The Navier-Stokes equations in the critical Lebesgue space

Abstract

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $u\in L_\infty^tL_{d}^x((0,T)\times {\mathbb R}^d)$ is a Leray-Hopf weak solution, then $u$ is smooth and unique in $(0,T)\times \bR^d$. This generalizes a result by Escauriaza, Seregin and \v{S}ver\'ak. Additionally, we show that if $T=\infty$ then $u$ goes to zero as $t$ goes to infinity.