Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Regularity for the Stokes Equations on a 2D Wedge Domain Subject to Navier Boundary Conditions

Matthias Köhne, Jürgen Saal, Laura Westermann

Abstract

We consider the Stokes equations subject to Navier boundary conditions on a two-dimensional wedge domain with opening angle $θ_0 \in (0,\,π)$. We prove existence and uniqueness of solutions with optimal regularity in an $L^p$-setting. The results are based on optimal regularity results for the Stokes equations subject to perfect slip boundary conditions on a two-dimensional wedge domain that have been obtained by the authors in [15]. Based on a detailed study of the corresponding trace operator on anisotropic Sobolev-Slobodeckij type function spaces on a two-dimensional wedge domain we are able to generalize the results proved in [15] to the case of inhomogeneous boundary conditions. Existence and uniqueness of solutions to the Stokes equations subject to (inhomogeneous) Navier boundary conditions are then obtained using a perturbation argument.

Optimal Regularity for the Stokes Equations on a 2D Wedge Domain Subject to Navier Boundary Conditions

Abstract

We consider the Stokes equations subject to Navier boundary conditions on a two-dimensional wedge domain with opening angle . We prove existence and uniqueness of solutions with optimal regularity in an -setting. The results are based on optimal regularity results for the Stokes equations subject to perfect slip boundary conditions on a two-dimensional wedge domain that have been obtained by the authors in [15]. Based on a detailed study of the corresponding trace operator on anisotropic Sobolev-Slobodeckij type function spaces on a two-dimensional wedge domain we are able to generalize the results proved in [15] to the case of inhomogeneous boundary conditions. Existence and uniqueness of solutions to the Stokes equations subject to (inhomogeneous) Navier boundary conditions are then obtained using a perturbation argument.

Paper Structure

This paper contains 9 sections, 19 theorems, 162 equations.

Table of Contents

  1. Introduction and Main Result
  2. Notation
  3. The Laplace Equation subject to Neumann Boundary Conditions
  4. The Stokes Equations subject to Navier Boundary Conditions
  5. Inhomogeneous Perfect Slip Boundary Conditions
  6. Inhomogeneous Free and Perfect Slip Boundary Conditions
  7. Proof of Theorem \ref{['thm_navierslip']}
  8. Hardy's inequality on the 2D wedge domain
  9. Density of Smooth Functions in $\widehat{W}^{m, p}(\Omega)$

Key Result

Theorem 1.2

Let $J = (0,T)$ with $0 < T < \infty$ and let $G \subset {\mathbb R}^2$ be defined as in def_wedge with $\theta_0 \in (0,\,\pi)$. Let $p \in (1, \infty) \setminus \{ \frac{2 \theta_0}{3 \theta_0- \pi},\ \frac{2 \theta_0}{3 \theta_0 - 2\pi},\ \frac{3}{2},\ 2,\ 3\, \}$. Let $\alpha \in BUC^1(\Gamma)$ and the compatibility conditions as well as If the boundary condition is posed based on $D_+$, th

Theorems & Definitions (38)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • ...and 28 more