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

Abstract

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