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On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System

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Abstract

In this paper we extend and complement the results in Chiodaroli et al. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ) = ρ γ, γ ≥ 1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417–1436, 2009), (Arch Ration Mech Anal 195:225–260, 2010). Moreover we prove that for some of these Riemann problems and for 1 ≤ γ < 3 such solutions have a greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in (J Diff Equ 14:202–212, 1973) does not favour the classical self-similar solutions.

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References

  1. Bressan, A.: Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. Oxford Lecture Series in Mathematics and its Applications, vol. 20. Oxford University Press, Oxford, (2000)

  2. Buckmaster, T., De Lellis, C., Székelyhidi, L.J.: Transporting microstructure and dissipative Euler flows. Preprint (2013)

  3. Chiodaroli, E.: A counterexample to well-posedeness of entropy solutions to the compressible Euler system. JHDE (to appear, 2014)

  4. Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Comm. Pure Appl. Math. (to appear, 2014)

  5. Chiodaroli, E., Feireisl, E., Kreml, O.: On the weak solutions to the equations of a compressible heat conducting gas. Annales IHP-ANL (to appear, 2014)

  6. Cordoba D., Faraco D., Gancedo F.: Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. 200(3), 725–746 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dafermos C.M.: The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Differ. Equ. 14, 202–212 (1973)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. Dafermos C.M.: Maximal dissipation in equations of evolution. J. Differ. Equ. 252, 567–587 (2012)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, vol. 325 of Grundleheren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 3rd edn. Springer, Berlin (2010)

  10. Daneri S.: Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations. Comm. Math. Phys. 329(2), 745–786 (2014)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. De Lellis C., Székelyhidi L.J.: The Euler equations as a differential inclusion. Ann. Math. 170(3), 1417–1436 (2009)

    Article  MATH  Google Scholar 

  12. De Lellis C., Székelyhidi L.J.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. De Lellis C., Székelyhidi L.J.: The h-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. (N.S.) 49(3), 347–375 (2012)

    Article  MATH  Google Scholar 

  14. De Lellis C., Székelyhidi L.J.: Dissipative continuous Euler flows. Invent. Math. 193(2), 377–407 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. De Lellis, C., Székelyhidi, L.J.: Dissipative Euler flows and Onsager’s conjecture. JEMS (to appear, 2014)

  16. Di Perna R.: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28(1), 137–188 (1979)

    Article  MathSciNet  Google Scholar 

  17. Feireisl, E.: Maximal dissipation and well-posedness for the compressible Euler system. J. Math. Fluid Mech. (to appear, 2014)

  18. Hsiao L.: The entropy rate admissibility criterion in gas dynamics. J. Differ. Equ. 38, 226–238 (1980)

    Article  MATH  ADS  Google Scholar 

  19. Lax, P.D.: Shock waves and entropy. Contrib. Nonlinear Funct. Anal., 603–634 (1971)

  20. Liu T.-P., Smoller J.A.: On the vacuum state for the isentropic gas dynamics equations. Adv. Appl. Math. 1(4), 345–359 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shvidkoy R.: Convex integration for a class of active scalar equations. J. Am. Math. Soc. 24(4), 1159–1174 (2011)

    Article  Google Scholar 

  22. Székelyhidi L.J.: Weak solutions to the incompressible Euler equations with vortex sheet initial data. C. R. Acad. Sci. Paris Ser. I 349(19–20), 1063–1066 (2011)

    Article  MATH  Google Scholar 

  23. Székelyhidi L.J.: Relaxation of the incompressible porous media equation. Ann. Sci. l’ENS 45(3), 491–509 (2012)

    MATH  Google Scholar 

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Authors and Affiliations

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

    Elisabetta Chiodaroli

  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Ondrej Kreml

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

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Communicated by C. Dafermos

The work of O.K. is part of the SCIEX project 11.152.

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Chiodaroli, E., Kreml, O. On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System. Arch Rational Mech Anal 214, 1019–1049 (2014). https://doi.org/10.1007/s00205-014-0771-8

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  • DOI: https://doi.org/10.1007/s00205-014-0771-8

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