Integral potential method for a transmission problem with Lipschitz interface in ${\mathbb R}^3$ for the Stokes and Darcy-Forchheimer-Brinkman PDE systems

TL;DR

This paper develops an integral potential method to establish existence and uniqueness for transmission problems involving the non-linear Darcy-Forchheimer-Brinkman and linear Stokes systems in Lipschitz domains in ${ m I extbf{R}}^3$, using layer potentials and fixed point theorems.

Contribution

It introduces a novel integral potential approach combined with fixed point techniques to solve transmission problems for complex PDE systems in Lipschitz domains.

Findings

Proves existence and uniqueness of solutions under small data conditions.

Extends layer potential methods to non-linear Darcy-Forchheimer-Brinkman systems.

Provides a framework for transmission problems in Lipschitz geometries.

Abstract

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in ${\mathbb R}^3$, one of them is a bounded Lipschitz domain $\Omega $ with connected boundary, and another one is the exterior Lipschitz domain ${\mathbb R}^3\setminus \overline{\Omega }$. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces.