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Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains
Author(s) -
Groşan Teodor,
Kohr Mirela,
Wendland Wolfgang L.
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3302
Subject(s) - mathematics , dirichlet problem , lipschitz continuity , lipschitz domain , sobolev space , mathematical analysis , bounded function , nonlinear system , dirichlet boundary condition , domain (mathematical analysis) , boundary value problem , physics , quantum mechanics
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 inR n ( n = 2 , 3 ) , with small boundary datum in L 2 ‐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 inR n ( n ≥ 2 ) , with Dirichlet boundary condition and data in L 2 ‐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.