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

Le résumé fourni par la source

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 $1 0$, 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.

Ce résumé expose les affirmations des auteurs. BNTIC ne l’interprète pas comme une validation indépendante des résultats.