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We show that the Stokes operator defined on $\mathrm{L}^p_{\sigma} (\Omega)$for an exterior Lipschitz domain $\Omega \subset \mathbb{R}^n$ $(n \geq 3)$admits maximal regularity provided that $p$ satisfies $| 1/p - 1/2| < 1/(2n) +\varepsilon$ for some $\varepsilon > 0$. In particular, we prove that thenegative of the Stokes operator generates a bounded analytic semigroup on$\mathrm{L}^p_\sigma (\Omega)$ for such $p$. In addition,$\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the Stokes semigroup andits gradient with optimal decay estimates are obtained. This enables us toprove the existence of mild solutions to the Navier--Stokes equations in thecritical space $\mathrm{L}^{\infty} (0 , T ; \mathrm{L}^3_{\sigma} (\Omega))$(locally in time and globally in time for small initial data).

The Navier--Stokes equations in exterior Lipschitz domains: $\mathrm{L}^p$-theory