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

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.