Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in , with small boundary datum in L2-based Sobolev spaces. A useful intermediary result is the well-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in , with Dirichlet boundary condition and data in L2-based Sobolev spaces. We obtain this well-posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well-posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.

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