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

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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.

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Authors and Affiliations

  1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Dagmar Medková

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  1. Dagmar Medková
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Correspondence to Dagmar Medková.

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The work was supported by RVO: 67985840 and GAČR Grant No. 17-01747S.

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Medková, D. The Robin problem for the Brinkman system and for the Darcy–Forchheimer–Brinkman system. Z. Angew. Math. Phys. 69, 132 (2018). https://doi.org/10.1007/s00033-018-1020-z

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