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

TL;DR

This paper establishes optimal regularity, existence, and uniqueness of solutions for the Stokes equations with Navier boundary conditions on a 2D wedge domain, extending previous results to inhomogeneous boundary conditions.

Contribution

It generalizes previous regularity results for the Stokes equations on wedge domains to include inhomogeneous Navier boundary conditions.

Findings

Proves existence and uniqueness of solutions with optimal regularity.

Extends regularity results to inhomogeneous boundary conditions.

Utilizes detailed analysis of trace operators on anisotropic Sobolev spaces.

Abstract

We consider the Stokes equations subject to Navier boundary conditions on a two-dimensional wedge domain with opening angle $\theta_0 \in (0,\,\pi)$. 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.