Approximation of divergence-free vector fields vanishing on rough planar sets

Giacomo Del Nin, Bian Wu
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Abstract

Given any divergence-free vector field of Sobolev class $W^{m,p}_0(\Omega)$ in bounded open subset $\Omega \subset \mathbb{R}^2$, we are interested in approximating it in $W^{m,p}$ with divergence-free smooth vector fields compactly supported in $\Omega$. We show that this approximation property holds in the following cases. For $p>2$, this holds given that $\partial \Omega$ has zero Lebesgue measure (a weaker but more technical condition is sufficient); For $p \leq 2$, this holds if $\Omega^c$ can be decomposed into finitely many disjoint closed set, each of which is connected or $d$-Ahlfors regular for some $d\in[0,2]$. This has links to the uniqueness of weak solutions to the Stokes equation in $\Omega$. For H\"older spaces, we prove this property in general bounded domains.
粗糙平面集上消失的无发散向量场的近似值
给定在有界开放子集 $\Omega \子集 \mathbb{R}^2$中任何索波列夫类 $W^{m,p}_0(\Omega)$ 的无发散向量场,我们感兴趣的是在 $W^{m,p}$ 中用紧密支撑在 $\Omega$ 中的无发散光滑向量场来近似它。我们将证明这一近似性质在以下情况下成立。对于$p>2$,只要$\partial \Omega$的Lebesgue度量为零(一个较弱但技术性更强的条件就足够了),这个近似性质就成立;对于$p (leq 2$),如果$\Omega^c$可以分解为有限多个二交闭集,其中每个都是连通的,或者对于[0,2]$中的某个$d$来说是$d$-Ahlfors正则的,这个近似性质就成立。这与$\Omega$中斯托克方程弱解的唯一性有关。对于 H\"older 空间,我们在一般有界域中证明了这一性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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