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

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