Remarks on local regularity of axisymmetric solutions to the 3D Navier--Stokes equations

TL;DR

This paper introduces a new local regularity criterion for axisymmetric solutions to the 3D Navier--Stokes equations, providing an upper bound on the oscillation of a key quantity near the axis, which advances understanding of solution regularity.

Contribution

It proposes a slightly supercritical regularity criterion that bounds the oscillation of the angular velocity component in axisymmetric Navier--Stokes solutions.

Findings

Establishes an exponential decay bound on the oscillation of mma near the axis.

Provides a new criterion that is slightly supercritical.

Enhances understanding of local regularity in axisymmetric flows.

Abstract

In this note, a new local regularity criteria for the axisymmetric solutions to the 3D Navier--Stokes equations is investigated. It is slightly supercritical and implies an upper bound for the oscillation of $\Gamma=r u^{\theta}$: for any $0< \tau<1$, there exists a constant $c>0$, $$ |\Gamma(r,x_{3},t)|\leq N e^{-c\, |\ln r|^{\tau}},\ 0<r\leq \frac{1}{4}. $$