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Finite-Energy Solutions for Compressible Two-Fluid Stokes System

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Abstract

We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions–Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier–Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via an unconventional approximation using the Lagrangian formulation, truncations, and a stability result of trajectories for rough velocity fields.

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Acknowledgements

The author D.B. is partly supported by the ANR- 13-BS01- 0003-01 project DYFICOLTI, by the ANR-16-CE06-0011-02 FRAISE and by the project TelluS-INSMI-MI (INSU) CNRS. The authors P.B.M. and E.Z. have been partly supported by National Science Centre grant 2014/14/M/ST1/00108 (Harmonia). The authors want to thank P.E. Jabin and the anonymous referee for their valuable remarks about the paper.

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Authors and Affiliations

  1. LAMA CNRS UMR 5127, Université Grenoble–Alpes, Université Savoie Mont–Blanc, Bat. Le Chablais, Campus scientifique, 73376, Le Bourget du Lac, France

    Didier Bresch

  2. Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warszawa, Poland

    Piotr B. Mucha

  3. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Ewelina Zatorska

Authors
  1. Didier Bresch
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  2. Piotr B. Mucha
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  3. Ewelina Zatorska
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Correspondence to Didier Bresch.

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Communicated by T.-P. Liu

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Bresch, D., Mucha, P.B. & Zatorska, E. Finite-Energy Solutions for Compressible Two-Fluid Stokes System. Arch Rational Mech Anal 232, 987–1029 (2019). https://doi.org/10.1007/s00205-018-01337-6

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  • DOI: https://doi.org/10.1007/s00205-018-01337-6

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