  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.
