The Navier-Stokes equations in the critical Lebesgue space

Details The Navier-Stokes equations in the critical Lebesgue space The Navier-Stokes equations in the critical Lebesgue space

2009-03-08

arxiv.org/abs/0903.1461v2

Mathematics

Analysis of PDEs

20 pages, to appear in Comm. Math. Phys

Scientific paper

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.

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