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

Abstract

Given any divergence-free vector field of Sobolev class $W_0^{m,p}(\hat{I}\text{c◯})$ in a bounded open subset $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields compactly supported in $\hat{I}\text{c◯}$. We show that this approximation property holds in the following cases: For $p>2$, this holds given that $\partial \hat{I}\text{c◯}$ has zero Lebesgue measure (a weaker but more technical condition is sufficient); For $p\le 2$, this holds if $\hat{I}{\text{c◯}}^c$ can be decomposed into finitely many disjoint closed sets, 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 $\hat{I}\text{c◯}$. For Hölder spaces, we prove this approximation property in general bounded domains.

31 pages