Regularity criterion of weak solutions to the Navier-Stokes equations

Abstract

In a domain $\Omega \subset \mathbb{R}^n $, consider a weak solution $u$ of the Navier-Stokes equations in the class $u \in L^{\infty}(0, T; L^n(\Omega))$. If $\limsup_{t\to t_*-0}\|u(t)\|_n^n -\|u(t_*)\|_n^n$ is small at each point of $t_*\in(0, T)$, then $u$ is regular on $\bar{\Omega}\times (0, T)$. As an application, we give a precise characterization of the singular time; i.e., we show that if a solution $u$ of the Navier-Stokes equations is initially smooth and loses its regularity at some later time $T_* < T$, then either $\limsup_{t\to T_*-0}\|u(t)\|_{L^n(\Omega)} =+ \infty$, or $u(t)$ oscillates in $L^n(\Omega)$ around the weak limit w-$\lim_{t\to T_*-0}u(t)$ with sufficiently large amplitude. Furthermore, we prove that every weak solution $u$ of bounded variation on $(0, T)$ with values in $L^n(\Omega)$ becomes regular.