Remarks on regularity conditions of the Navier-Stokes equations

Abstract

Let $v$ and $ø$ 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 $v\times \frac{ø}{|ø|}\in L_{x,t}^{\gamma ,\alpha }(Q_{z_0,r})$ or $ø\times \frac{v}{|v|}\in L_{x,t}^{\gamma ,\alpha }(Q_{z_0,r})$, where $L_{x,t}^{\gamma ,\alpha }$ 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.

13 pages