Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain

Abstract

In this note we prove that the solution of the stationary and the instationary Stokes equations subject to perfect slip boundary conditions on a 2D wedge domain admits optimal regularity in the $L^p$-setting, i.p. it is $W^{2,p}$ in space. This improves known results in the literature to a large extend. For instance, in [21, Theorem 1.1 and Corollary 3] it is proved that the Laplace and the Stokes operator in the underlying setting have maximal regularity. In that result the range of p admitting $W^{2,p}$ regularity, however, is restricted to the interval $1<p<1+\delta$ for small $\delta>0$, depending on the opening angle of the wedge. This note gives a detailed answer to the question, whether the optimal Sobolev regularity extends to the full range $1<p<\infty$. We will show that for the Laplacian this does only hold on a suitable subspace, but, depending on the opening angle of the wedge domain, not for every $p\in(1,\infty)$ on the entire $L^p$-space. On the other hand, for the Stokes operator in the space of solenoidal fields $L^p_{\sigma}$ we obtain optimal Sobolev regularity for the full range $1<p<\infty$ and for all opening angles less that $\pi$. Roughly speaking, this relies on the fact that an existing $ bad$ part of $L^p$ for the Laplacian is complementary to the space of solenoidal vector fields.