Optimal regularity results for the Stokes--Dirichlet problem

TL;DR

This paper establishes sharp maximal regularity results for the Stokes--Dirichlet problem in low-regularity domains, covering a broad range of function spaces including endpoint cases, and provides explicit operator descriptions.

Contribution

It develops a comprehensive regularity theory for the Stokes--Dirichlet problem on rough domains, extending classical $L^p$-theory to Besov and Sobolev spaces with endpoint cases.

Findings

Resolved estimates in half-space for endpoint spaces.

Boundedness of the $ extbf{H}^oldsymbol{ ext{infty}}$-calculus for the Stokes--Dirichlet operator.

Explicit description of the Stokes--Dirichlet operator on $L^oldsymbol{ ext{infty}}(oldsymbol{ ext{R}}^n_+)$.

Abstract

We develop a sharp maximal regularity theory for the resolvent and evolution Stokes equations with no-slip boundary conditions, focusing on bounded domains of low regularity. Our framework covers the full scales of Besov and Sobolev spaces, $B^s_{p,q}$ and $H^{s,p}$, including endpoint cases such as $L^\infty$. Our approach also allows extending the classical $L^p$-theory for $1\leqslant p\leqslant\infty$, giving a complete picture that includes both Bessel potential spaces $H^{s,p}$ and Besov spaces $B^s_{p,q}$, $p,q\in[1,\infty]$.\\ Our first main result establishes resolvent estimates in the half-space encompassing endpoint function spaces, while the second addresses bounded domains of minimal boundary regularity. In both cases we derive resolvent bounds, prove boundedness of the $\mathbf{H}^\infty$-functional calculus for the Stokes--Dirichlet operator, and characterize precisely the domains of its fractional powers.\\ In the half space setting, we work with homogeneous Sobolev and Besov spaces following the notion due to Bahouri, Chemin and Danchin, further refined by the second author. The analysis of solenoidal function spaces provides here a complete toolkit for the study of incompressible fluid flows. As a consequence of our analysis, we obtain an explicit description for the Stokes--Dirichlet operator on $L^\infty(\mathbb R^n_+)$, which seems completely new.\\ For bounded domains, we obtain sharp results for a wide class of rough domains under minimal assumptions on boundary regularity. To this end, we rely on Sobolev multiplier theory. The assumptions coincide with those of Maz'ya--Shaposhnikova, already shown to be optimal in the case of the Laplace equation with Dirichlet boundary conditions.