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

TL;DR

This paper establishes boundary regularity criteria for solutions to the Navier-Stokes equations in critical Lebesgue spaces, extending known results to higher dimensions and boundary domains, with implications for solution smoothness and decay.

Contribution

It proves new boundary regularity conditions for Navier-Stokes solutions in critical Lebesgue spaces, generalizing previous results to higher dimensions and boundary domains.

Findings

Solutions are regular up to the boundary under certain Lebesgue space conditions.

Solutions tend to zero as time approaches infinity in unbounded domains.

Hölder continuity is established for solutions in half-cylinder domains.

Abstract

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times \mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and the pressure $p$ satisfies a local condition $\|p\|_{L_{2-1/d}(Q(z_0,1)\cap (0,T)\times \mathbb{R}^d_+)}\leq K$ 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 $t\rightarrow \infty$. We also study the local problem in half unit cylinder $Q^+$ and prove that if $u\in L^t_{\infty}L^x_d(Q^+)$ and $ p\in L_{2-1/d}(Q^+)$, then $u$ is H\"{o}lder continuous in the closure of the set $Q^+(1/4)$. This generalizes a result by Escauriaza, Seregin, and \v{S}ver\'{a}k to higher dimensions and domains with boundary.