Some Remarks on Regularity Criteria of Axially Symmetric Navier-Stokes Equations

TL;DR

This paper establishes new regularity criteria for axially symmetric Navier-Stokes equations using log supercritical and subcritical assumptions, and provides explicit Green function estimates to aid in analyzing solution regularity.

Contribution

It introduces novel regularity criteria under specific log-critical conditions and derives explicit Green function formulas for the associated operator.

Findings

Proves regularity under log supercritical conditions on radial and vertical velocity components.

Provides explicit Green function and weighted L^p estimates for the operator involved.

Establishes regularity results based on critical assumptions on the angular vorticity component.

Abstract

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^\theta$ of the velocity. Second, the precise Green function for the operator $-(\Delta-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^\theta$ of the vorticity.