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

Abstract

We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let $v$ and $\omega$ be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote $\{f{\}}_{+}=\max \{f,0\}$ , $Q_T={\mathbb{R}}^3\times (0,T)$. If ${\left\{\left(v\times \frac{\omega }{|\omega |}\right)\cdot \frac{{\Lambda }^{\beta }v}{|{\Lambda }^{\beta }v|}\right\}}_{+}\in L_{x,t}^{\gamma ,\alpha }(Q_T)$ with $3/\gamma +2/\alpha \le 1$ for some $\gamma >3$ and $1\le \beta \le 2$, then the local smooth solution $v$ of the Navier-Stokes equations on $(0,T)$ can be continued to $(0,T+\delta )$ for some $\delta >0$. We also prove localized version of a special case of this. Let $v$ be a suitable weak solution to the Navier-tokes equations in a space-time domain containing $z_0=(x_0,t_0)$, 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 ${\left\{\left(v\times \frac{\omega }{|\omega |}\right)\cdot \frac{\nabla \times \omega }{|\nabla \times \omega |}\right\}}_{+}\in L_{x,t}^{\gamma ,\alpha }(Q_{z_0,r})$ with $\frac{3}{\gamma }+\frac{2}{\alpha }\le 1$, or ${\left\{\left(\frac{v}{|v|}\times \omega \right)\cdot \frac{\nabla \times \omega }{|\nabla \times \omega |}\right\}}_{+}\in L_{x,t}^{\gamma ,\alpha }(Q_{z_0,r})$ with $\frac{3}{\gamma }+\frac{2}{\alpha }\le 2$, ($\gamma \ge 2$, $\alpha \ge 2$), then $z_0$ is a regular point for $v$. This improves previous local regularity criteria for the suitable weak solutions.

12 pages