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

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