On Endpoint Regularity Criterion of the 3D Navier-Stokes equations

Abstract

Let $(u,\pi )$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in ${\mathbb{R}}^3\times [0,T]$. Denote by ${\dot{\mathcal{B}}}_{\infty ,\infty }^{-1}$ the closure of $C_0^{\infty }$ in ${\dot{B}}_{\infty ,\infty }^{-1}$. We prove that if $u\in L^{\infty }(0,T;{\dot{B}}_{\infty ,\infty }^{-1})$, $u(x,T)\in {\dot{\mathcal{B}}}_{\infty ,\infty }^{-1})$, and $u_3\in L^{\infty }(0,T;L^{3,\infty })$ or $u_3\in L^{\infty }(0,T;{\dot{B}}_{p,q}^{-1+3/p})$ with $3<p,q<\infty$, then $u$ is smooth in ${\mathbb{R}}^3\times [0,T]$. Our result improves a previous result established by Wang and Zhang [Sci. China Math. 60, 637-650 (2017)].