Skip to main content
SpringerLink
Log in

Low Mach and thin domain limit for the compressible Euler system

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer \(\Omega _{\delta }=(0,\delta )\times {\mathbb {R}}^2, \ \ \delta >0\). In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and \(\delta \rightarrow 0\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alibert, J.J., Bouchitté, G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4(1), 129–147 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Asano, K.: On the incompressible limit of the compressible Euler equation. Jpn. J. Appl. Math. 4(3), 455–488 (1987)

    Article  MathSciNet  Google Scholar 

  3. Ball, J. M.: A version of the fundamental theorem for Young measures. PDEs and continuum models of phase transitions (Nice: Lecture Notes in Phys., 344. Springer, Berlin 1988, 207–215 (1988)

  4. Ball, J.M., Murat, F.: Remarks on Chacons biting lemma. Proc. Am. Math. Soc. 107, 655–663 (1989)

    MathSciNet  MATH  Google Scholar 

  5. Bella, P., Feireisl, E., Novotný, A.: Dimension reduction for compressible viscous fluids. Acta Appl. Math. 134, 111–121 (2014)

    Article  MathSciNet  Google Scholar 

  6. Blanc, X., Danchin, R., Ducomet, B. Nečasová, Š.: The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings, arxiv:1906.08075

  7. Bellout, H., Bloom, F., Nečas, J.: Young measure-valued solutions for non-Newtonian incompressible fluids. Commun. Partial Differ. Equ. 19(11–12), 1763–1803 (1994)

    Article  MathSciNet  Google Scholar 

  8. Breit, D., Feireisl, E., Hofmanova, M.: Generalized solutions to models of inviscid fluids, arXiv:1907.00757

  9. Březina, J., Feireisl, E.: Measure-valued solutions to the complete Euler system revisited, Z. Angew. Math. Phys., 69, Art. 57, 17 pp (2018)

  10. Březina, J., Feireisl, E.: Measure-valued solutions to the complete Euler system. J. Math. Soc. Jpn. 70, 1227–1245 (2018)

    Article  MathSciNet  Google Scholar 

  11. Březina, J., Kreml, O., Mácha, V.: Dimension reduction for the full Navier–Stokes–Fourier system. J. Math. Fluid Mech. 19(4), 659–683 (2017)

    Article  MathSciNet  Google Scholar 

  12. Bruell, G., Feireisl, E.: On a singular limit for stratified compressible fluids. Nonlinear Anal. Real World Appl. 44, 334–346 (2018)

    Article  MathSciNet  Google Scholar 

  13. Caggio, M., Donatelli, D., Nečasová, Š., Sun, Y.: Low Mach number limit on thin domains. Nonlinearity 33, 840–863 (2020)

    Article  MathSciNet  Google Scholar 

  14. Chiodaroli, E.: A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11, 493–519 (2014)

    Article  MathSciNet  Google Scholar 

  15. Chiodaroli, E., Feireisl, E., Kreml, O., Wiedemann, E.: \(\cal{A}\)-free rigidity and applications to the compressible Euler system. Ann. Mat. Pura Appl. 196, 1557–1572 (2017)

    Article  MathSciNet  Google Scholar 

  16. Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68(7), 1157–1190 (2015)

    Article  MathSciNet  Google Scholar 

  17. Chiodaroli, E., Kreml, O.: On the energy dissipation rate of solutions to the compressible isentropic Euler system. Arch. Ration. Mech. Anal. 214(3), 1019–1049 (2014)

    Article  MathSciNet  Google Scholar 

  18. DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88(3), 223–270 (1985)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Demoulini, S., Stuart, D., Tzavaras, A.: Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal. 205(3), 927–961 (2012)

    Article  MathSciNet  Google Scholar 

  21. Ducomet, B., Caggio, M., Nečasová, Š., Pokorný, M.: The rotating Navier–Stokes–Fourier–Poisson system on thin domains. Asymptot. Anal. 109(3–4), 111–141 (2018)

    Article  MathSciNet  Google Scholar 

  22. Ebin, D.B.: The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math. 105, 141–200 (1977)

    Article  MathSciNet  Google Scholar 

  23. Feireisl, E., Tang, T.: On a singular limit for the stratified compressible Euler system. Asymptot. Anal. 114, 59–72 (2019)

    Article  MathSciNet  Google Scholar 

  24. Feireisl, E., Klingenberg, C., Markfelder, S.: On the low Mach number limit for the compressible Euler system. SIAM J. Math. Anal. 51(2), 1496–1513 (2019)

    Article  MathSciNet  Google Scholar 

  25. Feireisl, E., Gwiazda, P., Swierczewska-Gwiazda, A., Wiedemann, E.: Dissipative measure-valued solutions to the compressible Navier–Stokes system, Calc. Var. Partial Differential Equations, 55, Art. 141, 20 pp (2016)

  26. Feireisl, E., Jin, J.B., Novotný, A.: Inviscid incompressible limits of strongly stratified fluids. Asymptot. Anal. 89, 307–329 (2014)

    Article  MathSciNet  Google Scholar 

  27. Feireisl, E., Jin, J.B., Novotný, A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14, 717–730 (2012)

    Article  MathSciNet  Google Scholar 

  28. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel (2009)

    Book  Google Scholar 

  29. Grassin, M., Serre, D.: Global smooth solutions to Euler equations for an isentropic perfect gas. C. R. Acad. Sci. Paris 325, 721–726 (1997)

    Article  Google Scholar 

  30. Grassin, M.: Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J. 47, 1397–1432 (1998)

    Article  MathSciNet  Google Scholar 

  31. Gwiazda, P., Swierczewska-Gwiazda, A., Wiedemann, E.: Weak-strong uniqueness for measure-valued solutions of some compressible fluid models. Nonlinearity 28, 3873–3890 (2015)

    Article  MathSciNet  Google Scholar 

  32. Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)

    Article  MathSciNet  Google Scholar 

  33. Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, London (1996)

  34. Maltese, D., Novotný, A.: Compressible Navier-Stokes equations on thin domains. J. Math. Fluid Mech. 16(3), 571–594 (2014)

    Article  MathSciNet  Google Scholar 

  35. Métivier, G., Schochet, S.: The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158, 61–90 (2001)

    Article  MathSciNet  Google Scholar 

  36. Neustupa, J.: Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids. Math. Nachr. 163, 217–227 (1993)

    Article  MathSciNet  Google Scholar 

  37. Oliver, M.: Classical solutions for a generalized Euler equation in two dimensions. J. Math. Anal. Appl. 215, 471–484 (1997)

    Article  MathSciNet  Google Scholar 

  38. Strichartz, R.S.: A priori estimates for the wave equation and some applications. J. Functional Analysis 5, 218–235 (1970)

    Article  MathSciNet  Google Scholar 

  39. Ukai, S.: The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26(2), 323–331 (1986)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of Š.N. leading to these results has received funding from the Czech Sciences Foundation (GAČR), GA19-04243S and in the framework of RVO: 67985840. The research of T.T. is supported by the NSFC Grant No. 11801138. The research of B.D. is partially supported by the ANR project INFAMIE (ANR-15-CE40-0011). The research of M.C. is supported by the Institute of Mathematics, CAS and in the framework of RVO: 67985840. Last modifications have been written during his stay in the Department of Mathematics, Faculty of Science, University of Zagreb, where M.C. is fully supported by the Croatian Science Foundation under the project MultiFM IP-2019-04-1140.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567, Praha 1, Czech Republic

    Matteo Caggio & Šárka Nečasová

  2. LAMA, Université Paris-Est, 61 Avenue du Général de Gaulle, 94010, Créteil, France

    Bernard Ducomet

  3. Department of Mathematics, College of Sciences, Hohai University, Nanjing, 210098, People’s Republic of China

    Tong Tang

Authors
  1. Matteo Caggio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernard Ducomet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Šárka Nečasová
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tong Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Šárka Nečasová.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caggio, M., Ducomet, B., Nečasová, Š. et al. Low Mach and thin domain limit for the compressible Euler system. Annali di Matematica 200, 1469–1486 (2021). https://doi.org/10.1007/s10231-020-01045-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10231-020-01045-7

Keywords

Mathematics Subject Classification

Search

Navigation