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

摘要

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized DarcyForchheimer-Brinkman system in a bounded Lipschitz domain in R-n(n = 2, 3), with small boundary datum in L-2-based Sobolev spaces. A useful intermediary result is thewell-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in R-n(n >= 2), with Dirichlet boundary condition and data in L-2-based Sobolev spaces. We obtain this well-posedness result by showing that thematrix type operator associated with the Poisson problem is an isomorphism. Then, we combine thewell-posedness result fromthe linear case with a fixed point theoremin 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 (C) 2014 JohnWiley & Sons, Ltd.