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Boundary Value Problems of Robin Type for the Brinkman and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains

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The purpose of this paper is to study boundary value problems of Robin type for the Brinkman system and a semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system, on Lipschitz domains in Euclidean setting. In the first part of the paper, we exploit a layer potential analysis and a fixed point theorem to show the existence and uniqueness of the solution to the nonlinear Robin problem for the Darcy–Forchheimer–Brinkman system on a bounded Lipschitz domain in \({\mathbb{R}^n}\) \({(n \in \{2,3\})}\) with small data in L 2-based Sobolev spaces. In the second part, we show an existence result for the mixed Dirichlet–Robin problem for the same semilinear Darcy–Forchheimer-Brinkman system on a bounded creased Lipschitz domain in \({\mathbb{R}^3}\) with small L 2-boundary data. We also study mixed Dirichlet–Robin problems and boundary value problems of mixed Dirichlet–Robin and transmission type for Brinkman systems on bounded creased Lipschitz domains in \({\mathbb{R}^n}\) (n ≥ 3). Finally, we show the well-posedness of the Navier problem for the Brinkman system with boundary data in some L 2-based Sobolev spaces on a bounded Lipschitz domain in \({\mathbb{R}^3}\) .

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Mirela Kohr

  2. Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, Padua, 35121, Italy

    Massimo Lanza de Cristoforis

  3. Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring, 57, 70569, Stuttgart, Germany

    Wolfgang L. Wendland

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  1. Mirela Kohr
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  2. Massimo Lanza de Cristoforis
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  3. Wolfgang L. Wendland
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Correspondence to Massimo Lanza de Cristoforis.

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Communicated by M. Feistauer

The work of Mirela Kohr was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0994. Some of the research for this paper was done in August, 2013, while Mirela Kohr visited the Department of Mathematics of the University of Toronto. She is grateful to the members of this department for their hospitality during that visit.

Massimo Lanza de Cristoforis acknowledges the support of ‘GNAMPA-Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni’ and of the project ‘Singular perturbation problems for differential operators’ of the University of Padua.

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Kohr, M., de Cristoforis, M.L. & Wendland, W.L. Boundary Value Problems of Robin Type for the Brinkman and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains. J. Math. Fluid Mech. 16, 595–630 (2014). https://doi.org/10.1007/s00021-014-0176-3

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