Unconditional Axis-Regularity in the 5D Corridor

TL;DR

This paper investigates axis regularity for 3D axisymmetric Navier--Stokes equations by transforming the problem into a 5D setting, establishing key weighted estimates and a Morrey iteration approach.

Contribution

It introduces a novel 5D radial lift formulation and derives new weighted estimates to analyze axis regularity in the Navier--Stokes equations.

Findings

Established Hardy--Campanato decay estimate for the parabolic core

Derived weighted Friedrichs--Poincaré estimate for vorticity

Proved a localized weighted quartic estimate for the swirl source

Abstract

We study axis regularity for the three-dimensional axisymmetric incompressible Navier--Stokes equations through a five-dimensional radial lift with weighted measure \[ d\mu_5=r^3\,dr\,dz. \] 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--Poincar\'e estimate for the renormalized vorticity branch, and a localized weighted quartic estimate for the swirl source. The distinguished corridor \[ \alpha\in\left(\frac34,1\right) \] 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.