The Stokes operator in two-dimensional bounded Lipschitz domains

TL;DR

This paper establishes $ ext{L}^p$-resolvent estimates and maximal regularity for the Stokes operator in two-dimensional Lipschitz domains, advancing the understanding of Navier-Stokes solutions in irregular geometries.

Contribution

It proves new $ ext{L}^p$-resolvent bounds, shows the Stokes operator has maximal regularity, and characterizes fractional powers, extending regularity results to Lipschitz domains.

Findings

Established $ ext{L}^p$-resolvent estimates for the Stokes operator.

Proved the Stokes operator admits maximal regularity.

Applied results to the regularity of weak Navier-Stokes solutions.

Abstract

We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $\Omega$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutions to the Navier-Stokes equations in bounded planar Lipschitz domains.