The Dirichlet problem for the Brinkman system is studied in \(W^{1,q}(\Omega ,{\mathbb {R}}^m) \times L^q_\textrm{loc}(\overline{\Omega })\) for unbounded domains \(\Omega \subset {\mathbb {R}}^m\) with compact Lipschitz boundary. If the boundary of \(\Omega \) is of class \({\mathcal {C}}^{k,1}\) we are able to study this problem in \(W^{k+1,q}(\Omega ,{\mathbb {R}}^m)\times D^{k,q}(\Omega )\) . Here \(W^{k,q}(\Omega )\) denotes Sobolev space and \(D^{k,q}(\Omega )\) 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.
