Very weak solution for the exterior stationary Stokes equations with Navier slip boundary condition

TL;DR

This paper investigates the existence and uniqueness of very weak solutions for the stationary Stokes equations with Navier slip boundary conditions in unbounded exterior domains, accommodating irregular data in weighted Sobolev spaces.

Contribution

It establishes a Hilbertian framework for very weak solutions of the exterior stationary Stokes problem with Navier boundary conditions, addressing irregular data and unbounded domains.

Findings

Proves existence of very weak solutions in weighted Sobolev spaces.

Establishes uniqueness of solutions under Navier slip conditions.

Provides a mathematical foundation for fluid flow with slip boundary in exterior domains.

Abstract

In some problems of fluid mechanics, it is possible to be confronted with data that are not regular, that is why we are interested here in the search for the so-called very weak solutions for the stationary Stokes problem with Navier-type boundary conditions in a three-dimensional exterior domain. The problem describes the flow of a viscous and incompressible fluid past an obstacle where we assume that the fluid may slip on the boundary of the obstacle. Because the flow domain is unbounded, we set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the solutions. Our purpose is to prove the existence and the uniqueness of a very weak solution in a Hilbertian framework.