Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains. (21st October 2014)

• Record Type: Journal Article
• Authors: Groşan, Teodor; Kohr, Mirela; Wendland, Wolfgang L.
• Is Part Of: Mathematical methods in the applied sciences. Volume 38:Number 17(2015:Nov. 30)
• Page Start: 3615
• Page End: 3628
• Publication Date: 2014-10-21
• Subjects: nonlinear generalized Darcy–Forchheimer–Brinkman system -- Stokes and Brinkman systems -- Lipschitz domain -- Dirichlet problem -- layer potentials -- nonlinear operator -- fixed point theorem; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3302
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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 R 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 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 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.