Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions

TL;DR

This paper establishes the well-posedness and regularity of the stationary Stokes equations on an infinite wedge with Navier-slip boundary conditions, addressing challenges from multiple scalings in unbounded domains.

Contribution

It introduces a novel approach to handle multiple scalings in boundary conditions for the Stokes system in wedge-shaped domains, using weighted Sobolev spaces and iterative regularity methods.

Findings

Proved well-posedness in weighted Sobolev spaces.

Achieved higher regularity up to the wedge tip.

Developed a method applicable to other multi-scale variational problems.

Abstract

We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.