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The Stokes problem with Navier boundary conditions in irregular domains

Dominic Breit, Sebastian Schwarzacher

TL;DR

This work establishes a sharp maximal-regularity theory for the steady Stokes system with Navier boundary conditions in irregular domains. By developing a Sobolev-multiplier framework and a careful domain parametrisation, the authors reduce local boundary issues to half-space problems and derive $W^{1,p}$ and $W^{2,p}$-type estimates for the velocity and pressure; crucially, they show that exactly one derivative more is required in boundary charts than in the no-slip case. The main contributions include the divergence- and non-divergence-form Stokes results under minimal boundary regularity, the precise multiplier-type conditions on the boundary, and a demonstration of sharpness via explicit constructions. The techniques combine localization, reflection, duality, and very-weak formulations, enabling maximal-regularity results in Lipschitz domains with boundary charts in $\mathscr M^{3-1/p,p}$ (or $\mathscr M^{2-1/p,p}$ for lower-order control).

Abstract

We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for $1<p<\infty$ for the velocity field) in bounded domains of minimal regularity. Interestingly, exactly one derivative more is required for the local boundary charts compared to the case of no-slip boundary conditions. We demonstrate the sharpness of our results by a propos examples.

The Stokes problem with Navier boundary conditions in irregular domains

TL;DR

This work establishes a sharp maximal-regularity theory for the steady Stokes system with Navier boundary conditions in irregular domains. By developing a Sobolev-multiplier framework and a careful domain parametrisation, the authors reduce local boundary issues to half-space problems and derive and -type estimates for the velocity and pressure; crucially, they show that exactly one derivative more is required in boundary charts than in the no-slip case. The main contributions include the divergence- and non-divergence-form Stokes results under minimal boundary regularity, the precise multiplier-type conditions on the boundary, and a demonstration of sharpness via explicit constructions. The techniques combine localization, reflection, duality, and very-weak formulations, enabling maximal-regularity results in Lipschitz domains with boundary charts in (or for lower-order control).

Abstract

We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces and for for the velocity field) in bounded domains of minimal regularity. Interestingly, exactly one derivative more is required for the local boundary charts compared to the case of no-slip boundary conditions. We demonstrate the sharpness of our results by a propos examples.

Paper Structure

This paper contains 17 sections, 9 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Sharpness of the boundary regularity.
  3. The structure of the paper.
  4. Preliminaries
  5. Conventions
  6. Classical function spaces
  7. Fractional differentiability and Sobolev mulitpliers
  8. Parametrisation of domains
  9. The problem in the half space
  10. The problem in divergence form
  11. The problem in non-divergence form
  12. Duality
  13. Duality from $L^{p'}$ to $L^p$
  14. Very weak solutions
  15. Duality from $W^{2,p'}$ to $L^{p}$ and $W^{2+s,p}$ to $(W^{s,p}_n)'$
  16. ...and 2 more sections

Key Result

Theorem 2.2

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • proof : Proof of Theorem \ref{['thm:stokessteadyF']}
  • Corollary 3.6
  • proof
  • ...and 11 more