Maximal regularity for the Stokes operator in bounded $C^1$ domains

Abstract

Let $Ω$ be a bounded $C^1$ domain in $\mathbb{R}^d$, $d\geq2$. We establish vector-valued resolvent estimates for the Dirichlet Stokes system in $L^q(Ω;\ell^2)$ for every $1<q<\infty$. The estimates are uniform in the number and choice of spectral parameters in any fixed sector avoiding the negative real axis. These estimates imply that the Stokes operator is $\mathcal R$-sectorial of angle zero and has maximal $L^r$-regularity for every $1<r<\infty$. Our proof builds on the perturbation approach of Geng and Shen, with a different half-space argument to handle independently varying spectral parameters.