Resolvent Estimates for the Stokes Operator in Bounded and Exterior $C^1$ Domains

We establish resolvent estimates in $L^q$ spaces for the Stokes operator in a bounded $C^1$ domain $\mathrm{\Omega }$ in ${\mathbb{R}}^d$. As a corollary, it follows that the Stokes operator generates a bounded analytic semigroup in $L^q(\mathrm{\Omega };{\mathbb{C}}^d)$ for any $1<q<\mathrm{\infty }$ and $d\ge 2$. The case of an exterior $C^1$ domain is also studied.