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

TL;DR

This paper proves that the Stokes equations on a 2D wedge domain have optimal regularity in the full range of p-values, extending previous results limited to a smaller p-range, especially for solenoidal fields.

Contribution

It establishes full-range $W^{2,p}$ regularity for the Stokes operator on a wedge domain, improving upon prior partial results and clarifying the role of solenoidal fields.

Findings

Optimal Sobolev regularity holds for all $1<p< $ in the solenoidal space.

Regularity results depend on the wedge opening angle, with less regularity for the Laplacian.

The full $p$-range regularity is achieved for the Stokes operator in solenoidal spaces.

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.