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On the Oseen–Brinkman flow around an \((m-1)\)-dimensional solid obstacle

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Abstract

The purpose of this paper is to develop a layer potential analysis in order to show the well-posedness result of a transmission problem for the Oseen and Brinkman systems in open sets in \({\mathbb R}^m\) (\(m\in \{2,3\}\)) with compact Lipschitz boundaries and around a lower dimensional solid obstacle, when the boundary data belong to some \(L^q\)-spaces. If \(m=3\) or if the Brinkman system is given on bounded open set then there exists a solution of the transmission problem for arbitrary data. If \(m=2\) and the Brinkman system is given on exterior open set then necessary and sufficient conditions for the existence of a solution of the transmission problem are stated. A solution of the transmission problem is not unique. All solutions of the problem are found.

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Notes

  1. \(\partial ^\beta \) means the differential operator of order \(\beta \), \(\partial ^\beta :=\frac{\partial ^{|\beta |}}{\partial ^{\beta _1}\ldots \partial ^{\beta _m}}\), where \(\beta =(\beta _1,\ldots ,\beta _m)\) and \(|\beta |=\beta _1+\cdots +\beta _m\).

  2. The conormal derivatives below exist a.e. on \(\partial \Omega \) and are understood in the sense of nontangential limits.

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Acknowledgments

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. The work of Dagmar Medková was supported by RVO: 67985840 and GAČR Grant No. 16-03230S. The work of Wolfgang L. Wendland was supported by the SimTech Cluster of Excellence at the University Stuttgart, and he also gratefully acknowledges advice by Professor Rainer Helmig. Part of this work was done in 2014, when 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.

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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. Institute of Mathematics of the Czech Academy of Sciences, Z̆itná 25, 115 67, Prague 1, Czech Republic

    Dagmar Medková

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

    W. L. Wendland

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  1. Mirela Kohr
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Correspondence to Dagmar Medková.

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Communicated by A. Constantin.

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Kohr, M., Medková, D. & Wendland, W.L. On the Oseen–Brinkman flow around an \((m-1)\)-dimensional solid obstacle. Monatsh Math 183, 269–302 (2017). https://doi.org/10.1007/s00605-016-0981-2

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