On the Geometric Regularity Conditions for the 3D Navier-Stokes Equations

Dongho Chae, Jihoon Lee

Published 2016-06-27Version 1

Let $v$ and $\omega$ be the velocity and the vorticity of a suitable weak solutions of the 3D Navier-Stokes equations in a space-time domain containing $z_0= (x_0, t_0)$, and let $Q_{z_0, r}=B_{x_0, r} \times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if either $\left(v \times \frac{\omega}{|\omega|}\right) \cdot \frac{\nabla \times \omega}{|\nabla \times \omega|} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 1$, or $\left(\omega \times \frac{v}{|v|}\right) \cdot \frac{\nabla \times \omega}{|\nabla \times \omega|} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 2$, ($\gamma \geq 2$, $\alpha \geq 2$), where $L^{\gamma, \alpha}_{x,t}$ denote the Serrin type of class, then $z_0$ is a regular point for $v$. This improves previous local regularity criteria for the suitable weak solutions.