Mathematische Annalen

, Volume 356, Issue 2, pp 683–702 | Cite as

Suitable weak solutions: from compressible viscous to incompressible inviscid fluid flows

  • Eduard Feireisl
  • Antonín Novotný
  • Hana Petzeltová


We establish the asymptotic limit of the compressible Navier–Stokes system in the regime of low Mach and high Reynolds number on unbounded spatial domains with slip boundary condition. The result holds in the class of suitable weak solutions satisfying a relative entropy inequality.


Stokes System Slip Boundary Condition Euler System Dispersive Estimate Suitable Weak Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Alazard, T.: Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions. Adv. Differ. Equ. 10(1), 19–44 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Caffarelli, L., Kohn, R.V., Nirenberg, L.: On the regularity of the solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Clopeau, T., Mikelić, A., Robert, R.: On the vanishing viscosity limit for the \(2{\rm D}\) incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    D’Ancona, P., Racke, R.: Evolution equations in non-flat waveguides. arXiv:1010.0817 (2010)Google Scholar
  5. 5.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fefferman, C.L.: Existence and smoothness of the Navier-Stokes equation. In: Carlson, J., Jaffe, A., Wiles, A. (eds.) The Millenium Prize Problems, pp. 57–67. Clay Math. Inst., Cambridge (2006)Google Scholar
  7. 7.
    Feireisl, E.: Low Mach number limits of compressible rotating fluids. J. Math. Fluid Mech. 14, 61–78 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Feireisl, E., Novotný, A., Petzeltová, H.: Low Mach number limit for the Navier–Stokes system on unbounded domains under strong stratication. Commun. Partial Differ. Equ. 35, 68–88 (2010)zbMATHCrossRefGoogle Scholar
  9. 9.
    Feireisl, E., Novotný, A., Sun, Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–632 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Germain, P.: Weak-strong uniqueness for the isentropic compressible Navier–Stokes system. J. Math. Fluid Mech. 13, 137–146 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grenier, E.: Oscillatory perturbations of the Navier–Stokes equations. J. Math. Pures Appl. 76(6), 477–498 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Isozaki, H.: Singular limits for the compressible Euler equation in an exterior domain. J. Reine Angew. Math. 381, 1–36 (1987)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kato, T.: Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Chern, S.S. (ed.) Seminar on PDE’s. Springer, New York (1984)Google Scholar
  14. 14.
    Kelliher, J.P.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006). (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lions, P.-L.: Mathematical topics in fluid dynamics, vol. 2, compressible models. Oxford Science Publication, Oxford (1998)Google Scholar
  16. 16.
    Masmoudi, N.: Incompressible inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré. Anal. Nonlinéaire 18, 199–224 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Neustupa, J., Penel, P.: Local in time strong solvability of the non-steady Navier–Stokes equations with Navier’s boundary condition and the question of the inviscid limit. C. R. Math. Acad. Sci. Paris 348(19–20), 1093–1097 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ukai, S.: The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26(2), 323–331 (1986)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Wang, S., Jiang, S.: The convergence of the Navier–Stokes-Poisson system to the incompressible Euler equations. Comm. Partial Differ. Equ. 31(4–6), 571–591 (2006)zbMATHCrossRefGoogle Scholar
  20. 20.
    Wang, X.: Examples of boundary layers associated with the incompressible Navier–Stokes equations. Chin. Ann. Math. 31B, 781–792 (2010)CrossRefGoogle Scholar
  21. 21.
    Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Comm. Pure Appl. Math. 60(7), 1027–1055 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eduard Feireisl
    • 1
  • Antonín Novotný
    • 2
  • Hana Petzeltová
    • 1
  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic
  2. 2.IMATH, Université du Sud Toulon-VarLa GardeFrance

Personalised recommendations