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 resolvent estimates in $L^p_\sigma (\Omega)$ for the Stokes operator in a bounded Lipschitz domain $\Omega$ in $\R^3$ for any $(3/2)-\e<p\le \infty$, where $\e>0$ depends on $\Omega$. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in $L^p_\sigma (\Omega)$. 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 Appendices we prove a theorem on interpolation between $L^2_\sigma(\Omega)$ and $L^\infty_\sigma(\Omega)$. Given $1<p<\infty$ and a bounded Lipschitz domain $\Omega$ in $\R^d, d\ge 2$, we show that the resolvent estimate cannot hold in $L^p(\Omega; \C^d)$ unless the Helmholtz projection is bounded on $L^p(\Omega; \C^d)$. This explains the counter-example constructed in \cite{Deuring-2001}.