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A Schauder theory for the Stokes equations in rough domains

Dominic Breit

Abstract

We consider the steady Stokes equations in a bounded domain with forcing in divergence form supplemented with no-slip boundary conditions. We provide a maximal regularity theory in Campanato spaces (inlcuding $\mathrm{BMO}$ and $C^{0,α}$ for $0<α<1$ as special cases) under minimal assumptions on the regularity of the underlying domain. Our approach is based on pointwise multipliers in Campanto spaces.

A Schauder theory for the Stokes equations in rough domains

Abstract

We consider the steady Stokes equations in a bounded domain with forcing in divergence form supplemented with no-slip boundary conditions. We provide a maximal regularity theory in Campanato spaces (inlcuding and for as special cases) under minimal assumptions on the regularity of the underlying domain. Our approach is based on pointwise multipliers in Campanto spaces.
Paper Structure (12 sections, 7 theorems, 109 equations)

This paper contains 12 sections, 7 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. Function spaces
  5. Campanato spaces
  6. Parametrisation of domains
  7. The Sokes equations in the half space
  8. Decay estimates
  9. Campanto estimates
  10. The Stokes equations in rough domains
  11. The problem in divergence form
  12. The problem in non-divergence form

Key Result

Theorem 3.1

Suppose that we have $\bfF\in L^2(\mathbb H)$ and that $(\bfu,\pi)\in \mathcal{D}^{1,2}_{0,\mathop{\mathrm{div}}\nolimits}(\mathbb H)\times L^2(\mathbb H)$ is the solution to eq:Stokeshalf with $h=0$. Then we have for all $x_0\in \partial\mathbb H$, all $r>0$ and all $\theta\in(0,1)$.

Theorems & Definitions (16)

  • Definition 2.1
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • proof : Proof of Corollary \ref{['cor:camH']}
  • Corollary 3.5
  • ...and 6 more