The Dirichlet problem for the Brinkman system in exterior domains

Abstract The Dirichlet problem for the Brinkman system is studied in $$W^{1,q}(\Omega ,{\mathbb {R}}^m) \times L^q_\textrm{loc}(\overline{\Omega })$$ W 1 , q ( Ω , R m ) × L loc q ( Ω ¯ ) for unbounded domains $$\Omega \subset {\mathbb {R}}^m$$ Ω ⊂ R m with compact Lipschitz boundary. If the boundary of $$\Omega $$ Ω is of class $${\mathcal {C}}^{k,1}$$ C k , 1 we are able to study this problem in $$W^{k+1,q}(\Omega ,{\mathbb {R}}^m)\times D^{k,q}(\Omega )$$ W k + 1 , q ( Ω , R m ) × D k , q ( Ω ) . Here $$W^{k,q}(\Omega )$$ W k , q ( Ω ) denotes Sobolev space and $$D^{k,q}(\Omega )$$ D k , q ( Ω ) denotes homogeneous Sobolev space. We use the integral equation method. So, we are interested about behavior of volume potentials and also boundary layer potentials. As a consequence of results for the Brinkman system we prove the existence of a solution of the Dirichlet problem for the Darcy–Forchheimer–Brinkman system in the same function spaces.