A note on a transmission problem for the Brinkman system and the generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R^3

Abstract

The purpose of this paper is to treat a nonlinear transmission-type problem for a generalized version of the Darcy-Forchheimer-Brinkman system and the classical Brinkman system in complementary Lipschitz domains in R^3. First of all, we define the required spaces in which we seek our solution. Next, we describe the generalized Brinkman and the generalized Darcy-Forchheimer-Brinkman systems. Further, we give important lemmas that allow us to introduce the trace and conormal derivative operators that appear in the formulation of our transmission problem. We invoke a result regarding the well-posedness of the (linear) transmission problem for the generalized and classical Brinkman systems in complementary Lipschitz domains in R^3. The above mentioned well-posedness result in the linear case combined with Banach's fixed point theorem will allow us to establish the main result of the paper, the well-posedness of the transmission problem for the Brinkman system and the nonlinear generalized Darcy-Forchheimer-Brinkman system in Lipschitz domains in R^3.