A Note On Local Regularity of Axisymmetric Solutions to the Navier-Stokes Equations

TL;DR

This paper introduces a new slightly supercritical condition that ensures local regularity of axisymmetric solutions to the 3D Navier-Stokes equations, extending existing results in the field.

Contribution

It proposes a novel supercritical condition that broadens the class of solutions known to be locally regular in axisymmetric Navier-Stokes flows.

Findings

The new condition guarantees local regularity under broader circumstances.

It generalizes nearly all known local regularity results for weak axisymmetric solutions.

The approach advances understanding of regularity criteria in fluid dynamics.

Abstract

In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local regularity theory of weak axisymmetric solutions.