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

TL;DR

This paper establishes geometrically improved regularity criteria for the 3D Navier-Stokes equations, enabling the continuation of solutions under new conditions involving vorticity and velocity orientations, thus advancing understanding of solution regularity.

Contribution

The paper introduces novel geometric regularity conditions involving velocity and vorticity orientations that extend and improve previous criteria for solution regularity in 3D Navier-Stokes equations.

Findings

Proved a generalized blow-up criterion based on geometric conditions.

Established localized regularity criteria for weak solutions.

Improved previous regularity conditions for suitable weak solutions.

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.