A Poisson Problem of Transmission-type for the Stokes and Generalized Brinkman Systems in Complementary Lipschitz Domains in $\mathbb{R}^3$

Abstract

The purpose of this paper is to give a well-posedness result for a boundary value problem of transmission-type for the Stokes and generalized Brinkman systems in two complementary Lipschitz domains in R 3 . In the first part of the paper, we have introduced the classical and weighted L 2 -based Sobolev spaces on Lipschitz domains in R 3 . Afterwards, the trace and conormal derivative operators are defined in the case of both Stokes and generalized Brinkman systems. Also, a summary of the main properties of the layer potential operators for the Stokes system, is provided. In the second part of the work, we exploit the well-posedness of another transmission problem concerning the Stokes system on two complementary Lipschitz domains in R 3 which is based on the Potential Theory for the Stokes system. Then, certain properties of Fredholm operators will allow us to show our main well-posedness result in L 2 -based Sobolev spaces.

On transmission-type problems for the generalized Darcy-Forchheimer-Brinkman and stokes systems in complementary lipschitz domains in R3

“…Proof. In order to show the statement of our theorem, we use the following arguments (see also, [2,Theorem 4.5]). First, we note that, the Poisson problem of transmission-type for the Stokes system in the bounded Lipschitz domain Γ + and the Stokes system in the complementary Lipschitz set Γ − in R 3 :…”