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Measure-valued solutions to the complete Euler system revisited

  • Jan Březina
  • Eduard Feireisl
Article
  • 64 Downloads

Abstract

We consider the complete Euler system describing the time evolution of a general inviscid compressible fluid. We introduce a new concept of measure-valued solution based on the total energy balance and entropy inequality for the physical entropy without any renormalization. This class of so-called dissipative measure-valued solutions is large enough to include the vanishing dissipation limits of the Navier–Stokes–Fourier system. Our main result states that any sequence of weak solutions to the Navier–Stokes–Fourier system with vanishing viscosity and heat conductivity coefficients generates a dissipative measure-valued solution of the Euler system under some physically grounded constitutive relations. Finally, we discuss the same asymptotic limit for the bi-velocity fluid model introduced by H.Brenner.

Keywords

Euler system Measure-valued solution Weak-strong uniqueness Vanishing dissipation limit 

Mathematics Subject Classification

Primary 35L45 Secondary 35Q35 76N15 

References

  1. 1.
    Ball, J.M.: A version of the fundamental theorem for Young measures. In: Lecture Notes in Physics, vol. 344, Springer, pp. 207–215 (1989)Google Scholar
  2. 2.
    Bechtel, S.E., Rooney, F.J., Forest, M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Differential Equations. First Order Systems and Applications. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  4. 4.
    Brenner, H.: Kinematics of volume transport. Physica A 349, 11–59 (2005)CrossRefGoogle Scholar
  5. 5.
    Brenner, H.: Navier–Stokes revisited. Physica A 349(1–2), 60–132 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brenner, H.: Fluid mechanics revisited. Physica A 349, 190–224 (2006)CrossRefGoogle Scholar
  7. 7.
    Bresch, D., Desjardins, B., Zatorska, E.: Two-velocity hydrodynamics in fluid mechanics: Part II. Existence of global \(\kappa \)-entropy solutions to the compressible Navier–Stokes systems with degenerate viscosities. J. Math. Pures Appl. (9) 104(4), 801–836 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bresch, D., Giovangigli, V., Zatorska, E.: Two-velocity hydrodynamics in fluid mechanics: Part I. Well posedness for zero Mach number systems. J. Math. Pures Appl. (9) 104(4), 762–800 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Březina, J., Feireisl, E.: Measure-valued solutions to the complete Euler system (to appear in J. Math. Soc. Japan) (2017) arxiv preprint no. arXiv:1702.04878
  10. 10.
    Cai, X., Cao, Z., Sun, Y.: Global regularity to the two dimensional compressible Navier–Stokes equations with mass diffusion. Acta Appl. Math. 136, 63–77 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, G.-Q., Frid, H.: Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations. Trans. Am. Math. Soc. 353(3), 1103–1117 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4th edn, vol. 325. Springer, Berlin (2016)Google Scholar
  14. 14.
    Feireisl, E.: Vanishing dissipation limit for the Navier–Stokes–Fourier system. Commun. Math. Sci. 14(6), 1535–1551 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Feireisl, E., Klingenberg, C., Kreml, O., Markfelder, S.: On oscillatory solutions to the complete Euler system (2017) arxiv preprint no. arXiv:1710.10918
  16. 16.
    Feireisl, E., Mucha, P., Novotný, A., Pokorný, M.: Time periodic solutions to the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 745–786 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser, Basel (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Feireisl, E., Novotný, A.: Weak-strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feireisl, E., Vasseur, A.: New perspectives in fluid dynamics: mathematical analysis of a model proposed by Howard Brenner. In: New Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech. Birkhäuser-Verlag, Basel, pp. 153–179 (2010)Google Scholar
  20. 20.
    Fjordholm, U.K., Käppeli, R., Mishra, S., Tadmor, E.: Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws. Found. Comput. Math. 17(3), 763–827 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fjordholm, U.S., Mishra, S., Tadmor, E.: On the computation of measure-valued solutions. Acta Numer. 25, 567–679 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Oettinger, H.C., Struchtrup, H., Liu, M.: Inconsistency of a dissipative contribution to the mass flux in hydrodynamics. Phys. Rev. E 80, 056303 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tokyo Institute of TechnologyTokyoJapan
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrague 1Czech Republic

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