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The Robin problem for the Brinkman system and for the Darcy–Forchheimer–Brinkman system

  • Dagmar MedkováEmail author
Article
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Abstract

In this paper, we study the Neumann problem and the Robin problem for the Darcy–Forchheimer–Brinkman system in \(W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )\) for a bounded domain \(\Omega \subset {\mathbb {R}}^m\) with Lipschitz boundary. First, we study the Neumann problem and the Robin problem for the Brinkman system by the integral equation method. If \(\Omega \subset {\mathbb {R}}^m\) is a bounded domain with Lipschitz boundary and \(2\le m\le 3\), then we prove the unique solvability of the Neumann problem and the Robin problem for the Brinkman system in \(W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )\), where \(3/2<q<3\). Then we get results for the Darcy–Forchheimer–Brinkman system from the results for the Brinkman system using the fixed point theorem. If \(\Omega \subset {\mathbb {R}}^m\) is a bounded domain with Lipschitz boundary, \(2\le m\le 3\), \(3/2<q<3\), then we prove the existence of a solution of the Neumann problem and the Robin problem for the Darcy–Forchheimer–Brinkman system in \(W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )\) for small given data.

Keywords

Brinkman system Neumann problem Robin problem Darcy–Forchheimer–Brinkman system Boundary layer potentials 

Mathematics Subject Classification

35Q35 

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of Mathematics of the Czech Academy of SciencesPraha 1Czech Republic

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