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An overview of some recent results on the Euler system of isentropic gas dynamics

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Abstract

This overview is concerned with the well-posedness problem for the isentropic compressible Euler equations of gas dynamics. The results we present are in line with the programof investigatingthe efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis and Székelyhidi for the incompressible Euler equations. Mainly following [5], we investigate the role of the maximal dissipation criterion proposed by Dafermos in [6]: we prove how, for specific pressure laws, some non-standard (i.e. constructed via convex integration methods) solutions to the Riemann problem for the isentropic Euler system in two space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions.

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Authors and Affiliations

  1. EPFL Lausanne, Station 8, CH-1015, Lausanne, Switzerland

    Elisabetta Chiodaroli

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

    Ondřej Kreml

Authors
  1. Elisabetta Chiodaroli
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  2. Ondřej Kreml
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Correspondence to Elisabetta Chiodaroli.

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Chiodaroli, E., Kreml, O. An overview of some recent results on the Euler system of isentropic gas dynamics. Bull Braz Math Soc, New Series 47, 241–253 (2016). https://doi.org/10.1007/s00574-016-0135-0

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  • DOI: https://doi.org/10.1007/s00574-016-0135-0

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