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Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

  • Eduard Feireisl
  • Mária Lukáčová-Medvid’ováEmail author
  • Hana Mizerová
Article

Abstract

The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy producing) weak solutions. This suggests that there might be sequences of approximate solutions that develop fine-scale oscillations. Accordingly, the concept of measure-valued solution that captures possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Here dissipative means that a suitable form of the second law of thermodynamics is incorporated in the definition of the measure-valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

Keywords

Compressible Euler equations Entropy stable finite volume scheme Entropy stability Convergence Dissipative measure-valued solution 

Mathematics Subject Classification

65M08 76N10 35L65 35R06 

Notes

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Copyright information

© SFoCM 2019

Authors and Affiliations

  • Eduard Feireisl
    • 1
  • Mária Lukáčová-Medvid’ová
    • 2
    Email author
  • Hana Mizerová
    • 1
  1. 1.Institute of Mathematics of the Czech Academy of SciencesPraha 1Czech Republic
  2. 2.Institute of MathematicsJohannes Gutenberg-University MainzMainzGermany

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