The Robin problem for the Brinkman system and for the Darcy–Forchheimer–Brinkman system

Abstract

In this paper, we study the Neumann problem and the Robin problem for the Darcy–Forchheimer–Brinkman system in $$W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )$$ for a bounded domain $$\Omega \subset {\mathbb {R}}^m$$ with Lipschitz boundary. First, we study the Neumann problem and the Robin problem for the Brinkman system by the integral equation method. If $$\Omega \subset {\mathbb {R}}^m$$ is a bounded domain with Lipschitz boundary and $$2\le m\le 3$$ , then we prove the unique solvability of the Neumann problem and the Robin problem for the Brinkman system in $$W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )$$ , where $$3/2<q<3$$ . Then we get results for the Darcy–Forchheimer–Brinkman system from the results for the Brinkman system using the fixed point theorem. If $$\Omega \subset {\mathbb {R}}^m$$ is a bounded domain with Lipschitz boundary, $$2\le m\le 3$$ , $$3/2<q<3$$ , then we prove the existence of a solution of the Neumann problem and the Robin problem for the Darcy–Forchheimer–Brinkman system in $$W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )$$ for small given data.