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 $u\in L_{\infty }^t{L}_d^x((0,T)\times {\mathbb{R}}^d)$ 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 Šverák. Additionally, we show that if $T=\infty$ then $u$ goes to zero as $t$ goes to infinity.

20 pages, to appear in Comm. Math. Phys

References in corpus (2)

• Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations
• Partial Regularity of solutions to the Four-dimensional Navier-Stokes equations at the first blow-up time

Cited by in corpus (4)

• A profile decomposition approach to the $L_t^{\infty }(L_x^3)$ Navier-Stokes regularity criterion
• Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces
• A minimum critical blowup rate for the high-dimensional Navier-Stokes equations
• On local Type I singularities of the Navier-Stokes equations and Liouville theorems