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Euler System with a Polytropic Equation of State as a Vanishing Viscosity Limit

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Abstract

We consider the Euler system of gas dynamics endowed with the incomplete \((e - \varrho - p)\) equation of state relating the internal energy e to the mass density \(\varrho \) and the pressure p. We show that any sufficiently smooth solution can be recovered as a vanishing viscosity-heat conductivity limit of the Navier–Stokes–Fourier system with a properly defined temperature. The result is unconditional in the case of the Navier type (slip) boundary conditions and extends to the no-slip condition for the velocity under some extra hypotheses of Kato’s type concerning the behavior of the fluid in the boundary layer.

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Notes

  1. Note, that we use the index n both for the sequence and the normal component. Throughout this paper \(\mathbf{u}_n\) denotes the nth element of the sequence \((\mathbf{u}_n)_{n=1}^{\infty }\) and \((\mathbf{u}_n)_n\) its normal component.

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Authors and Affiliations

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

    Eduard Feireisl

  2. Institute of Mathematics, Würzburg University, Emil-Fischer-Str. 40, 97074, Würzburg, Germany

    Christian Klingenberg

  3. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Simon Markfelder

Authors
  1. Eduard Feireisl
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  2. Christian Klingenberg
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  3. Simon Markfelder
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Correspondence to Eduard Feireisl.

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The authors declare that they have no conflict of interest.

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Communicated by T. Ozawa.

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This article is part of the topical collection “Yoshihiro Shibata” edited by Tohru Ozawa.

The research of Eduard Feireisl leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 21-02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

Simon Markfelder acknowledges financial support by the Alexander von Humboldt Foundation.

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Feireisl, E., Klingenberg, C. & Markfelder, S. Euler System with a Polytropic Equation of State as a Vanishing Viscosity Limit. J. Math. Fluid Mech. 24, 67 (2022). https://doi.org/10.1007/s00021-022-00690-7

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