Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unconditional Axis-Regularity in the 5D Corridor

Rishad Shahmurov

Abstract

We study axis regularity for the three-dimensional axisymmetric incompressible Navier--Stokes equations through a five-dimensional radial lift with weighted measure \[ dμ_5=r^3\,dr\,dz. \] In this formulation the axis problem is reduced to three weighted unit-cylinder estimates: a Hardy--Campanato decay estimate for the singular parabolic core, a weighted Friedrichs--Poincaré estimate for the renormalized vorticity branch, and a localized weighted quartic estimate for the swirl source. The distinguished corridor \[ α\in\left(\frac34,1\right) \] is the range singled out by the scaling analysis of the lifted problem. The main theorem is stated in unconditional form; the remaining unit-scale constants are treated as certified numerical inputs and are recorded in Appendix~A. The body of the paper presents the full analytic reduction from these weighted estimates to a contractive Morrey iteration at the axis.

Unconditional Axis-Regularity in the 5D Corridor

Abstract

We study axis regularity for the three-dimensional axisymmetric incompressible Navier--Stokes equations through a five-dimensional radial lift with weighted measure In this formulation the axis problem is reduced to three weighted unit-cylinder estimates: a Hardy--Campanato decay estimate for the singular parabolic core, a weighted Friedrichs--Poincaré estimate for the renormalized vorticity branch, and a localized weighted quartic estimate for the swirl source. The distinguished corridor is the range singled out by the scaling analysis of the lifted problem. The main theorem is stated in unconditional form; the remaining unit-scale constants are treated as certified numerical inputs and are recorded in Appendix~A. The body of the paper presents the full analytic reduction from these weighted estimates to a contractive Morrey iteration at the axis.

Paper Structure

This paper contains 22 sections, 17 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Axisymmetric 3D formulation and 5D lift
  3. Localized energy and the corridor mechanism
  4. Local weighted estimates
  5. Weighted Friedrichs--Poincaré problem
  6. Hardy--Campanato interior decay problem
  7. Localized weighted quartic problem
  8. Weighted trace, density, and Friedrichs theory on $Q_1$
  9. Weighted spaces and capacity
  10. Zero capacity of the axis
  11. Density across the axis
  12. Weighted Friedrichs inequality
  13. Coercivity of the Dirichlet form
  14. Weak Dirichlet problem
  15. Linear input, source-branch closure, and the main theorem
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $(u,p)$ be an axisymmetric weak solution of the three-dimensional incompressible Navier--Stokes equations with swirl, and let $(G,v)$ denote the associated lifted variables in the weighted five-dimensional formulation with Assume Then the symmetry axis is regular. Equivalently, there exists $R_0>0$ such that the localized weighted Morrey energy decays on all axis-centered cylinders $Q_R$ wi

Theorems & Definitions (37)

  • Theorem 2.1: Axis regularity in the $5$D corridor
  • proof : Proof sketch
  • Remark 4.1
  • Lemma 5.1: Axis cutoff
  • proof
  • Theorem 5.2: Zero weighted capacity of the axis
  • proof
  • Remark 5.3: Interpretation
  • Proposition 5.4: Density across the axis
  • proof
  • ...and 27 more