Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces

Abstract

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\in L_{\infty }^t{L}_d^x((0,T)\times {\mathbb{R}}_{+}^d)$ is a Leray-Hopf weak solution vanishing on the boundary and the pressure $p$ satisfies a local condition $\Vert p{\Vert }_{L_{2-1/d}(Q(z_0,1)\cap (0,T)\times {\mathbb{R}}_{+}^d)}\le K$ for some constant $K>0$ uniformly in $z_0$, then $u$ is regular up to the boundary in $(0,T)\times {\mathbb{R}}_{+}^d$. Furthermore, when $T=\infty$, $u$ tends to zero as $t\to \infty$. We also study the local problem in half unit cylinder $Q^{+}$ and prove that if $u\in L_{\infty }^t{L}_d^x(Q^{+})$ and $p\in L_{2-1/d}(Q^{+})$, then $u$ is Hölder continuous in the closure of the set $Q^{+}(1/4)$. This generalizes a result by Escauriaza, Seregin, and Šverák to higher dimensions and domains with boundary.

32 pages, submitted. arXiv admin note: text overlap with arXiv:0903.1461

Cited by in corpus (1)

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