Boundary Value Problems of Robin Type for the Brinkman and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains

Abstract

The purpose of this paper is to study boundary value problems of Robin type for the Brinkman system and a semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system, on Lipschitz domains in Euclidean setting. In the first part of the paper, we exploit a layer potential analysis and a fixed point theorem to show the existence and uniqueness of the solution to the nonlinear Robin problem for the Darcy–Forchheimer–Brinkman system on a bounded Lipschitz domain in $${\mathbb{R}^n}$$ $${(n \in \{2,3\})}$$ with small data in L 2-based Sobolev spaces. In the second part, we show an existence result for the mixed Dirichlet–Robin problem for the same semilinear Darcy–Forchheimer-Brinkman system on a bounded creased Lipschitz domain in $${\mathbb{R}^3}$$ with small L 2-boundary data. We also study mixed Dirichlet–Robin problems and boundary value problems of mixed Dirichlet–Robin and transmission type for Brinkman systems on bounded creased Lipschitz domains in $${\mathbb{R}^n}$$ (n ≥ 3). Finally, we show the well-posedness of the Navier problem for the Brinkman system with boundary data in some L 2-based Sobolev spaces on a bounded Lipschitz domain in $${\mathbb{R}^3}$$ .