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Optimal regularity results for the Stokes--Dirichlet problem

Dominic Breit, Anatole Gaudin

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.

Optimal regularity results for the Stokes--Dirichlet problem

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, and , including endpoint cases such as . Our approach also allows extending the classical -theory for , giving a complete picture that includes both Bessel potential spaces and Besov spaces , .\\ 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 -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 , 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.

Paper Structure

This paper contains 88 sections, 149 theorems, 1334 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Historical overview and first motivations
  3. Homogeneous function spaces
  4. Domains with minimal regularity
  5. Illustration by the Dirichlet Laplacian
  6. Towards the Stokes--Dirichlet problem
  7. Precise framing of the main questions and the related state of the art
  8. Main results
  9. Homogeneous Sobolev and Besov estimates on the half space
  10. Loo--regularity results on the half space
  11. Lp--Sobolev and Besov regularity theory for domains of minimal regularity
  12. Loo-regularity theory on bounded C1a-domains
  13. L1-type regularity theory on bounded C1alpha-domains
  14. The Leray projection for bounded domains of low regularity, including the endpoint spaces
  15. Technical biases, technical issues and subsequent additional results
  16. ...and 73 more sections

Key Result

Theorem 2.1

Let $(\mathrm{X}_0,\mathrm{X}_1)$ and $(\mathrm{Y}_0,\mathrm{Y}_1)$ be two pairs of compatible couples, and let be two bounded linear operators such that $\mathfrak{R}\mathfrak{E}=\mathrm{I}_{\mathrm{X}_0+\mathrm{X}_1}$, i.e. $\mathfrak{E}$ is a right bounded inverse for $\mathfrak{R}$. Then for all $\theta\in(0,1)$, $p\in[1,\infty]$, one has The result still holds, replacing $(\cdot,\cdot)_{\th

Figures (6)

  • Figure 1: This figure shows the local Lipschitz parametrisations of the boundary $\partial\Omega$ of some Lipschitz domain $\Omega$. The boundary can be covered neighbourhoods such that their intersection with $\partial\Omega$ arises -- after translating and rotating -- as the graph of a Lipschitz function.
  • Figure 2: Function spaces for which $\nabla\mathcal{T}\varphi$ has membership in the corresponding multiplier class. Borderline cases of regularity $s=\alpha+1/p$, $p< r$, and $s=\frac{r\alpha+1}{p}$, $p> r$, might not be reached. When $s\geqslant 0$, it corresponds to \ref{['eq:ProofMultiplierEstLpRange p < r']} and \ref{['eq:ProofMultiplierEstLpRange q >r']}. For negative $s$ this is just the dual range.
  • Figure 3: Representation of $(s,1/p)$ for the isomorphism property in point (i) of above Theorem \ref{['thm:DirLapC1Domains']}.
  • Figure 4: Representation of $(s,1/p)$, for Bessel potential spaces that coincides with the domain of an appropriate fractional power of Dirichlet Laplacian, the parallelogram such that $p\in(1,\infty)$ and $|\frac{1}{p}-\frac{1}{2}|+2|s|<2$. It also includes the first iteration below (the first bullet point).
  • Figure 5: Stabilitised function spaces for a given boundary $\partial\Omega$ in the multiplier class $\mathcal{M}^{1+\alpha,r}_\mathrm{W}$ and associated maximal gain of Sobolev regularity for the solutions of the Stokes--Dirichlet (resolvent) system.
  • ...and 1 more figures

Theorems & Definitions (289)

  • Theorem 2.1: bookTriebel1978
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proof 1
  • Corollary 2.5
  • Proof 2
  • Theorem 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 279 more