Advertisement

Communications in Mathematical Physics

, Volume 359, Issue 2, pp 733–763 | Cite as

An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

  • Theodore D. Drivas
  • Gregory L. Eyink
Article

Abstract

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier–Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L3-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Onsager L.: Statistical hydrodynamics. Nuovo Cim. Suppl. VI, 279–287 (1949)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Eyink G.L.: Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Physica D 78(3–4), 222–240 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Constantin P., Weinan E., Titi E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13(1), 249 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eyink G.L., Sreenivasan K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87–135 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Lellis C., Székelyhidi L. Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, 225–260 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Lellis C., Székelyhidi L. Jr: The h-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. 49(3), 347–375 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buckmaster T.: Onsager’s conjecture almost everywhere in time. Commun. Math. Phys. 333(3), 1175–1198 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Isett P.: A proof of Onsager’s conjecture. arXiv preprint arXiv:1608.08301 (2016)
  10. 10.
    Feireisl E., Gwiazda P., Świerczewska-Gwiazda A., Wiedemann E.: Regularity and energy conservation for the compressible Euler equations. Arch. Ration. Mech. Anal. 223(3), 1375–1395 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Landau L., Lifshitz E.: Fluid Mechanics, 2nd edn. Pergamon Press, New York (1987)Google Scholar
  12. 12.
    de Groot S., Mazur P.: Non-equilibrium Thermodynamics. Dover, New York (1984)zbMATHGoogle Scholar
  13. 13.
    Gallavotti G.: Foundations of Fluid Dynamics. Springer, Berlin (2013)zbMATHGoogle Scholar
  14. 14.
    Feireisl E.: Dynamics of Viscous Compressible Fluids, vol. 26. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  15. 15.
    Feireisl E., Novotnyˋ A.: Inviscid incompressible limits of the full Navier–Stokes–Fourier system. Commun. Math. Phys. 321(3), 605–628 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lions P.-L.: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  17. 17.
    Martin-Löf, A.: Statistical mechanics and the foundations of thermodynamics. Lecture Notes in Physics. Springer, Berlin (1979)Google Scholar
  18. 18.
    Ruelle D.: Statistical Mechanics: Rigorous Results. World Scientific, Singapore (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Callen H.: Thermodynamics and an Introduction to Thermostatistics. Wiley, London (1985)zbMATHGoogle Scholar
  20. 20.
    Evans, L.C.: Entropy and Partial Differential Equations. http://math.berkeley.edu/evans/entropy.and.PDE.pdf (2004)
  21. 21.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2015)zbMATHGoogle Scholar
  22. 22.
    Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (2015)zbMATHGoogle Scholar
  23. 23.
    Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1987)zbMATHGoogle Scholar
  24. 24.
    Johnson B.M.: Closed-form shock solutions. J. Fluid Mech. 745, R1 (2014)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Eyink, G.L., Drivas, T.D.: Cascades and dissipative anomalies in compressible fluid turbulence. arXiv preprint arXiv:1704.03532 (2017)
  26. 26.
    Kim J., Ryu D.: Density power spectrum of compressible hydrodynamic turbulent flows. Astrophys. J. Lett. 630(1), L45 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    Oberguggenberger M.: Multiplication of Distributions and Applications to Partial Differential Equations, Volume 259 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, London (1992)Google Scholar
  28. 28.
    Triebel H.: Theory of Function Spaces III. Birkhäuser, Basel (2006)zbMATHGoogle Scholar
  29. 29.
    Aluie H.: Scale decomposition in compressible turbulence. Physica D 247(1), 54–65 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Eyink, G.L., Drivas, T.D.: Cascades and dissipative anomalies in relativistic fluid turbulence. arXiv preprint arXiv:1704.03541 (2017)
  31. 31.
    Isett, P.: Regularity in time along the coarse scale flow for the incompressible Euler equations. arXiv preprint arXiv:1307.0565 (2013)
  32. 32.
    Isett, P.: Hölder continuous Euler flows in three dimensions with compact support in time. arXiv preprint arXiv:1211.4065 (2012)
  33. 33.
    Isett P., Oh S.-J.: On nonperiodic Euler flows with Hölder regularity. Arch. Ration. Mech. Anal. 221((2), 725–804 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ziemer W.: Weakly Differentiable Functions. Graduate Text in Mathematics 120. Springer, Berlin (1989)CrossRefGoogle Scholar
  35. 35.
    Showalter R.: Hilbert Space Methods in Partial Differential Equations. Dover, New York (2011)zbMATHGoogle Scholar
  36. 36.
    Rudin W.: Functional Analysis. McGraw-Hill, New York (2006)zbMATHGoogle Scholar
  37. 37.
    Huang K.: Introduction to Statistical Physics. CRC Press, Boca Raton (2009)Google Scholar
  38. 38.
    Stuart A., Ord K.: Kendall’s Advanced Theory of Statistics: Volume 1: Distribution Theory. Wiley, London (2009)zbMATHGoogle Scholar
  39. 39.
    Favre, A.: Statistical equations of turbulent gases. In: Lavrentiev, M.A. (ed.) Problems of Hydrodynamics and Continuum Mechanics, pp. 37–44. SIAM, Philadelphia (1969)Google Scholar
  40. 40.
    Eyink G.L.: Turbulent general magnetic reconnection. Astrophys. J. 807(2), 137 (2015)ADSCrossRefGoogle Scholar
  41. 41.
    Eyink, G.L.: Turbulence Theory. Course notes. http://www.ams.jhu.edu/~eyink/Turbulence/notes/ (2015)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Department of Physics and AstronomyThe Johns Hopkins UniversityBaltimoreUSA
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations