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

## TL;DR

This paper establishes maximal regularity and semigroup generation for the Stokes operator in exterior Lipschitz domains within certain $ ext{L}^p$ spaces, enabling existence results for Navier--Stokes solutions in critical spaces.

## Contribution

It proves the Stokes operator admits maximal regularity and generates an analytic semigroup in $ ext{L}^p$ spaces for exterior Lipschitz domains, extending the theory to critical spaces.

## Key findings

- Maximal regularity for the Stokes operator in specified $ ext{L}^p$ spaces.
- Generation of a bounded analytic semigroup by the Stokes operator.
- Existence of mild solutions to Navier--Stokes equations in critical $ ext{L}^3$ space.

## Abstract

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 the negative 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 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 $\mathrm{L}^{\infty} (0 , T ; \mathrm{L}^3_{\sigma} (\Omega))$ (locally in time and globally in time for small initial data).