On regularity and singularity for $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ solutions to the Navier-Stokes equations

Abstract

We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier-Stokes equations. We present a new regularity criterion for the weak solution $u$ satisfying the condition $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ without any smallness assumption on that scale, where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time $t$. The condition that the weak Lebesgue space norm of the veclocity field $u$ is bounded in time is encompassing type I singularity and significantly weaker than the end point case of the so-called Ladyzhenskaya-Prodi-Serrin condition proved by Escauriaza-Sergin-Šverák.