Regularity of Solution to Axis-symmetric Navier-Stokes Equations with a Slightly Supercritical Condition

TL;DR

This paper proves the regularity of axis-symmetric solutions to 3D Navier-Stokes equations under a slightly supercritical condition, extending previous critical regularity results.

Contribution

It introduces a new regularity criterion for axis-symmetric Navier-Stokes solutions under supercritical assumptions, broadening the understanding of solution behavior.

Findings

Regularity established under supercritical conditions

Extends previous critical regularity results

Applicable to solutions with nontrivial swirl

Abstract

Consider an axis-symmetric suitable weak solution of 3D incompressible Navier-Stokes equation with nontrivial swirl. If the solution satisfies a slightly supercritical assumption, we will prove that v is regular. This extends the results of Chen-Strain-Tsai-Yau, Koch-Nadirashvili-Seregin-Sverak and Lei-Zhang where regularities under critical assumptions were proven.