a temporary prescription for the outer radial profile; Section B replaces its inner part by a regular axis. We seek an exterior power Epow = c∞X−A, with c∞> 0 independent of η, and the moment identities
1942. u1942 (A.8) Their role is to remove the two integration constants that would otherwise remain in the exterior stress. We now define each stage of the profile.
1943. u1943 Recall that l = X∂X log H, so prescribing l means integrating (log E)′ = l −1/2. Reducing the axial component to zero. Starting at x = 1, retain U = 4η and use l = (3/5)(1 −σ(y)) for one unit of logarithmic radial length. On the next interval set
1944. u1944 Here |k′| ≤ed/(1 + y), with ed = 4∥σ′∥∞/Md. The function k is zero for the final eleven units of this stage. We write P1 ≍P∗for the value of E/ f at the start of the axial transition, so E = P1 f e−y/2 throughout that interval.
1945. u1945 The intermediate power-law interval. With U = 0, set l = −λσ(y) on a unit interval and then retain l = −λ for
1946. u1946 (A.9) In the local logarithmic radial coordinate on this interval, reserve (Tw −25, Tw −20), (Tw − 20, Tw −15), (Tw −14, Tw −9), and (Tw −8, Tw −3). They will support, respectively, the correction after cone realization, heat compensation, moment conditions for higher-order background corrections, and moment conditions for the phase-averaged corrections. We choose λ small enough that they lie in this interval.
1947. u1947 The axial pulse. An axial pulse supplies the adjustable contribution to the total S-moment; two smaller corrections at its end set M = J = 0. Let Xp be the radius at the end of the intermediate power-law interval and write E(Xp, η) = eb f. For 0 ≤y ≤13/λ, keep l = −λ and set U = ERb. With ξb = λy, put
1948. u1948 Rb = Amp(η)R0(ξb) + c1β1(y) + c2β2(y).
1949. u1949 Extend φb by zero to ξb < 0. The bumps β1, β2 are identical translates of width .3, centered at 13/λ −3 and 13/λ −1. Their coefficients c1, c2 set M = J = 0 at the end. These coefficients are affine functions of Amp; the amplitude itself will be determined by the first equality in (A.8).
1950. u1950 Profile interpolation and the angular moment correction. We next remove the parameter dependence of the azimuthal profile and adjust its angular moment while preserving the pressure integral. Set U = 0 from now on. Write Xend for the radius at pulse end and E(Xend, η) = eend f. Prescribe log E = log eend −(1/2+ λ)y −ϑf (y)J0 −(1−ϑf (y)) log 2,
1951. u1951 ϑf (y) = 1−σ(y/Tf ). (A.10) A fixed sufficiently large Tf gives −λ −.1 ≤l ≤−λ. The factor depending on η decreases to the constant 1/2, so the amplitude cannot increase. Then keep the resulting η-independent exponential for 30 log(1/λ), with two relative E bumps of width .3 centered three and one units before its end. They impose
1952. u1952 (A.11) Both endpoint values of E are unchanged. This part of the construction uses E only and is independent of Amp.
1953. u1953 Transition to the exterior power law. Let Xrel be the radius at exterior transition start and erel the unaltered uniform value of E there. Vary l smoothly from −λ to −1 on a unit interval, retain l = −1 for 4 log(1/h), and vary it from −1 to −h on a unit interval. Each transition uses σ. On the terminal radial interval 0 ≤y ≤3, fix
1954. u1954 ψo(y) = 1 −σ((y −1)/2),
1955. u1955 , (A.12) where co > 0 is fixed small enough that 0 ≤f ′
1956. u1956 o/ ′ fo < h/4. Extend fo constantly to the left and by 1 to the right of this interval. Define
1957. u1957 (A.13) After the transition to −h, retain l = −h until the solution of Q′ + (1 + l)Q = −l −h, initially Q = (λ −h)/(1 −λ) at exterior transition start, has decreased to Qp. Apply the terminal transition over three units of logarithmic radial length, and retain l = −h at all larger radii. On and beyond that interval E = Epow fo.
1958. u1958 Proposition A.4. There are finite choices in the order (A.6) for which the profile specified in Section A.2 defines smooth U, E for X > 0, η ∈[−1, 1], with E > 0, and satisfies (A.8). All stage boundaries divided by XR are independent of XR and η.
1959. u1959 After the compact axial pulse, U = M = J = 0; after the terminal interval, E = Epow. There are four disjoint unaltered patches, listed after (A.9), on which U = 0 and E is a positive multiple of f X−1/2−λ. For each fixed
1960. u1960 x−∈(0, 1], a sufficiently large XR, allowed to depend on x−, makes the reference profile satisfy the strict relaxed cone condition from x = x−through the first half-unit of its terminal tail. It satisfies the admissible stress cone throughout the intermediate power-law interval, axial pulse, profile interpolation, and exterior transition, and as soon as l < 0 on the transition into that constant-slope interval. These assertions concern a finite logarithmic interval ending at tail coordinate 1/2; the heat construction below treats the remaining endpoint collar.
1961. u1961 A.3. Closing the moment integrals. The radial pieces have been specified in Section A.2, with amplitudes still to be chosen in the axial pulse and moment-correction bumps. In this subsection we choose them to satisfy the moment identities in Proposition A.4. The estimates also control the sizes and parameter derivatives of these corrections, which will be needed when we verify the stress cone inequalities in Section A.5.
1962. u1962 We write Cpre for constants depending on the fixed choices Md, Td, P∗and the fixed steps. They are uniform for small λ and then sufficiently small h. A prescribed finite number of parameter derivatives may change these constants. No such constant depends on the later radius XR or the normalization C.
1963. u1963 At the start of the pulse, direct integration of the axial transition and intermediate powerlaw interval gives
1964. u1964 (A.14)
1965. u1965 Indeed M is constant after the axial transition, whereas XE grows at rate 1/2 −λ in the intermediate power-law interval. The denominator XHE for the J radial moment grows at rate 1/2 −2λ, giving the same type of small bound for its moment discrepancy. Also XE2 is constant during the axial transition and changes by the factor e−2λy in the intermediate power-law interval, which remains bounded above and below on 0 ≤y ≤Tw. Integrating over its logarithmic length proves the last bound, including the fixed reference profile on the inner interval. The shape f and its reciprocal have bounded fixed derivatives on [−1, 1], so differentiating these integrals proves the stated parameter bounds.
1966. u1966 For the two pulse corrections, divide by the respective normalizing factors Xpeb f and XpH(Xp)eb f from M, J. The weights in dy are es1y, es2y with s1 = 1/2 −λ, s2 = 1/2 −2λ. Normalize each at the first bump center. The main pulse ends at 11/λ, at least 2/λ −3 before that center, and both slopes exceed .4. The normalized main-pulse moment discrepancy is therefore bounded by C(1 + |Amp|)e−c/λ; the contribution accumulated before the pulse acquires the factor e−si(13/λ−3), times the bounds just proved. Formula (A.1) gives the bound on the inverse matrix O(λ−1). Absorbing that loss into the exponential yields
1967. u1967 (A.15) Here and in the amplitude argument the estimate means that the affine coefficients of 1 and Amp have the stated parameter bounds. Every fixed y derivative satisfies the same estimate; each fixed ξb derivative introduces only a power of λ−1 and is still exponentially small.
1968. u1968 The axial corrections have now fixed M and J for each trial pulse amplitude. We next set the angular moment required at the start of the exterior transition. Put rI = I/(XH); it obeys Before the exterior transition, 1 + l is bounded below, and the explicit shapes give a C1 η
1969. u1969 η bound Cpre at the end of profile interpolation. On the interval with uniform profile, rI −
1970. u1970 (1 −λ)−1 decays at rate 1 −λ. The moment discrepancy at the first correction center is thus OC1η(Cpreλ28). For a relative change E ↦→E(1 + c1β1 + c2β2), the I row is linear with slope 1 −λ and the linearized pressure row has slope −1 −2λ in y. Divide these rows by XH and E2 at the first center, respectively. Their inverse stays bounded by (A.1); the only nonlinear term is the quadratic pressure increment. Lemma A.2 gives coefficients OC1η(Cpreλ28) and the exact conditions (A.11). The derivatives of these corrections with respect to y are much smaller than λ, so E > 0 and l ≤−λ/2 persist.
1971. u1971 At the start of the exterior transition, U = M = J = 0 and I = XH/(1 −λ) is independent of η. The integral formula for Qs (4.16) therefore gives Qs = (λ −h)/(1 −λ) exactly. The subsequent source −l −h is nonnegative until the terminal interval. The first transition preserves a lower bound cλ; during the constant-slope interval with l = −1, Q′ s
1972. u1972 s = 1 −h, so its end value is comparable to log(1/h). The next transition changes that size by bounded factors. On the other hand, Qp ≍ρo ≍h, by (A.13) on a fixed interval. The intervening constant-slope interval with l = −h therefore has a positive finite logarithmic radial length. On the terminal interval the exact solution is
1973. u1973 (A.16) It vanishes at the endpoint and remains zero afterward. Throughout the exterior transition I is independent of η, and then Qs = −1 + (1 −h)I/(XH). Hence
1974. u1974 The subtracted moment is integrable at zero because Hpow ∝X−h and h < 1, and its integrand is identically zero sufficiently far out. The angular correction, its endpoint, and all exterior-transition lengths depend only on E and are independent of the axial amplitude.
1975. u1975 The angular moment is now fixed independently of the pulse amplitude. The remaining scalar condition is S(∞) = 0. To solve it, we first bound the contribution from the transition to the exterior profile. Since (XE2)′ = 2lXE2, its constant-slope interval with l = −1 gives the exact factors
1976. u1976 (A.17) Here “hold start” and “hold end” denote the endpoints of the constant-slope interval with l = −1. The integration label “release” below denotes the entire transition to the exterior power law. The first transition and the constant-slope interval contribute at most CXrele2
1977. u1977 rel in
1978. u1978 XE2 dy. From the end of the constant-slope interval onward l ≤−3h/4, so the remaining rel ∫︁ integral is bounded by CXrele2
1979. u1979 (A.18) with an h-uniform constant. The h8 estimate controls this integral; the separate h6 estimate will control the ratio w = Ns/(EQs) near the outer endpoint. Profile interpolation and the subsequent constant-slope interval contribute at most C(1 + log(1/λ))Xende2
1980. u1980 end. First η derivatives have the same bounds, since the exterior transition is uniform and the earlier log shapes have bounded derivatives.
1981. u1981 On the pulse XE2 = Xpe2 b
1982. u1982 b f 2e−2λy. Changing to ξb = λy, and using (A.14), (A.15), and (A.18), gives the exact form
1983. u1983 (A.19) where on [.9, 1.2] × [−1, 1],
1984. u1984 All terms contributed by the outer profile E in this estimate were chosen independently of Amp; the exponentially small affine end bumps account for its remaining dependence. Since R0 ≤ξb, Kb ≤∫︁∞ 0
1985. u1985 be−2ξb dξb = 1/4. On .02 ≤ξb ≤9 one has R0 ≥ξb −.02, and integration gives Kb > .20. The principal expression in (A.19) is below −.047 at .9 and above .038 at 1.2, and its derivative in the amplitude is at least .36 on the bracket. For small λ the full expression has a unique root there. Its linearization is nonsingular, so the root is smooth and
1986. u1986 (A.20) Both equalities of (A.8) now hold. The remaining task is to verify that these moment corrections preserve the stress cone. Only finitely many orders of η-derivatives must satisfy prescribed smallness bounds below. If bounds on additional finite derivative orders are prescribed, the differentiated integrals and Lemma A.2 provide them before the parameters are fixed. Derivatives of every other fixed order have finite bounds after the parameters are chosen.
1987. u1987 A.4. Fixing the pressure datum for the axis profile. The azimuthal profile E now determines the axis pressure Π0 by the normalization that pressure vanish at radial infinity, as in (4.25). The analytic axis construction of Proposition B.2 requires this datum to extend holomorphically near [−1, 1], with the sign and size bounds stated in Lemma A.5. In this subsection we verify these properties before attaching the axis, and show that the datum is independent of the later choice of the radial scale XR. For the pressure integrals in this subsection, we use the global logarithmic coordinate y = log(X/XR) on the complete reference profile.
1988. u1988 Lemma A.5. Let Eid,sched be the reference inner profile (A.7) followed by the outer profile E specified in Section A.2, with the azimuthal corrections that preserve the total pressure increment omitted and all other transition intervals unchanged. The function
1989. u1989 (A.21) is independent of XR and analytic on a complex neighborhood of [−1, 1]. It is even, Π′
1990. u1990 0(η) has the sign of η for η̸ = 0, and
1991. u1991 (A.22) For the complete reference radial profile, integration of ΠX = E2/(2X) from this datum gives
1992. u1992 (A.23) Proof. Without the angular bumps, every part of E has the form c(y) f (η)ϑ(y), where c and the transition lengths are independent of η and 0 ≤ϑ ≤1. The reference inner profile, axial transition, intermediate power-law interval, and pulse have ϑ = 1; profile interpolation decreases it to zero; exterior transition and tail have ϑ = 0. Choose one simply connected
1993. u1993 complex neighborhood of the real interval avoiding the poles and zeros of f, and fix an analytic logarithm there. Then f ϑ = exp(ϑ log f ) is holomorphic with a bound uniform in 0 ≤ϑ ≤1. The squared integrand is bounded by Cey/5 as y →−∞and by Ce−(1+2h)y as y →+∞, and the intermediate interval is finite. Uniform integration of holomorphic functions proves analyticity.
1994. u1994 For real η, differentiating each factor gives ∂η f 2ϑ = −2ϑJ′ 0
1995. u1995 0 f 2ϑ. Thus every contribution to Π′
1996. u1996 0 has the sign of η, and the reference inner profile makes it strict. Its contribution is exactly −(5/2)P2
1997. u1997 ∗f 2, proving (A.22). The construction in the logarithmic radial coordinate contains no XR, proving its independence. Finally, (A.11) says that reinstating the angular bumps changes the total pressure increment by zero. Its total increment is therefore −Π0, which
1998. u1998 □ proves (A.23).
1999. u1999 We can now use this pressure datum in the axis construction. Subsequent profile changes must preserve the total pressure increment to keep it fixed. Matching all five radial integrals also preserves the radial velocity and the functions Qs, Ns beyond the correction interval. If an edit has zero total integral of (E2
2000. u2000 old)/(2X), the forward pressure is unchanged before and after its support, though it can change inside that support. Accordingly, a correction supported strictly to the right of X that preserves the total pressure increment may be omitted in the backward integral (A.23). Equality of all five radial moment functions at a radius where the fields rejoin has the stronger consequence of Lemma 4.4: it restores the radial velocity and the functions Qs, Ns on the entire later interval, including their parameter derivatives. The axis construction will use the already fixed analytic function Π0 and integrate pressure forward. Later smooth, possibly nonanalytic, edits of E cannot change that datum; their total pressure increment is either zero or is restored by the correction matching all five radial moments. The heat replacement preserves the required convergent moment differences while changing the exterior profile.
2001. u2001 A.5. Uniform admissible stress cone inequalities on the outer interval. The moment identities and the axis pressure datum are now fixed. It remains to verify the stress cone inequalities on the outer intervals specified in Proposition A.4. We estimate the ratios entering the sufficient cone test of Lemma 4.5 before choosing the large radial scale XR; this final choice then gives uniform strict inequalities on the required finite interval.
2002. u2002 Write w = Ns/(EQs) wherever Qs > 0. The sufficient test of Lemma 4.5 is
2003. u2003 (A.24) We establish uniform positive lower bounds for a −bsw and 2 −2bsw −b2
2004. u2004 s/a −(a −2)w2, with a, bs, w in fixed compact ranges and Qs > 0. Increasing XR afterward then makes ps,1 = XQs/L large enough on the designated finite logarithmic interval. Where vs ≤2, it is enough to have Pc = ps,1(1 −bsw/a) > 2.
2005. u2005 The reference inner profile and axial transition. On (A.7), direct substitution into the equation for Qs gives
2006. u2006 Qs = (3/5)(4L −1) −h(1 8/5 −8η2) + (D + 4d)ηJ′ 0
2007. u2007 (A.25) There bs = 0 and a = 4/5. In the equation for Ns, split the source into its geometric and pressure parts. At the start of the first transition their values are respectively O(|η|) and O(P2
2008. u2008 ∗|η|); one parameter derivative removes the factor |η|. These estimates follow by integrating the constant ideal geometric source and the bounded pressure source against the
2009. u2009 kernel in the integral representation of Ns. Evenness of the pressure and the ideal shapes supplies the vanishing at η = 0.
2010. u2010 During the first transition U = 4η, W = 1 −4L < 0 and l ≥0, so the angular source is at least −h. The finite transition therefore preserves a positive lower bound for Qs, and bs = 0, a ≤2. In the axial transition, the angular source Sq is
2011. u2011 (A.26) The second estimate follows from variation of constants, since ηJ′ 0
2012. u2012 0 ≍η2, D + dk ≥D, the value of Qs at the inner endpoint is positive, and h ≪e−Td.
2013. u2013 In this interval the geometric axial source is O(|η|).
2014. u2014 By (A.23), its pressure source is O(E2|η|): the unmodified profile at larger radii decays at least as a constant times e−∆y/2 and satisfies |∂η log E| ≤|J′
2015. u2015 0|. The subsequent azimuthal-profile correction may be omitted because it preserves the total pressure increment. Solving N′ s
2016. u2016 s + Ns = Sn and using E2 = P2 1
2017. u2017 Now bs = 2k′η/E, so (A.26) gives |bsw| ≤Ced. Also b2 s
2018. u2018 s ≪1 because P∗> eTd. As a = 2, choosing Md first to make ed small and P∗afterward proves both strict inequalities in (A.24). The intermediate power-law interval. During the axial transition, write AX(U) = ¯kη. Then ¯k′ = k −¯k, and the final eleven units with k = 0 give L¯k < 1/2. Hence W = 1 −L¯k ≥1/2 on the next transition. Put cη = DηJ′ 0
2019. u2019 0 ≍η2. With U = 0 the angular source on the transition is (−l)W −h + cη. A positive lower bound for Qs, independent √ of λ, persists. The bound on w depends only on the parameters fixed before λ. Thus
2020. u2020 In the local logarithmic radial coordinate on the constant-slope interval ¯k(y) = ¯k(0)e−y, so the angular source is exactly λ −h + cη −λL¯k(0)e−y. Integrating the scalar equation for Qs gives
2021. u2021 (A.27)
2022. u2022 For the last line, U = 0 eliminates the geometric source and the pressure source remains O(E2|η|); convolution with e−y proves the bound. One η derivative obeys the same estimate without |η|. Using |η|/(η2 + e−(1−λ)y) ≤1
2023. u2023 Since a = 2 + 2λ and bs = 0, this proves the admissible stress cone there. On the preceding transition the same argument gives the admissible stress cone condition whenever l < 0, and the relaxed condition at its initial endpoint.
2024. u2024 The pulse. Here both shear components vary, so the cone test requires a more precise relation between the averaged axial velocity and the pulse profile. Set m = AX(U)/E and β = 1/2 −λ. The exact equation for m is m′ + βm = Rb. Extending the main pulse by zero on the left and expanding its exponential convolution twice gives
2025. u2025 (A.28)
2026. u2026 Indeed Taylor’s formula bounds the convolution remainder by Cλ2∥∂2
2027. u2027 0 e−βvv2 dv. The start is flat, and the end bumps have exponentially small fixed ξb derivatives. The same expansion holds after one η derivative. The initial value is bounded by (A.14).
2028. u2028 In the angular equation, W = 1 −2DηEm −d(Em)η and (A.20) show that Sq = λ −h + cη + O(CpreE). Equations (A.14) and (A.27) therefore imply
2029. u2029 (A.29) The pressure normalized to vanish at radial infinity and its first η derivative are O(E2). Furthermore, integrating XE2(R2 b
2030. u2030 b −1/2) and the moment discrepancy accumulated before the pulse gives S/(XE2) = OC1η(Cpree26/λ) throughout the pulse. Formula (4.16) expresses Qs, Ns in terms of the radial moment functions and their η-derivatives, without radial derivatives of E, U. Substitution yields
2031. u2031 E = −Rb + (D + cη)m −Dηmη + O(Cpreλ27).
2032. u2032 (A.30) Here all constants are independent of C. Because |η|/(λ + cη) ≤C/
2033. u2033 λ, (A.20) makes the contribution of ηmη/Qs an o(1) term. Combining Equations (A.28) to (A.30) gives
2034. u2034 The estimate for w now reduces the cone test to bounds on the pulse shape and its derivative. For the coefficient of Rb use D + cη −β = λ −h + cη; the bound on Cd follows by maximizing the quotient at cη = 0, with h/λ →0. Also a = 2 + 2λ and bs = −Rb + O(λ). Apart from exponentially small end bumps, Rb ≥0 and ∂ξbRb ≤1.2. Thus
2035. u2035 Only the upper bound on the pulse derivative is used in these quadratic cross terms; its negative derivative near the cutoff decreases the left-hand sides. All derivatives remain bounded by fixed constants. The first margin gives a −bsw > 1, and vs ≥a > 2, proving the admissible stress cone on the whole pulse.
2036. u2036 Profile interpolation and exterior transition. After the pulse, U = M = J = 0 and hence W = 1. During the profile interpolation the angular source is −l −h + ϑf cη, so Qs ≥cλ; the small C1 η
2037. u2037 η angular correction preserves this bound. Since S(∞) = 0,
2038. u2038 (A.31) Until the exterior transition, l ≤−λ/2, and the remaining exterior transition integral has the bound (A.18). Thus (|S| + |Sη|)/X ≤CE2/λ, including the angular edit, while |Π| + |Πη| ≤CE2. Formula (4.16) now gives |w| ≤CE/λ2 ≪1, since pulse end made E exponentially small in 1/λ. Here bs = 0 and 2 < a ≤2 + 2λ + .2 + o(1), so both cone margins persist.
2039. u2039 Once the exterior transition has started, its fields and remaining radial moment functions are uniform in η except for the explicit coefficients in the identity for Qs, Ns. Combining
2040. u2040 (A.23) and (A.31) gives
2041. u2041 Throughout the exterior transition and at larger radii, l ≤−3h/4. Since (log E)′ = l −1/2, for v ≥y we have
2042. u2042 Consequently
2043. u2043 With A = 1/2 + h and |η| ≤1, these estimates give the stated bound for Ns uniformly in h. Before the steep-decay interval Qs ≥cλ. After it, and through the first half-unit of the terminal interval, Qs ≥ch: in the last constant-slope interval it decreases only to Qp ≍h, and (A.16) gives the same lower bound for 0 ≤y ≤1/2. By (A.17), E ≤Cerelh6 after the steep-decay interval, so E/Qs ≪1 even when h is much smaller than erel. Consequently |w| ≪1, bs = 0, and 2 < a ≤4 on the exterior transition interval under consideration. This proves the strict true cone there.
2044. u2044 Every lower bound for Qs used here is on a fixed compact logarithmic interval after all choices in (A.6). The fields and the functions Qs, Ns on such an interval are independent of XR under the normalizations (A.4). Hence one final increase of XR makes the large-ps,1 test uniform. This finishes the proof of Proposition A.4.
2045. u2045 A.6. Replacing the exterior power by a heat flow. The reference profile (U, E) of Proposition A.4 has the required moments and satisfies U = 0, E = Epow = c∞X−A beyond the terminal interval. The exterior power law Epow still has a viscous residual. In this subsection we will replace it beyond the annulus by the exact radial swirl heat flow K of Lemma A.6. This will make the exterior momentum residual vanish and give the field smooth limits at the singular time away from the axis. This replacement changes three moments; a correction in the second reserved interval I2 restores them (Proposition A.7), including the pressure datum Π0 already prescribed for the axis in (A.21).
2046. u2046 For a profile regular at the axis, write R(0) θ
2047. u2047 for its leading azimuthal and axial momentum residuals. Thus R(0) j
2048. u2048 in the notation of (4.12); these include radial viscosity and omit the higher-order axial viscosity. The stress components are Tθ(r) = −r−2 ∫︁r 0
2049. u2049 z (s) ds, as in Proposition 4.2. In the following subsection we will use the corrected moment conditions to rewrite these integrals as integrals from r to infinity (Lemma A.8) and determine the sign and limiting direction of the stress at the outer endpoint (Proposition A.10).
2050. u2050 Throughout this subsection 0 < h < 1/2, A = 1/2 + h, and D = 1/2 −h. We retain d = 1 −η2, τ = 1 −t, and L = 1 −2hη2 from (4.1). All parameters in the construction of the outer radial profile are fixed in the order (A.6) before XR is increased. In the local logarithmic radial coordinate for the tail of (A.12), write y = log(X/Xtail), Xb = e3Xtail, and Epow = c∞X−A, where c∞> 0 is independent of η. Thus the reference tail is Epow fo(y).
2051. u2051 Set aK = 1 + h, and define the heat factor by
2052. u2052 (A.32) Here Epow(X)H(2d/X) Γ(aK) = ∫︁∞ 0
2053. u2053 0 e−vvaK−1 dv is the gamma integral, so H(0) = 1. Write Eheat(X, η) = Epow(X)H(2d/X) for the heat profile.
2054. u2054 Lemma A.6. The function H is positive and smooth up to Z = 0 from the right. Every fixed derivative is bounded on bounded nonnegative intervals. The field
2055. u2055 (A.33) is independent of z, satisfies ∂tK = (∂rr + r−1∂r −r−2)K, and has Kr < 0.
2056. u2056 Its profile is Epow(X)H(2d/X), smooth with all η-derivatives extending continuously to η = ±1 from the interior.
2057. u2057 Proof. For the rising factorial (b)m = b(b + 1) · · · (b + m −1), with (b)0 = 1, dominated differentiation gives
2058. u2058 (A.34)
2059. u2059 (A.35) The integrable majorant obtained by deleting the last factor works also at Z = 0. In particular, for each fixed derivative order the usual finite Taylor formula is valid there; no convergent Taylor series is needed. On a fixed small nonnegative interval it gives
2060. u2060 (A.36) Here the last constant can be uniform for 0 < h ≤h0 < 1/2. Indeed (h)m(1 + h)m ≤Cmh for every fixed m ≥1. Integration of h∂v[e−vvaK(1 + Zv)−aK] gives
2061. u2061 Z2H′′ + (1 + 2aKZ)H′ + aK(aK −1)H = 0;
2062. u2062 (A.37) both boundary terms vanish. Direct differentiation of (A.33), with Z = 2τ/s, gives
2063. u2063 Z2H′′ + (2A + 1)ZH′ + (A2 −1/4)H Since 2A + 1 = 2aK and A2 −1/4 = aK(aK −1), (A.37) proves the heat equation with its swirl term. Moreover, −ZH′/H is the average of hZv/(1 + Zv) against the positive probability density proportional to the integrand in (A.32). Consequently
2064. u2064 The composition with Z = 2(1 −η2)/X uses only Z ≥0. Each nonzero η-derivative is a finite sum of terms H(k)(2d/X)(2/X)k times polynomials in d′ = −2η and d′′ = −2. Thus no division by d occurs. This proves the claimed closed-interval smoothness. The integral formula is used only on its stated nonnegative domain. In particular the large-X expansion, with its remainder after every fixed application of ((X∂X)j∂m η
2065. u2065 (A.38)
2066. u2066 □
2067. u2067 The heat profile approaches the original power law at large X, with the derivative bounds in (A.38). Denote by Ecl the reference profile constructed in Proposition A.4. We use this small change to replace the terminal profile and restore the three affected moments by earlier corrections. The other two moments are unchanged because both modified regions have U = 0.
2068. u2068 Proposition A.7. Let XK = e.2Xtail, and choose a smooth step 0 ≤χK(y) ≤1 equal to zero for y ≤.2 and to one for y ≥.5. The terminal replacement
2069. u2069 (A.39) can be compensated by three additive E-bumps in the second reserved patch in the intermediate powerlaw interval. The complete edit preserves the functions M, J, the totals S(∞), Cp(∞) from (4.15), and the XR, renormalized it preserves E angular > 0 and moment the strict ∫︁∞ 0
2070. u2070 0 (H −Hpow) dX, pointwise on [−1, 1]. For sufficiently large XR, it preserves E > 0 and the strict admissible stress cone from that patch through y = .5. The original axis pressure datum and the profiles before the compensation patch are unchanged. Proof. We estimate the three changes caused by the heat factor, cancel them with three earlier angular bumps, and use the normalized radial moment formulas to preserve the admissible stress cone.
2071. u2071 Put x = X/XK and let eK be the reference amplitude at XK. The reference tail is comparable to eKx−A. Equations (A.34) to (A.36) and the chain rule show that, for all fixed integers j, m ≥0,
2072. u2072 The constants may depend on the already fixed schedule. In particular, all derivatives in this estimate are bounded at d = 0. Writing ∆for the change caused just by (A.39), the three discrepancies satisfy⃓⃓⃓⃓
2073. u2073 (A.40)
2074. u2074 (A.41)⃓⃓⃓⃓
2075. u2075 (A.42) Here H =
2076. u2076 2XEpow; the pure power is unchanged. The final integral is 1/h. It is finite because h > 0 has already been fixed before the large-radius choice.
2077. u2077 The three moment discrepancies are small after normalization by their natural radial scales. We cancel them on the second reserved patch, where the power-law profile gives three independent moment weights. At the start X∗of this patch, put x∗= X/X∗. Its
2078. u2078 , U = 0, where f = (1 + η2)−1 ≥1/2. Choose three unmodified profile is E = e∗f (η)x−1/2−λ ∗ nonnegative bumps βi with ordered disjoint supports inside this patch and add
2079. u2079 ∑ Normalize the pressure, S, and angular rows respectively by
2080. u2080 (A.43) Their derivatives at c = 0 are the integrals against the weights
2081. u2081 The exponents are distinct, so Lemma A.1 makes this matrix invertible. Its inverse is uniform in η ∈[−1, 1]. The first two rows have quadratic remainders and the last is linear. Since X∗/XK and e∗/eK are fixed, dividing Equations (A.40) to (A.42) by (A.43) gives Om(X−1 K
2082. u2082 K ) after every fixed number of η-derivatives. The local existence result of Lemma A.2 gives a smooth solution with ∥∂m η
2083. u2083 K . Choose XR sufficiently large that the moment equations have a unique solution near zero correction coefficients, with invertible Jacobian. Implicit differentiation then gives a finite bound at each fixed higher order of η-derivatives. Only the finitely many derivative bounds used to preserve the cone must meet prescribed smallness thresholds.
2084. u2084 The three altered moments have now been restored, and M, J remain unchanged because both edited regions have U = 0. We must still check that the local changes in the profiles and cumulative integrals preserve the cone between the compensation patch and the terminal interval.
2085. u2085 The normalized radial moment formulas (4.16), in X/XR, transfer the small profile and moment changes to Qs, Ns on the fixed compact log interval from this patch to y = .5. More explicitly, I = X3/2 R
2086. u2086 M, ˆ︁ in and H. S The = XR comparison ˆ︁S; insertion costs in one the identities for Qs, Ns cancels every XR-factor, including those in H. The comparison costs one further η-derivative and gives O(X−1 K
2087. u2087 K ) changes in the required η-derivatives of Qs, Ns. The positive lower bounds for E, Qs, the slope bounds, and the admissible stress cone margins of Proposition A.4 consequently persist for large XR. The intermediate values of I and the transition value of Qs may vary while these admissible stress cone inequalities persist.
2088. u2088 Finally, zero total pressure discrepancy means that integration of ΠX = E2/(2X) forward from the unchanged Π0 still gives the pressure normalized to vanish at radial infinity. Before the compensation patch this forward solution is unchanged. In particular the terminal heat
2089. u2089 □ factor, which need only be smooth in η, does not alter the analytic axis datum. A.7. Recovering the stress from the exterior. In this subsection we will use the corrected moments to prove that the stress vanishes beyond the annulus (Lemma A.8) and to determine its direction as the outer edge is approached (Proposition A.10). We will first show that the moment identities make the total weighted integrals of the leading residual zero. This will allow us to compute the stress by integration from the current radius to infinity. This representation depends only on the exterior profile once the regular axis and matching moment conditions have been supplied by Proposition B.2 and Corollary B.10. Lemma A.8. Suppose the leading axisymmetric field is smooth at the axis, has U = 0 and E = Epow fo[1 + χK(H(2d/X) −1)] for y ≥0, and satisfies the exact moment conditions
2090. u2090 M(∞) = J(∞) = S(∞) = 0, The leading tangential stresses vanish for X ≥Xb. On y ≥1/2, where χK = 1, they are given by the weighted integrals of the leading residual from r to infinity, with positive integration sign, in (A.46). Proof. The radial moment functions in the tail admit the following representations by integrals from X to infinity. Since U = M = J = 0 there, W = 1, and
2091. u2091 The first identity uses h < 1 to integrate Hpow from zero. The others follow directly from the exact moments and the integral identities for Qs, Ns (4.16).
2092. u2092 We verify that no boundary term obstructs the backward stress formula. Let Kpow = c∞s−A. At large radius, uniformly for τ in bounded nonnegative intervals,
2093. u2093 (A.44) These follow from (A.36) and its fixed derivative versions. In particular, at any sufficiently large fixed physical radius R∗,
2094. u2094 Thus the subtracted angular moment is a convergent integral. Its time derivative is justified by the same integrable majorant, uniformly on compact time intervals. We obtain
2095. u2095 (A.45) At the axis the subtracted power has weight r1−2h, also integrable. Its time derivative is zero. The angular transport integral is the sum of the radial boundary [r2uruθ]∞ 0
2096. u2096 0 and the axial regularity derivative and by of U ∫︁
2097. u2097 r2uzuθ dr = q3/2−2AJ(∞) = 0. The radial boundary vanishes by axis regularity and by U = M = 0 on the exterior, which also gives V0 = 0. The radial viscosity
2098. u2098 0 = 0 by (A.44). Thus the leading angular residual R(0) θ integrates to [r2∂ruθ −ruθ]∞ 0
2099. u2099 has zero integral against r2dr.
2100. u2100 For uθ(r′)2 the axial dr′/r′ component, ∫︁
2101. u2101 The pressure identity p(r) = −∫︁∞ r
2102. u2102 Indeed the exact pressure calculation is
2103. u2103 The exterior decay p = O(r−2−4h) ensures convergence and r2p →0. The boundary terms from radial transport and viscosity in the axial equation = q−A−1/2T0. also vanish. Hence ∫︁
2104. u2104 dr = 0.
2105. u2105 Write Tθ, Tz for the physical stress components, so T = q−A−1/2T0. Their defining integrals from the axis can now be rewritten as
2106. u2106 (A.46) For X ≥Xb, the field is (ur, uθ, uz) = (0, K, 0) with z-independent pressure normalized to vanish at radial infinity. Lemma A.6 makes both leading residuals zero there. Thus these
2107. u2107 □ stress components vanish for X ≥Xb.
2108. u2108 The backward integrals contain the cutoff factor that vanishes to every order at the outer edge. The next lemma separates this factor from a smooth coefficient, so that the stress direction can be estimated even where the stress itself tends to zero.
2109. u2109 Lemma A.9. Let c > 0, let j be a fixed nonnegative integer, and let b(u, η) be smooth for 0 ≤u ≤δ0 and for η in a compact interval K. There exists a coefficient B such that, for 0 < δ ≤δ0,
2110. u2110 (A.47)
2111. u2111 The coefficient B is smooth on [0, δ0] × K, has B(0, η) = b(0, η)/c, and has bounded mixed derivatives of every fixed order there. The same assertion holds with any finite collection of smooth compact parameters in place of η.
2112. u2112 2δ3(1 + δ2v)−3/2, and c/u2 = c/δ2 + Proof. Substitute u = δ/(1 + δ2v)1/2. The Jacobian is −1 cv. This proves (A.47) with
2113. u2113 Every fixed mixed derivative of the final integrand is bounded by e−cv times a polynomial in v, uniformly on the closed parameter rectangle. Dominated differentiation proves smoothness and the derivative bounds. Setting δ = 0 under the integral gives the stated endpoint
2114. u2114 □ value.
2115. u2115 The stress vanishes at the outer edge, so its limiting direction depends on the relative rates at which its two components vanish. The backward formula and the flat integral lemma give these rates, with a positive leading coefficient in the angular component.
2116. u2116 Proposition A.10. Under the regular axis and exact moment hypotheses of Lemma A.8, the profile on the terminal interval satisfies the admissible stress cone on .5 ≤y < 3. Its stress direction extends smoothly to the outer endpoint, and 2 −(a −2)(Tz/Tθ)2, interpreted through this extension, has a uniform positive lower bound. More precisely, for δ = 3 −y sufficiently small,
2117. u2117 (A.48)
2118. u2118 (A.49)
2119. u2119 (A.50) The coefficients are smooth on a closed outer collar, and bθ has a positive lower bound there. For every fixed mixed profile derivative ∂I, there are finite constants CI, NI such that
2120. u2120 (A.51) These constants may depend on all fixed profile choices and on I, but not on the physical scale q. Proof. We first derive a positive angular stress and bound the ratio of axial to angular stress. We then factor the flat endpoint terms to prove smoothness of the limiting direction.
2121. u2121 On y ≥1/2, where χK = 1, let f = fo(y) and write f ′ = ∂y f. Here uθ = K f and ur = uz = 0. The heat equation from Lemma A.6 and yt = (qL)−1, yz = −2η/(qDL) yield
2122. u2122 (A.52)
2123. u2123 (A.53) Integrating the viscous terms of (A.52) by parts in (A.46) gives the exact formula
2124. u2124 (A.54) All three terms are nonnegative: K > 0, Kr < 0, and f ′ ≥0. On the remaining log interval, whose length is at most 2.5, the ratios of radii and heat factors are bounded above and below.
2125. u2125 Since ∫︁3 y
2126. u2126 y f ′(v) dv = ρoψo(y), the last term gives
2127. u2127 Similarly (A.53) and the integral representation of Tz imply
2128. u2128 (A.55) The constants here are uniform for sufficiently large XR and small h. The last equality uses 1 −D = A; the amplitude reduction in the exterior transition of (A.17) makes its last expression small. Meanwhile bs = 0, and
2129. u2129 (A.56) because f ′/ f < h/4 and, for large XR, −ZH′/H < h/4. Also a ≤2 + 2h. Thus vs = a, Pc −vs = T0,θ/F > 0, and Jc = T0,z/F. The directional cone inequality is (a −2)(Tz/Tθ)2 < 2, with fixed strict slack by (A.55). This proves the admissible stress cone at each point before the outer endpoint X = Xb.
2130. u2130 The cone inequality now holds at every interior point of the terminal interval. It remains to prove that the stress direction extends smoothly to the endpoint. We factor the common vanishing part of the two stress components and show that the angular coefficient stays positive. The specified smooth step gives, near δ = 0,
2131. u2131 and hence We apply Lemma A.9 to the backward stress formulas.
2132. u2132 Multiplying the boundary term in (A.54) by qA+1/2 gives
2133. u2133 The two integral terms contain f ′, so (A.47) with j = 3, c = 4 gives a factor e−4/δ2 times a smooth coefficient. They therefore vanish to order at least δ3 relative to the boundary factor. It follows that (A.48) holds, with
2134. u2134 The positive lower bound is uniform for −1 ≤η ≤1, since H has a positive minimum on [0, 2/Xb].
2135. u2135 The pressure integral (A.53) has one f ′-factor, so the same identity first gives pz = e−4/δ2 times a smooth coefficient, apart from its factor q−D−2A. The integral representation of Tz requires one further integration with smooth weights. Applying (A.47) with j = 0 gives precisely (A.49). In this normalization the powers cancel as qA+1/2q1/2−D−2A = q1−D−A = 1. All remaining coefficients contain only smooth functions of X, η, H(2d/X), and L−1; here L ≥1 −2h > 0. This also verifies closed-parameter smoothness and the absence of a physical scale in both flat factors.
2136. u2136 Equation (A.50) follows by division by the positive coefficient bθ. Its limiting direction is (1, 0), where the normalized directional quadratic expression equals 2. The stress itself is zero at the endpoint. Finally each fixed profile derivative of the factorizations only introduces a finite inverse power of δ, proving (A.51). A smooth positive factor from the other endpoint
2137. u2137 □ can multiply this outer weight without changing these local conclusions.
2138. u2138 Appendix B. Analytic profiles near the axis and their continuation
2139. u2139 In this appendix we construct the inner part of the leading profile (U, E, Π), where the leading residual stress T0 must vanish (Proposition B.2). The outer profile has already fixed the pressure datum Π0 at the axis (Lemma A.5). With that datum, we choose the remaining axis data U∗, ϕ∗and solve the nonlinear profile equations (4.13) on a short radial interval. The solution must be regular at the axis and have the strict shear inequality (B.19) at its outer endpoint, so that it can be continued toward the outer profile (Corollary B.10). A large parameter Λ in the azimuthal datum (B.3) gives both the analytic solution and the endpoint bounds needed for this continuation (Propositions B.2 and B.3). The corresponding velocity is smooth in Cartesian coordinates across the axis by Equations (4.4) to (4.5). B.1. Choice of axis data. The pressure datum Π0 is prescribed, while the azimuthal datum ϕ∗= ϕ(0, η) and the axial velocity U∗at the axis remain to be chosen. In this subsection we choose these data to keep the angular source Sq positive and enforce the endpoint inequality throughout the parameter interval (Proposition B.3). The azimuthal and axial terms will provide the required lower bound on complementary parts of that interval, as separated in (B.2).
2140. u2140 We use the datum Π0 of Lemma A.5; in particular it is real analytic near [−1, 1], is even, and satisfies
2141. u2141 All schedule parameters, including 0 < h ≤10−2, have already been fixed. Choose 0 < j0 ≤ .05 small and set
2142. u2142 (B.1)
2143. u2143 Z∗= −A(1 −2ηU∗)U∗−H∗U∗η −dΠ0η + 4AηΠ0. The function H∗has exactly one zero η0 ∈(−1, 0): on (−1, 1) the function H∗/d = Dη/d + 4η + j0 is strictly increasing from −∞to +∞. Its zero satisfies |η0| ≍j0. Moreover
2144. u2144 −W∗= 3 −8hη2 + (1 −2h)j0η > 2.8, For the second assertion, the term 4Aη0Π0(η0) is positive and bounded below by cj0P2
2145. u2145 ∗; the term −dΠ0η(η0) is nonnegative, the H∗term vanishes, and the remaining term is O(j0). The already large P∗absorbs this last term.
2146. u2146 Consequently one can first choose δ∗> 0 so that {|Z∗| ≤δ∗} avoids a neighborhood of η0, and then choose σ∗> 0 so small that
2147. u2147 (B.2) These choices are made on a slightly larger compact interval I ⊃[−1, 1]. Choose a bounded simply connected complex neighborhood Ωof I on whose closure L and H2
2148. u2148 ∗do not
2149. u2149 vanish and Π0 is holomorphic. For Λ ≥1 define
2150. u2150 (B.3) The integral is well defined on Ω. On the real interval ϕ∗> 0. The amplitude C will be chosen after Λ.
2151. u2151 B.2. An analytic coefficient space. The profile equations (4.13) contain parameter derivatives as well as radial derivatives, with a regular singular point at X = 0. In this subsection we introduce a weighted coefficient space Bρ with norm (B.4) for their integral form, in which increasing the radial degree compensates for a parameter derivative. The bounds in Lemma B.1 control multiplication and radial inversion in this space, including the mixed nonlinear terms needed in the contraction argument for Proposition B.2.
2152. u2152 For the analytic axis construction, we use the scalar radial variable Y = ΛX. Fix a small radius ρ > 0. For F(Y, η) = ∑α≥0 Fα(η)Yα put
2153. u2153 (B.4) Let Bρ consist of the sequences of smooth coefficients with finite norm. It is complete: a Cauchy sequence converges uniformly at each coefficient and derivative order, and uniform convergence of successive derivatives identifies those limits as the derivatives of the limiting coefficient. In the rescaled variable Y, the logarithmic radial derivative is DX = X∂X = Y∂Y. Let IF = = ∫︁Y 0
2154. u2154 Radial averaging is unchanged by the scaling Y = ΛX, so AX(F) = Y−1IF.
2155. u2155 Lemma B.1. Multiplication is bounded on Bρ. For ν = 1, 2, let JνF be the solution G of YGYY + νGY = F that extends smoothly to Y = 0 and satisfies G(0) = 0. Its coefficients are given by
2156. u2156 (B.5) The operator Jν is bounded on Bρ, as are radial averaging, multiplication by Y, and the operators I and ∂ηI. In addition,⃦⃦
2157. u2157 (B.6) The same estimate holds if either derivative is omitted or F is replaced by AX(F). Extra undifferentiated factors can be inserted, with one additional algebra constant for each factor.
2158. u2158 Proof. The elementary convolution estimate
2159. u2159 ∑
2160. u2160 ∑
2161. u2161 (B.7) follows by dividing the sum at N/2. In the Leibniz formula for the product coefficient, its derivative binomial cancels the factorials in the weights. The remaining binomial ratio is at most one, since
2162. u2162 (B.8) Indeed the left side counts some of the β-element subsets of a set split into blocks of sizes i + k and α −i + β −k. Applying (B.7) in both indices gives the algebra constant C2
2163. u2163 The two weight ratios needed for integration are
2164. u2164 (B.9)
2165. u2165 (B.10) The first proves the asserted bounds for Jν, I, and multiplication by Y; averaging divides degree α by α + 1. The second proves the bound for ∂ηI, because its integration divisor cancels i + 1.
2166. u2166 For the mixed derivative bound (B.6), take the degree N = α + 1 coefficient of its left side. Assign its extra degree to the factor carrying the parameter derivative, and write I1 = i + 1, J1 = α −i. After (B.10) and division by aNβ, its βth derivative is bounded by
2167. u2167 ∑
2168. u2168 ∑ The binomial ratio was dropped using (B.8) with radial indices I1, J1. The first fraction is at most one, and the two sums are bounded by (B.7). With an average on F, its divisor i + 1 cancels I1; the resulting fraction is again at most one. If the logarithmic radial derivative is omitted, replace J1 by one. A logarithmic radial derivative without an η-derivative is controlled directly by (B.9) and (B.5). For additional factors, the same allocation leaves their indices undifferentiated and repeated convolution gives the claim. This argument includes
2169. u2169 □ the endpoint indices and every β.
2170. u2170 B.3. The nonlinear analytic axis profile. We now solve the equations (4.13) with vanishing leading residual stress for the chosen axis data (Section B.1). After rescaling the radius by Y = ΛX, we will show that the solution is a small perturbation of an explicit profile as Λ grows ((B.13)). The scalar comparison function f0 below controls the normalized azimuthal profile ϕ/ϕ∗through f0(Yχ); we will verify that it remains positive throughout the required radial interval in (B.11). This will ensure that division by ϕ in the profile equations is valid.
2171. u2171 The analytic scalar comparison function is
2172. u2172 For 0 ≤z ≤4.1, set t = z/2. The alternating tail after the cubic term has decreasing magnitudes, and hence
2173. u2173 (B.11) The cubic is decreasing on [0, 2.05], giving its stated endpoint value. For the upper bound group the alternating tail after the constant term; its magnitudes decrease from the linear term onward.
2174. u2174 Proposition B.2. With the axis data of Section B.1, there are Λ0 and, for each Λ ≥Λ0, an amplitude threshold C0(Λ) such that every C ≥C0(Λ) admits an analytic axis profile with vanishing leading residual stress on 0 ≤Y = ΛX ≤4.1. It has the form
2175. u2175 (B.12)
2176. u2176 where Φ(0, η) = 1 and u(0, η) = 0. The profile is analytic in Y and in η on a common neighborhood of [0, 4.1] × I. For every fixed r, s ≥0,
2177. u2177 (B.13) Here the constants and a smaller common parameter neighborhood are independent of sufficiently large Λ and of C ≥C0(Λ). In unscaled radial derivatives, the corresponding bound is Cr,sΛr−1. Proof. We rewrite the equations in the rescaled radius so that the nonlinear remainders carry a factor Λ−1. The radial inverses then reduce the problem to a contraction about the explicit comparison profile. In the notation of the stress equations, put
2178. u2178 B = −2DηAX(u) −d∂ηAX(u),
2179. u2179 (B.14) The equations with vanishing leading residual stress (4.13), together with radial pressure balance, are
2180. u2180 (B.15) Substitution of (B.12) gives
2181. u2181 with the explicit remainders
2182. u2182 LR1 = [W + h(1 −2ηU) + duζ∗]Φ + WDXΦ + HcΦη,
2183. u2183 LR2 = [A(1 −4ηU∗) + dU∗η]u −2AηΛ−1u2 + WDXu + H∗uη + dΛ−1uuη
2184. u2184 The leading angular term has the stated sign because H∗ξ0/(ΛL) = −χ. The expanded axial remainder follows by subtracting the constant value −Z∗/L from L−1{A(1 −2ηU)U + WDXU + HcUη −4AηΠ + dΠη −2ηDXΠ} and multiplying by Λ.
2185. u2185 To apply the radial inverse bounds, we must control the differentiated products in R1 and R2. The only products containing both an unintegrated parameter derivative and a logarithmic radial derivative are
2186. u2186 Their derivatives act on distinct factors. Furthermore, ∂ηp = ∂ηI(g2Φ2) is bounded by Lemma B.1, and DXp = Yg2Φ2. Thus the maps (Φ, u) ↦→JνRi are bounded and locally Lipschitz on fixed balls of B2
2187. u2187 ρ. This last assertion follows quantitatively by the displayed algebra bounds; to estimate a difference, subtract each monomial by replacing its factors one at a time.
2188. u2188 The coefficient bounds just used are uniform in the large parameters.
2189. u2189 Indeed ζ∗, χ, U∗, Z∗/L are fixed holomorphic functions on Ω. Choose a smaller neighborhood and take ρ strictly below its distance to the boundary of Ω. After choosing Λ, require
2190. u2190 (B.16) Then |g| ≤1 on Ω, so Cauchy’s inequality bounds its derivatives on the smaller neighborhood uniformly in both large parameters. The strict choice of ρ absorbs the factor (β + 1)2 in (B.4).
2191. u2191 It is essential here that the bound on g holds on a complex neighborhood, since ϕ∗itself can grow exponentially with Λ.
2192. u2192 It remains to invert the order-one angular term without assuming that its coefficient is small. Let Mχ bound multiplication by χ on Bρ and set T = J2χ/2, where multiplication precedes integration. If F has no radial coefficient below degree b, (B.9) gives
2193. u2193 Multiplication by χ preserves the vanishing of coefficients below degree b, and J2 increases the minimum possible nonzero degree by one. Parameter differentiation introduces no lower-degree coefficients, so the estimate includes arbitrarily high parameter derivatives of the iterated products. The series ∑k≥0(−T)k therefore converges absolutely in operator norm and is a two-sided inverse of 1 + T.
2194. u2194 The unperturbed solution is Φ0 = (1 + T)−11 = f0(Yχ) and u0 = −YZ∗/(2L). Solve the equations by the map
2195. u2195 On a fixed ball about (Φ0, u0) the uniform bounds and Lipschitz constants established above make this ball invariant under the map and make the map a strict contraction for all sufficiently large Λ. Its fixed point has norm distance at most C/Λ from the center. Choosing C0(Λ) to include (B.16) implements the stated order of parameters.
2196. u2196 Finally, for R < 20,
2197. u2197 ∑ Choose 4.1 < R < 20. The coefficient norm therefore gives a common positive parameter radius, any sufficiently small number below ρ(1 −R/20), on |Y| ≤R. Cauchy estimates on a smaller Y disk give (B.13), including all fixed radial derivative orders. The original derivatives satisfy ∂r
2198. u2198 Y. Real data and uniqueness of the fixed point give a real solution on the real rectangle. √ Its analyticity in X makes the scalar profiles smooth at the axis. Together with E =
2199. u2199 2Xϕ/C, this gives a smooth Cartesian field by Equations (4.4) to (4.5). The scalar lower bound (B.11) gives Φ > 0 for large Λ, and hence ϕ > 0. Division
2200. u2200 □ by ϕ in the first equation of (B.15) is therefore valid throughout the real rectangle. B.4. Positive sources and the outer-endpoint alternative. The analytic solution must meet the quantitative conditions needed to begin its continuation toward the outer profile. We derive a positive angular source Sq and a strict shear inequality (B.19) at the end of the inner interval from the preceding approximation (B.13). The same estimates will bound the normalized profiles and their parameter derivatives for use in the joining construction (Proposition B.3 and Lemma B.4).
2201. u2201 Proposition B.3. Increase Λ and then C0(Λ) if necessary. The normalized azimuthal profile satisfies Φ ≥c0 > 0 on [0, 4.1] × [−1, 1]. On that rectangle its angular source satisfies
2202. u2202 (B.17) Let p1 = ps,1 = XQs/L, p2 = ps,2 = XNs/(LE), and ns = Ns/L. On 0 < X ≤4.1/Λ one has
2203. u2203 (B.18)
2204. u2204 At X0 = 4/Λ there is a fixed cex > 0 such that
2205. u2205 (B.19) The quantities Φ, log Φ, u, p1, and ns have bounded derivatives in every fixed parameter order, uniformly in subsequent C ≥C0(Λ); constants for derivatives in the original radial variable X may depend on Λ.
2206. u2206 Proof. The scalar bound (B.11) and the approximation (B.13) give Φ ≥c0 > 0 for large Λ. On a smaller common parameter neighborhood Φ is also nonzero, by compactness and the uniform first derivative bound. The parameter derivatives of log Φ are consequently uniformly bounded.
2207. u2207 The source bound depends on the logarithmic derivatives of Φ. Since |H∗χ′| ≤2∥H′
2208. u2208 ∗∥∞χ and f0 is bounded away from zero, DX log f0(Yχ) = O(χ) and H∗∂η log f0(Yχ) = O(χ). The approximation and positivity therefore imply for the full solution
2209. u2209 |DX log Φ| + |H∗∂η log Φ| ≤Cχ + C/Λ.
2210. u2210 (B.20) Using (B.14) and (B.3), this gives
2211. u2211 (B.21)
2212. u2212 √χ controls the first error. The second inequality follows For example, |H∗|/(H2 by increasing Λ to absorb Cχ and then applying Young’s inequality to the square-root term, allocating a total of .05ΛLχ.
2213. u2213 For the source itself use l = 1 + DX log Φ and the defining formula Sq = −Wl −h(1 − 2ηU) −Hc(log E)η. By j0 ≤.05 and (B.13), increasing Λ ensures |U| ≤4.1 and therefore |h(1 −2ηU)| ≤10h. The preceding bounds and W −W∗= O(Λ−1) imply
2214. u2214 Since h ≤10−2 and −W∗> 2.8, the same absorption with Λ large proves (B.17). In particular no division by χ is used near a zero of H∗.
2215. u2215 The function Qs extends smoothly to X = 0 and has the positive integral representation
2216. u2216 Uniform upper and lower bounds for Φ and (B.17) give p1/X ≥c1. Integrating the equations with vanishing leading residual stress from X = 0 and using regularity at the axis gives the identities The upper bound for p1 and the approximation for ns now follow from (B.13). These identities also give the claimed fixed bounds on η-derivatives.
2217. u2217 It remains to prove the endpoint inequality. The two parameter regions selected by (B.2) allow us to use either the azimuthal or the axial contribution. At Y = 4, first suppose χ > .99, so 1.98 < t = 2χ ≤2. The alternating series for f0(z) + z f ′
2218. u2218 The polynomial is decreasing on [1.98, 2]. At t = 1.98 its value is −75535511/400000000 < −.188. The decreasing alternating remainder has the required negative sign. By (B.11),
2219. u2219 −2z f ′
2220. u2220 0)/ ′ f0 > 2.36. Equation (B.13) therefore gives p1 > 2.3 at the outer endpoint for large Λ.
2221. u2221 In the complementary region χ ≤.99, (B.2) gives |Z∗| > δ∗. Thus (B.18) implies |ns| ≥ δ∗/2, after increasing Λ. At the outer endpoint,
2222. u2222 For the now fixed Λ, the denominator has a finite upper bound uniform in subsequent large C, whereas p1 ≤C1. Increasing C0(Λ) therefore makes p2
2223. u2223 2/p1 > 2.3 uniformly on this region.
2224. u2224 □ Both alternatives imply (B.19), for example with cex = .2.
2225. u2225 B.5. A reference continuation satisfying ps,1 + p2
2226. u2226 s,2/ps,1 > 2. We next continue the analytic axis profile of Proposition B.2 toward the outer profile of Section A. The continuation must introduce nonzero leading residual stress T0 satisfying the cone conditions and preserve the pressure datum Π0 of Lemma A.5. In this subsection, we first construct a reference continuation ϕr, Ur by gradually setting the logarithmic radial derivatives of ϕ and U to zero, as prescribed below in (B.22). Bounds on its integrated residual coefficients, proved in Lemma B.4, will control the subsequent change in shear (Proposition B.5).
2227. u2227 As in Proposition B.3, p1, p2 denote the components of ps, and ns = Ns/L. Recall that DX = X∂X, equivalently ∂y in a logarithmic radial coordinate. Bounds in Ck η
2228. u2228 η refer only to parameter derivatives, uniformly for −1 ≤η ≤1.
2229. u2229 Let X0 = 4/Λ, Xb = 100, and Xi = 110. We choose a fixed ¯t > 0 with 4e2¯t < 4.1. We use the smooth step σ of (A.5). For 0 < t1 ≤¯t, put y = log(X/X0). The reference profile agrees with the analytic axis profile for y ≤t1; for t1 < y < 2t1 we prescribe
2230. u2230 (B.22) and then extend both fields constantly in the logarithmic radial coordinate. We compute the reference pressure and the functions Qs, Ns by radial integration from the axis, using the datum of Lemma A.5.
2231. u2231 Lemma B.4. After the parameters of the outer and axis profiles are fixed, one can take Λ, then C, sufficiently large, and then t1 sufficiently small, so that on [X0, Xi] × [−1, 1]
2232. u2232 (B.23) Here p1,r > 0, and p1,r > 3 for Xb ≤X ≤Xi. For each fixed k, the parameter norms of CEr, its reciprocal, p1,r, and ns,r on [X0, Xi] have bounds independent of sufficiently large C and of 0 < t1 ≤¯t. Proof. The analytic axis profile has a positive source and a strict inequality at its outer endpoint by Propositions B.2 and B.3. We use its gradient bounds (B.20) and (B.21) and source bound (B.17) to follow these properties through the cutoff. Integrating (B.22) shows that Ur and its radial average stay within O(Λ−1) + ot1(1) of U∗in each required parameter norm. Indeed radial averaging is a contraction:
2233. u2233 The estimate is independent of the length of the interval on which the reference profile is constant. The same slope integration bounds log ϕr −log ϕ∗in each parameter norm,
2234. u2234 uniformly in C, t1; its real value is bounded above and below. Thus CEr =
2235. u2235 2Xϕr and its reciprocal have the asserted bounds away from X = 0. Moreover
2236. u2236 (B.24) The regular axis factor makes this integral finite at zero.
2237. u2237 For the source sign, its leading gradient contribution is −H∗ξ0 = LΛχ. The bound |ξ0| ≤
2238. u2238 √χ source shows sign, that replacing its leading H∗by gradient Hc,r contribution introduces an is error −H∗ξ0 bounded = LΛχ. by The Cσ∗
2239. u2239 √χ + |ξ0| ot1(1), ≤ LΛσ−1 ∗ after Λ, C are fixed. The logarithmic slope of the axis profile and the remaining gradient give
2240. u2240 √χ ≤εΛLχ gradient + Cε,σ∗/Λ. give Cχ + C/Λ. Absorb these errors into a small fraction of LΛχ, using Cσ∗ The constant contribution retains the slack from −W∗> 2.8, yielding the first inequality in (B.23). Since the axis profile satisfies p1 = a > 0, its logarithmic ϕ slope is nonpositive; cutting this slope to zero gives lr ≤1.
2241. u2241 The axial source satisfies Sn,r = Z∗+ O(Λ−1) + ot1(1) + O(C−2). Here DXUnat = O(Λ−1) on the short cutoff, and (B.24) controls pressure and its needed parameter derivative on the entire finite interval. The equations for p1,r, ns,r are
2242. u2242 (B.25) Their regular initial values and bounded coefficients give the claimed parameter bounds. On χ ≤.99, the axis alternative |Z∗| > δ∗therefore preserves the sign and a positive lower bound of |ns,r|.
2243. u2243 The first equation in (B.25) preserves positivity and gives both comparisons
2244. u2244 p1,r(y) ≥p1,r(0)e−y + 1.88χ(ey −e−y),
2245. u2245 On χ > .99, the first right-hand side remains above 2.3, since its initial value can be replaced by 2.3 and 1.88χ > 2.3/2. On the complement, p2,r = Xns,r/Er grows uniformly in absolute value with C, while p1,r remains positive and bounded. This proves vr > 2 + c. The second
2246. u2246 □ comparison gives p1,r > 3 from X = 100 onward.
2247. u2247 B.6. Continuation to nonzero leading residual stress. In this subsection, we use the reference bounds of Lemma B.4 to reduce the shear s while keeping the integrated residual coefficients ps close to their reference values as in (B.28). The resulting stress T0 = F(ps −s) of (4.11) must become nonzero smoothly at the end of the axis region, satisfy the admissible stress cone on a first collar, and retain the relaxed cone condition along the rest of the continuation (Proposition B.5).
2248. u2248 Proposition B.5. The analytic axis profile can be continued to Xi with positive E, remaining unchanged for X ≤X0. Its stress has the factorization T0 = eaB0, where the scalar factor ea vanishes to infinite order at X0 and the smooth vector coefficient B0 is nonzero there. The admissible stress cone holds on a first collar, and the strict relaxed cone condition holds on the rest of (X0, Xi]. At Xi, a = .8, DXU = 0, and p1 > 2.
2249. u2249 Proof. We reduce the reference shear on a short interval while ps changes by a controlled small error. The resulting residual stress is nonzero and satisfies the cone conditions stated above. We then verify its factorization by a function vanishing to infinite order and complete the continuation with small shear.
2250. u2250 Fix a sufficiently large finite C. Choose 0 < t∗≤¯t sufficiently small that the bounds of Lemma B.4 hold uniformly for every reference width 0 < t1 ≤t∗. The parameter bounds of Lemma B.4 and its positive lower comparison for p1,r give one finite upper bound Vmax for vr over this entire reference family. It may depend on C, but not on t1. Let 0 < κ0 < 1/2, to be chosen using that bound, and on 0 < y < t1 set
2251. u2251 ea = (1 −κ0)σ(y/t1),
2252. u2252 .
2253. u2253 (B.26) The reference agrees with the analytic axis profile on this interval. Subtracting its equations gives exactly
2254. u2254 (B.27) Since ea increases, each fixed parameter derivative of these field differences is O(yea). Constants in the activation comparisons may depend on the already fixed C, Λ, but are uniform as t1 and κ0 decrease within the admissible ranges. The reference family bounds control the integrated coefficients independently of both widths. The radial moment identities (4.16), using one additional parameter derivative, imply
2255. u2255 (B.28) The comparison holds for field values and parameter derivatives; radial derivatives of the narrow cutoffs may be large.
2256. u2256 The actual shear is κ(p1,r, p2,rEr/E). In particular the common κ cancels from the ratio ts = −bs/a before it is estimated. The definitions of Pc, Jc, vs yield
2257. u2257 (B.29) No inverse power of κ0 enters these error constants. Taking t1 sufficiently small gives Pc −vs ≥cea, Pc > 2 + c, and
2258. u2258 Indeed the ratio of its left side to (Pc −vs)2 is at most Cy2. The quadratic test of Lemma 4.5 proves the admissible stress cone condition when vs > 2; when vs ≤2, Pc > 2 gives the strict relaxed condition directly. Continuity from vr(0) > 2 + c gives one positive-width first collar on which vs > 2 + c/2, uniformly in η for the final fixed parameters.
2259. u2259 To establish smoothness after division by the flat factor, write ea = e−t2
2260. u2260 1/y2ga(y) near zero, where ga is smooth and positive. Lemma A.9, applied with c = t2
2261. u2261 1, j = 0, and coefficient gab, gives for every smooth coefficient b
2262. u2262 Division by the positive smooth ga(y) preserves smoothness and bounds on mixed derivatives of each fixed order. The differences in (B.27) belong to eay3C∞. Products, smooth compositions, and reciprocals of nonvanishing fields preserve this class. The moment and pressure differences vanish before X0; their additional integration puts them in eay6C∞. Substitution in (4.16) consequently justifies (B.28) after division by ea in every fixed mixed derivative, and gives the exact factorization
2263. u2263 (B.30)
2264. u2264 At the edge the limiting shear is ps,r. Therefore B0 · (−ts, 1) = 0 and B0 · (1, ts) > 0 there. At the endpoint, B0 · (1, ts) > 0 and 2(B0 · (1, ts))2 −(vs −2)(B0 · (−ts, 1))2 > 0. Both quantities have positive lower bounds on a smaller collar. In particular, for every fixed radial/parameter multi-index I,
2265. u2265 (B.31) The derivatives in (B.31) act in (y, η); fixed derivatives in X obey the same form because X0 > 0. The nonzero vector B0(0, η) determines the limiting stress direction.
2266. u2266 We now continue to Xi while preserving the relaxed cone condition. After the first transition keep (B.26) with κ = κ0 through the reference cutoff and constant-slope interval. For fixed C, Λ, integration over [X0, Xb] and (4.16) give O(t1 + κ0) differences of field values, logarithms, and the required derivatives with respect to η. Choose κ0 small compared to Vmax and the uniform family comparison constants, then choose t1 small. This order does not require a reference width to be fixed before κ0. Formula (B.29) now gives Pc > 2 + c/2 and vs < 1. The strict relaxed cone condition follows, regardless of the size of Jc.
2267. u2267 At Xb, multiply the axial prescription for DXU by a smooth factor β decreasing from one to zero in a short log interval. During this transition, Pc = p1,r + βp2
2268. u2268 2,r/p1,r + o(1) > 2, while vs < 1. Next interpolate a convexly from κ0p1,r to .8, keeping DXU = 0. Arrange first that κ0 sup p1,r < .8. The second transition has vs = a ≤.8, and Pc = p1 > 2 by continuity from the positive margin established at Xb. Both transitions fit strictly before Xi. On the final constant-slope interval a = .8, l = .6, and the same gradient absorption as in Lemma B.4 gives Sq > 1: its constant contribution is .6(−W∗) > 1.68, while its leading gradient is LΛχ. At a hypothetical downward crossing p1 = 2, DXp1 = XSq/L −.6p1 > X −1.2 > 0, a
2269. u2269 □ contradiction. This completes the continuation.
2270. u2270 The continuation remains analytic in η on the first collar. We reserve a smaller rectangle there, so that later modifications preserve this regularity as well as the inner stress factorization.
2271. u2271 Corollary √ B.6. There is tc > 0, with tc < t1 and 4etc < 4.1, such that the functions
2272. u2272 2X, U, V0/X, Π on [0, X0etc] × [−1, 1] are smooth in (X, η) through X = 0 and analytic in η E/ on one complex neighborhood. Every fixed radial derivative of these four functions has that same parameter neighborhood and finite bounds on a smaller neighborhood. On X0 < X ≤X0etc, the admissible stress cone condition and the factorization (B.30) hold. All subsequent profile modifications are supported strictly to the right of X0etc.
2273. u2273 Proof. Choose a compact complex neighborhood inside that of Proposition B.2, small enough that the positive function Φ of the analytic axis profile has no zeros. The formulas for Qs, Ns computed from the reference profile, and (B.26), involve only analytic parameter coefficients, their parameter derivatives, and forward integrals, multiplied by cutoffs in y alone. Their denominators are nonzero on a smaller neighborhood. Exponentiation gives nonvanishing ϕ, so one neighborhood works throughout the fixed first collar. Radial differentiation changes the cutoff bounds but not the parameter domain. Compactness and Proposition B.5 give an inner collar on which the admissible stress cone condition holds; choose tc strictly inside it and reserve the resulting rectangle from all later modifications. Neither tc nor radial derivative bounds are asserted uniform as C grows and the transition widths shrink. The radial moment and pressure integrals are forward, so changes supported later leave this
2274. u2274 □ rectangle unchanged.
2275. u2275 B.7. A radial transition length chosen before the amplitude. In this subsection, we construct the remaining transition (B.34) to E = P∗f (X/XR)1/10, with f = (1 + η2)−1, before correcting its radial moment functions in the next subsection (Proposition B.8). We first bound its logarithmic radial length Tsh independently of the amplitude C chosen later, using Lemma B.7 and (B.33). Increasing that amplitude can then move the matching radius XR outward without lengthening this transition, making the normalized inner moment discrepancies small (Equations (B.38) and (B.39)). The estimates below concern parameter derivatives; bounds for derivatives with respect to X need not be uniform in the radial transition widths. In the bounds below, κ0 ∈(0, 1/2) is the parameter controlling the shear reduction in (B.26). Lemma B.7. Fix the data for the outer profile and the axis, Λ, and a finite parameter order k. Put ℓi = log(CE(Xi, η)) and Gi = U(Xi, η). Let ωfin be the sum of the logarithmic radial lengths of the final axial cutoff and the interpolation of a to .8 in the proof of Proposition B.5. The continuations constructed there satisfy
2276. u2276 (B.32)
2277. u2277 The constant Cnat k
2278. u2278 is independent of both Λ and sufficiently large C, with the preceding axis data fixed. The bounds Bk and Cjoin k
2279. u2279 (Λ) may depend on Λ, but are independent of sufficiently large C and of smaller radial transition widths.
2280. u2280 Proof. Write L0 = log(Xi/X0). Let Mk, Nk be the amplitude-independent bounds for p1,r, ns,r from Lemma B.4. Before the final interpolation of a to .8, the actual logarithmic slope is −κp1,r/2, with 0 < κ ≤1; during that interpolation it varies convexly to −.4, after which it remains constant. Consequently
2281. u2281 ∥log ϕ(Xi)∥Ckη ≤∥log ϕ∗+ log Φ(4, ·)∥Ckη + 1 Adding 1 2
2282. u2282 1 2 log(2Xi) proves the first assertion, using (B.13). The first axial transition changes U by at most X0e¯tNkt1/2; the remainder contributes at most XiNkκ0/2, with the same bound for its transition to zero. The approximation error for the analytic axis profile is Cnat k
2283. u2283 k /Λ, with the constant independent of Λ, by (B.13). These estimates prove (B.32). Only integrals
2284. u2284 □ of bounded cutoff values were used, so inverse powers of their widths do not occur.
2285. u2285 The bounds just proved let us choose the length of the transition independently of the final amplitude. We choose
2286. u2286 (B.33) For yi = log(X/Xi) we prescribe
2287. u2287 10 yi + (1 −σ(yi/Tsh))ℓi + σ(yi/Tsh) log f,
2288. u2288 (B.34) Then l = .6 + T−1 sh
2289. u2289 sh σ′(yi/Tsh)(log f −ℓi) lies in [.55, .65], so a ∈[.7, .9] and bs = 0.
2290. u2290 Along the transition (B.34), the source still satisfies Sq > 1. To verify its sign, make the changes before the final .8 constant-slope interval small in the required Ck η
2291. u2291 η norms; that constant-slope interval has a parameter-independent slope. Thus ℓ′ i
2292. u2292 i = ξ0 + (log Φ(4, ·))η +
2293. u2293 o(1) and Gi −U∗= O(Λ−1) + o(1). The old gradient obeys
2294. u2294 For the second line use ηH∗= (D + 4d)η2 + dj0η and complete the square. For the first line use (B.20), which gives H∗∂η log Φ = O(χ) + O(Λ−1); multiplication of the O(Λ−1) field
2295. u2295 √χ, which multiplication is absorbed of as the before. O(Λ−1) The field two error by ξ0 introduces an error bounded by Cσ∗ gradients are combined convexly in (B.34). Since −W remains close to −W∗> 2.8, l ≥.55, and h, j0 are small, Sq > 1 follows. The barrier DXp1 = XSq/L −lp1 > 0 at p1 = 2 preserves p1 > 2. Thus this entire transition, and the subsequent constant-slope interval before restoration, continue to satisfy the strict inequalities defining the relaxed cone condition. B.8. Matching the five radial moment functions exactly. The azimuthal profile E now has the required reference form (A.7), and the axial profile U = Gi is close to its reference value 4η by (B.32) and (B.1). In this subsection, we match the five cumulative radial integrals M, I, J, S, Cp of (4.15) to attach the outer profile. We first choose the radial scale XR so that their normalized discrepancies are small, then correct them with five fixed bumps while preserving the relaxed cone inequalities (Proposition B.8). The common axis pressure datum Π0 and exact moment matching will then preserve every subsequent outer field by Lemma 4.4.
2296. u2296 We set We also set Xsep = XieTsh, the endpoint of the transition (B.34); it is fixed before the amplitude. Proposition B.8. For a sufficiently large amplitude C, followed by sufficiently small activation transitions, the profile constructed in Proposition B.5 and continued by (B.34) can be continued to log x = −5, preserving the strict relaxed cone condition, so that its fields, pressure Π, and all five radial moment functions M, I, J, S, Cp agree exactly with those of the reference inner profile (A.7). It can then follow the reference outer radial profile without changing its prescribed pressure datum or the subsequent values of Qs, Ns.
2297. u2297 Proof. We first normalize the radial moment map so that its inverse bounds do not depend on the later large radius. We then show that the discrepancy accumulated at the inner endpoint is small in those √ units and cancel it with five fixed bumps.
2298. u2298 We introduce ˆ︁H =
2299. u2299 2xE and normalized radial moment functions by
2300. u2300 M, ˆ︁ ˆ︁I, ˆ︁J, in ˆ︁S (4.16) are the gives integrals of U, ˆ︁H, U ˆ︁H, U2 −E2/2, respectively, with respect to x. Here ˆ︁ Substitution in (4.16) gives
2301. u2301 Qs = −W + (1 −h)ˆ︁I −Dηˆ︁Iη −dˆ︁Jη + 2(h −D)ηˆ︁J
2302. u2302 (B.35)
2303. u2303 In particular XR has disappeared from the formulas for Qs, Ns.
2304. u2304 On the fixed interval R = [e−8, e−5], the ideal E0 = P∗f x1/10 and Qs,0 have positive minima, and the ideal Ns,0 is bounded; these are properties of the explicit formula (A.25). The map (B.35) is Lipschitz there in each prescribed finite parameter norm, with one further parameter derivative on its inputs, and with constants depending only on data of the outer profile.
2305. u2305 The same is true of the fixed restoration and moment correction operations. To make the cone tolerance explicit, set Qmin = minR×[−1,1] Qs,0 > 0, e∗= min E0 > 0, and w∗= 1 + max |Ns,0/(E0Qs,0)|. A tolerance determined by the exterior-profile data can ensure throughout both operations that
2306. u2306 (B.36) It follows that vs ≤.9 + .01/.7 < 1. Setting G = Qs −bsNs/(aE), we obtain
2307. u2307 (B.37) Thus Pc > 2 follows by increasing XR after fixing the tolerance. No moment correction tolerance has to shrink with XR.
2308. u2308 It remains to show that this fixed tolerance is achievable. In normalized coordinates, the endpoint Xsep has coordinate
2309. u2309 (B.38) On [0, Xsep], the previously established bounds for log(CE(Xi, ·)) and U(Xi, ·), together √ with (B.34) bound U and CE in each required parameter norm; near zero CE = O(
2310. u2310 X). Hence the contributions to the cumulative integrals from 0 ≤x ≤xsep satisfy
2311. u2311 (B.39)
2312. u2312 The corresponding integrals of the ideal profile with positive weights also vanish as positive powers of xsep. The pressure increment of the reference profile at xsep is (5/2)P2
2313. u2313 sep = O(C−2). All constants in these vanishing errors may depend on the already fixed axis and shape parameters.
2314. u2314 After Xsep, formula (B.34) is exactly the ideal angular profile because C−1 f (X/Xi)1/10 = P∗f x1/10. Choose the amplitude so large that xsep < e−8. On −8 < log x < −7, restore Gi smoothly to 4η, keeping E ideal. The discrepancy Gi −4η can be made as small as required by first choosing j0, then Λ, and finally the small transitions in (B.32). Integration over the remaining interval gives bounds depending only on the exterior-profile data: all relevant weights are integrable positive powers at zero. Therefore the five normalized moment discrepancies at the moment correction have size at most Ck∥Gi −4η∥Ck+1 η
2315. u2315 + oC(1). The same estimate controls field values, logarithmic radial derivatives, and Qs, Ns on the fixed restoration patch.
2316. u2316 On −6 < log x < −5, add two fixed radial bumps to U and three to E. At the ideal profile, replace the J row by J −4ηI, and the S row by S −8ηM. Up to fixed nonzero normalizing factors, the derivative of the normalized moment map has the two blocks
2317. u2317 Corollary A.3 makes this matrix uniformly invertible on the compact parameter interval. The nonlinear terms are exactly quadratic. Lemma A.2 therefore solves the five equations with smooth coefficients whose prescribed finite set of η-derivatives satisfy the required smallness bounds. All higher fixed derivatives exist with their own finite bounds. This proves (B.36), by first fixing the radius of its small-discrepancy neighborhood from data of the outer profile and then imposing it on the estimates above. The bumps are supported strictly inside their patch, so the fields equal the ideal near its right endpoint. All five rows and the common Π0 now give identical pressure and functions Qs, Ns by (B.35); Lemma 4.4 propagates this
2318. u2318 □ equality along the outer radial profile.
2319. u2319 B.9. Order of choices and the completed connection. In this subsection, we collect the parameter choices and state the completed connection. The matching tolerance in (B.36) depends only on the outer data, and the radial transition length Tsh in (B.33) has been bounded independently of the final amplitude C by Lemma B.7. These facts allow the amplitude to be chosen to reduce the moment discrepancies before the last transition widths are fixed, as summarized in (B.40).
2320. u2320 Remark B.9. Fix any finite list of η-derivative orders whose bounds are needed to preserve the profile inequalities and solve the finite moment equations, including the extra parameter derivative in (B.35). The choices above and in Section A can be made in the following order:
2321. u2321 Md, Td, P∗, λ, h −→tolerance for matching moments, j0 −→δ∗, σ∗, Λ
2322. u2322 (B.40)
2323. u2323 −→(Bk), Tsh −→C, XR −→κ0, t1, widths of final transitions.
2324. u2324 λ(1 + log(1/λ)) small, and then h ≪λ, The choices for the outer profile first make Cpre h ≪e−Td, with Cpre already fixed. The tolerance determined by the exterior-profile data is justified by (B.37); the choices of j0 and Λ make the contributions j0 and Cnat k
2325. u2325 k /Λ to the axial matching error small enough. Lemma B.7 fixes Tsh before the final amplitude. The amplitude then ensures p1,r + p2
2326. u2326 2,r/p1,r > 2 + c, the cone inequalities for the outer profile at large radius, Xsep/XR < e−8, and the bounds on the inner-interval contributions in (B.39). The final narrow transitions secure the activation comparisons and the remaining errors at X = Xi while preserving the previously established bound. Only after these finite choices is the radial frequency used for admissible-stress-cone realization selected. Smallness is required only for the prescribed finite set of η-derivative bounds. Derivatives of every other fixed order have finite bounds after the choices are made; radial cutoff derivatives may contain inverse powers of the final widths.
2327. u2327 Corollary B.10. The analytic axis profile of Proposition B.2 admits a smooth continuation with E > 0 for X > 0, matching the reference outer radial profile at log(X/XR) = −5, with exact fields, pressure, and radial moment functions. It has vanishing leading residual stress through X0, followed by an inner collar on which the profile remains analytic in η and satisfies the admissible stress cone condition, with the flat factor (B.30). It satisfies the strict inequalities of the relaxed cone condition thereafter up to the matching point. Its finite parameters can be chosen in the order (B.40). Proof. Combine Propositions B.5 and B.8, the preservation of p1 > 2 along the profile (B.34),
2328. u2328 □ and Corollary B.6.
2329. u2329 Appendix C. Realizing the admissible stress cone
2330. u2330 The inner and outer profiles E, U, Π have been joined with the required pressure datum and radial moments (Corollary B.10). On part of the joining interval, however, the shear
2331. u2331 (a, −bs) satisfies only the relaxed cone condition (4.21). The wave construction requires the additional inequality vs > 2 from Lemma 4.5. In this appendix, we will enforce it by a rapid radial variation of the shear, supported away from the axis and the exterior.
2332. u2332 The shear depends on radial derivatives (4.11), while the cumulative moments m = (M, I, J, S, Cp) depend on profile values (4.15). We will use this difference to obtain an order-one change in shear with a small change in the profiles and their moments: we will prescribe a periodic family of admissible shears (aL, −bL) whose mean is the original shear (Lemma C.1), then integrate it at high radial frequency. A correction on the next reserved interval listed after (A.9) will restore all five moments exactly and hence preserve the exterior fields (Proposition C.2).
2333. u2333 We take the profile supplied by Corollary B.10, Proposition A.4, and Proposition A.7. The matching of the axis profile’s radial moments to those of the reference outer profile in Proposition B.8 supplies the regular axis and exact moments required in Lemma A.8. These results therefore give the fixed input profile E, U, Π used below. We keep this profile and all its parameters fixed while carrying out the shear modulation and moment correction in this appendix. Its leading residual stress T0 is nonzero precisely on (Xa, Xb), with endpoints independent of η. The profile satisfies the strict inequalities of the relaxed cone condition throughout that interval and the admissible stress cone condition on an inner collar and from the intermediate power-law interval onward (Propositions A.4, A.7 and A.10). The two edge factorizations are (B.30) and Equations (A.48) to (A.49). We choose a compact interval I = [X−, X+] ⋐(Xa, Xb) containing every point where the admissible stress cone condition may fail, starting inside the first collar on which the admissible stress cone condition holds and ending in the intermediate power-law interval before its first reserved patch. Shorter collars at Xa and Xb will remain √ unchanged.
2334. u2334 We fix the rectangle near the axis from Corollary B.6, on which E/
2335. u2335 2X, U, V0/X, Π are smooth in X, η and analytic in η on one common complex neighborhood. We choose its radial endpoint in the shorter inner collar, before X−. We denote this endpoint by Xan.
2336. u2336 C.1. A loop with the prescribed mean. In this subsection, we will construct the family (aL, −bL) of shear vectors at each profile point, holding the integrated residual coefficient ps fixed (Lemma C.1). Their average must equal (a, −bs), so the difference has mean zero and hence a periodic antiderivative. In Proposition C.2 we will integrate this difference into a small change of the profiles. The family must depend smoothly on (X, η) and remain equal to the original shear near the radial boundaries, where no modification is needed. Lemma C.1. Let a, bs, ps be smooth on I × [−1, 1], with a > 0, and set ts = −bs/a, vs = a(1 + t2
2337. u2337 s). Suppose the strict inequalities of the relaxed cone condition in (4.21) hold everywhere, and that the admissible stress cone holds in neighborhoods of the two radial boundary components. There is a smooth loop (aL, −bL)(X, η, φ), of period one in φ, with aL > 0, such that
2338. u2338 (C.1) Let Ψ be the vector of cone gaps in (4.35). There are constants κL, δ∂> 0, depending on the fixed data a, bs, ps, I, such that
2339. u2339 (C.2)
2340. u2340 On the radial boundary collars,
2341. u2341 (C.3) All smoothness assertions include the closed parameter interval −1 ≤η ≤1.
2342. u2342 Proof. Write a shear vector as a positive multiple of (1, t). We will vary the ratio t with mean ts, choose a scalar v > 2, and seek admissible vectors of the form v(1, t)/(1 + t2). The variance of t will determine v. A change of parameter around the loop will then give these vectors the required mean (a, −bs).
2343. u2343 For an auxiliary angle θ′, write ⟨f ⟩θ′ = (2π)−1 ∫︁2π 0
2344. u2344 f (θ′) dθ′; this average is distinct from the physical angular and auxiliary-torus averages. Put Pc(t) = ps,1 + ps,2t and Jc(t) = ps,2 −ps,1t. Choose 0 < d0 < 1 2
2345. u2345 (C.4)
2346. u2346 (C.5) The apparent singularity at ps,2 = 0 is removable. Indeed, for a smooth numerator G(p) with all G(0) its = variables 0, G(p)/p and = the ∫︁1 0
2347. u2347 0 G′(up) du. Applying this identity to (C.5) gives joint smoothness in all its variables and the 0 value t = ts + d0µ sin θ′ when ps,2 = 0. The family has the identities
2348. u2348 At ps,2 = 0 the last expression is d2
2349. u2349 We show that a prescribed finite variance can be attained uniformly over the compact range of ps,2, including ps,2 = 0. Let g = log Me. Its second derivative is the variance of sin θ′ under the strictly positive tilted probability density, so g′′ > 0. For p̸ = 0 and µ > 0,
2350. u2350 ∂µ{g(2µp) −2g(µp)} = 2p{g′(2µp) −g′(µp)} > 0.
2351. u2351 (C.6) Thus V increases strictly with µ > 0, also when p = 0. Quadratic upper and lower bounds for sin θ′ near its maximum, and a uniform gap from that maximum on its complement, give
2352. u2352 |z| for |z| ≥1. Consequently Me(2z)/Me(z)2 ≍ Me(z) ≍e|z|/
2353. u2353 |z|, and V(µ, p) →∞for every fixed p, including zero. Continuity, monotonicity, and a finite cover of the compact range of ps,2 now give one finite µmax such that
2354. u2354 (C.7)
2355. u2355 We can therefore choose the variance of t throughout the compact parameter set. We next bound the allowed value of v so that all resulting shear vectors stay inside the cone. For 0 ≤µ ≤µmax, the functions t(θ′; µ) form a compact family with Pc(t) > 2. The upper admissible bound U for v from (4.21), the smaller root in v of 2(Pc −v)2 = (v −2)J2
2356. u2356 c, is continuous and strictly greater than two there. Choose 0 < δL < 1 so small that
2357. u2357 (C.8) Shrink δL further below the positive minimum of vs −2 on smaller neighborhoods of the radial boundaries. Choose δL after fixing µmax.
2358. u2358 Only the region where vs is near or below two needs a nonzero variance. We leave the admissible boundary data unchanged. Let ζL be a smooth nonnegative function of vs, at most one, equal to one for vs ≤2 + δL/8 and zero for vs ≥2 + δL/4. Set
2359. u2359 (C.9) where ρ is defined to be zero off the support of the cutoff. Its nonnegative square root where is smooth: ρ is defined wherever to the be zero cutoff off can the be support nonzero, of v∗−vs the cutoff. ≥δL/4, Its nonnegative so it is ζL(vs)√v∗−vs, square root extended by zero. Also 0 ≤ρ < 3. Solve
2360. u2360 (C.10) Existence and uniqueness follow from (C.7) and strict monotonicity. To verify smooth dependence where ρ = 0, the expansion of (C.4) gives, uniformly for bounded p,
2361. u2361 The signed square root of V is smooth and odd near µ = 0, with derivative d0/
2362. u2362 2. The The implicit signed function square theorem root of applied V is smooth to this and square odd near root µ and = 0, the with smooth derivative right side d0/ √ρ/√a 2. The proves smoothness through ρ = 0. Away from zero it follows from (C.6).
2363. u2363 The definition of ρ gives
2364. u2364 Since 0 ≤ζL ≤1, the three cutoff regimes satisfy
2365. u2365 Thus v > 2 everywhere, and v ≤v∗wherever the cutoff is nonzero. Where the cutoff is zero, µ = 0 and v = vs > 2, so the strict inequalities of the relaxed cone condition apply to the given data a, bs, ps. It follows from (C.8) that all the proposed states satisfy 2 < v < U(Pc(t), Jc(t)).
2366. u2366 The vectors now lie in the admissible cone. Their average depends on how long the loop spends at each value of t. To impose both prescribed shear averages, reparametrize θ′ by a lifted angle φ, with origin at θ′ = 0, by
2367. u2367 Since a⟨1 + t2⟩θ′ = vs + aV = v, the lift satisfies φ(θ′ + 2π) = φ(θ′) + 1. Its derivative has a positive lower bound on the compact family. It therefore defines a circle diffeomorphism with smoothly parameter-dependent inverse. Changing variables gives
2368. u2368 The loop satisfies −bL/aL = t and aL(1 + t2) = v, so the admissible stress cone inequalities follow from the preceding scalar inequalities. On the boundary neighborhoods the cutoff vanishes, hence the loop equals the original shear (a, −bs). Compactness gives the uniform margins (C.2) in aL, v −2, and the two strict inequalities of the quadratic cone test; the fixed
2369. u2369 □ boundary neighborhoods give (C.3).
2370. u2370 C.2. Radial modulation and exact restoration of the radial moment functions. The loop
2371. u2371 in Lemma C.1 was constructed with ps fixed. In this subsection, we will insert it into the profiles E, U and control the resulting change in ps, which depends on their radial moments through (4.16). We will evaluate periodic antiderivatives at N log X and divide by N, so that the profile values change by O(N−1) while the shear changes by order one. The proof of Proposition C.2 will establish these estimates, bound the resulting moment errors by O(N−1), and remove them on the reserved correction interval. The comparison with the fixed input profile E, U, Π in the following proposition does not assert small radial derivatives. Proposition C.2. For the fixed input profile E, U, Π selected at the beginning of this appendix, a sufficiently large finite integer N and a correction supported in the first reserved patch in the intermediate the admissible power-law stress cone interval throughout give smooth (Xa, Xb). profiles Both endpoint ˜︁E, ˜︁U, ˜︁Π with collars, ˜︁E > the 0 rectangle for X > near 0, satisfying the axis the admissible stress cone throughout (Xa, Xb). Both endpoint collars, the rectangle near the axis described above, and the two patches reserved for higher-order and mean corrections are unchanged. With ˜︁m, ˜︁Qs, ˜︁Ns defined from these profiles by (4.15) and (4.16),
2372. u2372 Before that point, profile values and each fixed number of η-derivatives differ from those of the input profile by O(N−1).
2373. u2373 Proof. The prescribed averages make the differences between the loop and the original shear integrable in the periodic variable. Let A , B be the unique zero-mean periodic antiderivatives
2374. u2374 (C.11) They exist by (C.1), are smooth, and vanish identically on the boundary neighborhoods of I. Extend them by zero radially and put
2375. u2375 (C.12) The extension agrees smoothly with the input profiles E, U. In the next identity, DX = X∂X differentiates a function of (X, η, φ) while holding φ fixed; all its terms are then evaluated at φ = N log X. The exact shears of (C.12) are
2376. u2376 (C.13) Indeed X∂X after substitution is DX + N∂φ; inserting (C.11) into aN = 1 −2X∂X log EN and bN = 2X∂XUN/EN proves both formulas, including the exponential denominator in the axial shear.
2377. u2377 On the fixed compact radial range from X−through the first correction interval, X, E, H, L have positive lower bounds. For each fixed m ≥0, the phase in (C.12) is independent of η, so
2378. u2378 ∑
2379. u2379 (C.14)
2380. u2380 The same estimate holds for the differences between the shears in (C.13) and their loop values. The derivative X∂X(EN −E) can have order one. In general, for fixed r ≥1, m ≥0, radial differentiation of (C.12) gives a bound Cr,mNr−1 for the mixed derivatives of the profile difference. To check the cone condition for the modified profiles, we need control of ps as well as of the shear. Its radial integral formulas use profile values and parameter derivatives, so the preceding estimates suffice even though radial derivatives need not be small.
2381. u2381 Let m = (M, I, J, S, Cp) and mN denote the five radial moment functions in (4.15) for E, U and EN, UN, respectively. Keep the same pressure value Π0(η) at the axis, so ΠN = Π0 + Cp,N. Integration of (C.14) on this fixed radial range, with the bounded weights in the radial moment definitions, gives
2382. u2382 ∑
2383. u2383 ∑
2384. u2384 (C.15)
2385. u2385 The formulas for Qs, Ns in (4.16) involve these radial moment functions and their first parameter derivatives, together with profile values. Their denominators are bounded away from zero here. Consequently
2386. u2386 ∑
2387. u2387 (C.16)
2388. u2388 The estimate at order m uses profile and radial moment estimates through order m + 1 in η. It uses no comparison of radial derivatives. The uniform positive lower bounds in the cone inequalities of Lemma C.1 and (C.13) now preserve the admissible stress cone on I for large N. Between I and the first correction interval, the fields equal E, U and ps,N −ps satisfies (C.16), so the uniform positive lower bounds in the input profile’s admissible stress cone inequalities persist there as well.
2389. u2389 The modified profiles now satisfy the cone inequalities through the joining interval. Their moments have errors of order N−1, which must be removed before reaching the exterior. We correct them on the first patch listed after (A.9); it precedes the terminal compensation patch and the two later correction patches. On this patch the input profiles satisfy U = 0, E = K(η)X−1/2−λ, where K is smooth and has a positive lower bound. Add two fixed compact bumps to U and three to E, with coefficients depending smoothly on η. Order the five radial moment functions at the right endpoint as (M, J; I, S, Cp). At zero coefficients their differential is block diagonal, with weights
2390. u2390 The exponents are distinct for the already fixed λ > 0. Choose the bumps with ordered disjoint supports within each block. Lemma A.1, or its specialization to these five moment functions Corollary A.3, gives an invertible matrix, with uniform inverse on [−1, 1]. Its bound can depend on λ and the fixed patch scales; the two U-weights coalesce when λ →0, so no uniformity in that limit is asserted.
2391. u2391 The map from correction coefficients to changes in (M, J, I, S, Cp) equals this linear map plus quadratic terms: J contains the product of the two field changes, and S, Cp contain their squares. The discrepancy accumulated before this correction is Om(N−1) by (C.15). Lemma A.2 therefore gives a smooth solution for the coefficients near zero which cancels that discrepancy. For every fixed parameter order its coefficients are Om(N−1). Choose N after λ, the patch scales, and all choices entering the fixed input profile so that the discrepancy lies in the neighborhood where this solution is defined and its derivatives through the finitely many η-orders needed for positivity and the cone inequalities meet their tolerances. Differentiating the implicit equation gives finite bounds for every further fixed η-derivative order; no simultaneous smallness condition for all derivative orders is needed. The fixed bump shapes also make the correction small in the radial derivatives used by the shear. Its partial radial
2392. u2392 moment functions have the same Om(N−1) bounds by integration, so (C.16) remains valid inside the correction interval. Thus the admissible stress cone persists throughout it.
2393. u2393 At the correction interval’s right endpoint Xrep, the corrected profiles satisfy
2394. u2394 (C.17) Equality of the five radial moment functions then propagates by integration. Lemma 4.4 gives equality of the complete pressure and the functions Qs, Ns for all X ≥Xrep, including their parameter derivatives. This equality preserves the terminal compensation already included in the input moment vector m. The exterior moments and pressure normalized to vanish at radial infinity are consequently retained. The first collar is unchanged because both modifications are supported later and pressure was integrated from the unchanged axis datum. The last collar is unchanged by the exact radial moment restoration. The two unused correction patches retain their prescribed reference power law and U = 0.
2395. u2395 Finally fix one such finite N. Every mixed radial and parameter derivative of the resulting profiles is finite, with constants allowed to depend on this N. This choice is made before any
2396. u2396 □ limit in the physical scale q or any later dyadic band or correction stage.
2397. u2397 C.3. The two edges and the completed profile. In this subsection, we write E, U, Π for the corrected profiles of Proposition C.2 and form all associated quantities from them. The interior cone inequalities and exact radial moments are now in place by Proposition C.2. In this subsection, we will prove Proposition C.3 by verifying the remaining assertions at the annular edges, where the stress T0 vanishes. Its direction n must extend smoothly and retain a strict cone margin, and the derivatives of T0 must satisfy the common exponential weight ζ in Theorem 4.6. Both endpoint collars were preserved, so we can deduce these properties from their original stress factorizations (B.30) and Equations (A.48) to (A.49). In the weight below, t1 > 0 is the fixed activation width in log(X/Xa) from Proposition 4.10. Proposition C.3. The profiles of Proposition C.2 satisfy all conclusions of Theorem 4.6, with the weight
2398. u2398 (C.18)
2399. u2399 extended by zero for X ≤Xa and X ≥Xb.
2400. u2400 Proof. The stress is zero up to Xa by the analytic axis profile and Proposition B.5, and beyond Xb by Lemma A.8. Proposition C.2 preserves these regions and makes the admissible stress cone inequalities hold everywhere between them. If the stress vanished at an interior point, the identity (4.11) would give ps = (a, −bs), hence Pc = vs, contradicting the strict quadratic cone test. This proves the stress-support assertion in part (ii) of Theorem 4.6; the unchanged axis profile also satisfies (4.13) for X ≤Xa.
2401. u2401 For the cone assertions in part (iii), set n = T0/|T0| on Xa < X < Xb. The bounds F > 0, a > 0, and vs > 2 follow in the interior from the positive profile construction and the admissible stress cone. At the inner endpoint a = p1,r > 0, while Proposition B.5 gives vs > 2 + c there and the analytic axis profile has F > 0. At the outer endpoint they follow from (A.56), with bs = 0, and positivity of the heat factor. Continuity and compactness give the asserted positive lower bounds on the closed annulus.
2402. u2402 The inner edge has the preserved factorization T0 = eaB0 in (B.30), with ea = e−t2
2403. u2403 aga(ya), ga(0) > 0, and B0(0, η) = Fps,r̸ = 0. Its direction is therefore smooth, and at the edge it is a
2404. u2404 positive scalar multiple of the limiting shear (a, −bs) = a(1, ts). In particular nz −tsnθ = 0 and nθ + tsnz > 0 there. At the outer edge, the preserved factors of Proposition A.10 give
2405. u2405 Thus the outer limiting direction is (1, 0), and the limiting shear ratio is ts = 0. All these statements are smooth through η = ±1; the heat factor is evaluated only on its nonnegative argument domain.
2406. u2406 The edge limits allow the interior cone inequalities to be made uniform on the closed annulus. In the interior, (4.11) implies
2407. u2407 The admissible stress cone makes 2 −(vs −2)(nz −tsnθ)2/(nθ + tsnz)2 positive there. Both this expression and its positive denominator extend continuously to the two edges; the former has value two at either edge by the factorizations just proved. Their compact positive minima yield, for a single 0 < κ < 2,
2408. u2408 (vs −2)(nz −tsnθ)2 ≤(2 −κ)(nθ + tsnz)2. Together with the endpoint directions and positive lower bounds above, this proves part (iii) of Theorem 4.6.
2409. u2409 For part (iv), it remains to prove the weighted estimates
2410. u2410 (C.19) for every fixed mixed profile derivative ∂I, with finite constants independent of physical q.
2411. u2411 On an inner collar ζ differs from e−t2
2412. u2412 a by a smooth positive factor bounded above and below; the bounds (B.31) therefore imply (C.19) there. On an outer collar it differs from e−4/y2 b
2413. u2413 b in the same way. The angular factor e−4/y2
2414. u2414 b bθ, with bθ bounded below, proves the lower bound, and (A.51) proves every fixed derivative upper bound. On the remaining compact interior interval, both ζ and |T0| have positive minima and all fixed derivatives are bounded. Combining the three regions proves (C.19). The exponential factors also show that extension of the stress by zero across both edges is smooth. The definition (C.18) also makes ζ positive in the interior and flat at both edges, completing part (iv) of Theorem 4.6.
2415. u2415 For the regularity in part (i), Proposition B.2 supplies the smooth axis profiles with F > 0, and Corollary B.6 preserves the common analytic rectangle [0, Xan] × [−1, 1]. The factors give Cartesian smoothness across the axis by Equations (4.4) to (4.5). The later modifications in Proposition C.2 are smooth and preserve positivity, with finite mixed derivative bounds on each fixed radial interval. The inner directional margin is part (iii), proved above. The heat profile is smooth through η = ±1 by Lemma A.6; no analyticity of its factor at zero is needed. The pressure normalization (4.25) follows from Lemma A.5 and Propositions A.7 and B.8 and is retained by (C.17). This completes part (i). Differentiating this pressure formula gives ΠX = E2/(2X) = F2, with its smooth extension at X = 0. The exact tangential residual identities follow from Proposition 4.2; together with the support and axis checks above, they complete part (ii).
2416. u2416 For part (v), the exact exterior moments (4.28) are supplied by Propositions A.7 and B.8 and retained by (C.17). The same restoration preserves the heat exterior (4.29), whose exact heat evolution and smoothness are proved in Lemma A.6. We take Xv = Xend ∈(Xa, Xb),
2417. u2417 the pulse-end radius defined before (A.10). Beyond it, U = M = 0 by Proposition A.4 and (C.17); thus AX(U) = M/X = 0, and (4.7) gives V0 = 0. This proves the remaining assertion of part (v).
2418. u2418 Finally, the third and fourth reserved intervals listed after (A.9) are the intervals Ipos, Imean in part (vi). They lie in (Xa, Xv), and their listed order gives sup Ipos < inf Imean. Proposition C.2 leaves them unchanged. The input profiles there satisfy U = 0 and E = K(η)X−1/2−λ, with K(η) = cpatch(1 + η2)−1 for a fixed positive constant cpatch, by Proposition A.4. Thus they retain (4.30), completing part (vi). The finite frequency N was fixed in Proposition C.2 before the physical scale q and all later bands and correction stages, so the constants have
2419. u2419 □ the independence asserted at the end of Theorem 4.6.
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