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 $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb R^3\times (0, T)$. If $\left\{\left( v \times \frac{\omega}{|\omega|} \right)\cdot \frac{\Lambda^{\beta}v}{|\Lambda^{\beta}v|}\right\}_+ \in L^{\gamma, \alpha}_{x,t} (Q_T)$ with $3/\gamma +2/\alpha \leq 1$ for some $\gamma >3$ and $1 \leq \beta \leq 2$, 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 $\left\{\left( v \times \frac{\omega}{|\omega|}\right) \cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 1$, or $\left\{\left(\frac{v}{|v|} \times \omega\right) \cdot \frac{\nabla \times \omega}{|\nabla \times \omega|}\right\}_{+} \in L^{\gamma, \alpha}_{x,t}(Q_{z_0, r})$ with $\frac{3}{\gamma}+\frac{2}{\alpha} \leq 2$, ($\gamma \geq 2$, $\alpha \geq 2$), then $z_0$ is a regular point for $v$. This improves previous local regularity criteria for the suitable weak solutions.