Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain

Abstract

By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish $L^p$ resolvent estimates for the Stokes operator in a bounded Lipschitz domain $Ω$ in $\R^3$ for any $1< p\le \infty$. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in $L^p_σ(Ω)$. The results are particularly surprising, as it is long believed that the $L^p$ resolvent estimates in three-dimensional Lipschitz domains may fail for large $p$. In the Appendix we prove a theorem on interpolation between $L^2_σ(Ω)$ and $L^\infty_σ(Ω)$.