Optimal regularity results for the Stokes-Dirichlet problem

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, $\mathrm{B}^s_{p,q}$ and $\mathrm{H}^{s,p}$, including endpoint cases such as $\mathrm{L}^\infty$. In particular, for the evolution problem, we show that in many situations the spatial regularity of $\partial_t \mathbf{u}$, $-\Delta \mathbf{u} +\nabla\mathfrak{p}$ in $\mathrm{B}^s_{p,q}$ can be controlled by the forcing term $\mathbf{f}$ in the same space with suitable time integrability, but also provide the exact amount of regularity for the boundary that allows to control separately $\nabla^2 \mathbf{u}$ and $\nabla \mathfrak{p}$. Our approach also allows extending the classical $\mathrm{L}^p$-theory for $1\leqslant p\leqslant\infty$, giving a complete picture that includes both Bessel potential spaces $\mathrm{H}^{s,p}$ and Besov spaces $\mathrm{B}^s_{p,q}$, $p,q\in[1,\infty]$.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. These spaces possess well-defined product laws, making them suitable for applications to the Navier-Stokes equations and other boundary value problems. Yet, we are due to analyze solenoidal subspaces and properties of the plain Laplacian in this framework. The analysis of solenoidal function spaces provides here a complete toolkit for the study of incompressible fluid flows. The resolvent estimate is then derived via Ukaï-type formulas, which yield new insights even in the classical half-space case. Nevertheless, new arguments are needed for the endpoint spaces, motivating the approach by homogeneous function spaces. As a consequence of our analysis, we obtain an explicit description for the Stokes-Dirichlet operator on $\mathrm{L}^\infty(\mathbb{R}^n_+)$, which seems completely new.For bounded domains, our method combines standard localization and flattening with the half-space estimates. 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. Specifically, we prove resolvent estimates in $\mathrm{B}^s_{p,q}$ and $\mathrm{H}^{s,p}$ provided the boundary charts have a small multiplier norm on the trace space $\mathrm{B}^{s+2-1/p}_{p,q}$. As a special case of our general strategy, we establish a full $\mathrm{L}^p$-theory for bounded $\mathrm{C}^{1,\alpha}$-domains for all $p\in(1,\infty)$, $\alpha>0$. In this case, solutions lie nearly in $\mathrm{H}^{1+\alpha+1/p,p}(\Omega)$. Moreover, for such domains, the resolvent problem in $\mathrm{L}^\infty(\Omega)$ produces solutions in $\mathrm{C}^{1,\alpha}(\overline{\Omega})$. Finally, we also obtain a $\mathrm{L}^1$-type theory in the Sobolev spaces $\mathrm{W}^{s,1}$.