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 $v\times \fracø{|ø|}\in L^{γ, α}_{x,t} (Q_{z_0, r})$ or $ø\times \frac{v}{|v|}\in L^{γ, α}_{x,t} (Q_{z_0, r})$, where $L^{γ, α}_{x,t}$ 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.