Resolvent estimates for the Stokes operator in bounded and exterior \(C^1\) domains

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