Remarks on regularity conditions of the Navier-Stokes equations

TL;DR

This paper refines local regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations by establishing conditions involving the cross product of velocity and vorticity in Serrin-type spaces, ensuring regularity at a point.

Contribution

It introduces new regularity conditions involving cross products of velocity and vorticity in Serrin spaces, improving previous criteria for weak solutions.

Findings

Regularity at a point is guaranteed under new cross product conditions.

Conditions involve Serrin-type integrability of velocity-vorticity cross products.

Refines previous local regularity criteria for Navier-Stokes solutions.

Abstract

Let $v$ and $\o$ be the velocity and the vorticity of the a suitable weak solution of the 3D Navier-Stokes equations in a space-time domain containing $z_0 =(x_0, t_0)$, and $Q_{z_0, r} =B_{x_0, r}\times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if $v\times \frac{\o}{|\o|}\in L^{\gamma, \alpha}_{x,t} (Q_{z_0, r})$ or ${\o}\times \frac{v}{|v|}\in L^{\gamma, \alpha}_{x,t} (Q_{z_0, r})$, where $L^{\gamma, \alpha}_{x,t}$ denotes the Serrin type of class, then $z_0$ is a regular point for $v$. This refines previous local regularity criteria for the suitable weak solutions.