Archive for Rational Mechanics and Analysis

, Volume 186, Issue 1, pp 77–107 | Cite as

The Low Mach Number Limit for the Full Navier–Stokes–Fourier System

  • Eduard FeireislEmail author
  • Antonín Novotný


We study the low Mach number asymptotic limit for solutions to the full Navier–Stokes–Fourier system, supplemented with ill-prepared data and considered on an arbitrary time interval. Convergencetowards the incompressible Navier–Stokes equations is shown.


Mach Number Entropy Production Variational Solution Singular Limit Entropy Production Rate 
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.


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  1. 1.
    Alazard, T.: Low mach number limit of the full Navier-Stokes equations. To appear in Arch. Ration. Mech. Anal. (2005)Google Scholar
  2. 2.
    Boccardo L., Dall’Aglio A., Gallouet T., Orsina L. (1997). Nonlinear parabolic equations with measure data. J. Funct. Anal. 147, 237–258zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bresch D., Desjardins B., Grenier E., Lin C.-K. (2002). Low Mach number limit of viscous polytropic flows: Formal asymptotics in the periodic case. Stud. Appl. Math. 109, 125–149zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buet, C., Després, B.: Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics. Preprint 2003Google Scholar
  5. 5.
    Danchin R. (2002). Zero Mach number limit for compressible flows with periodic boundary conditions. Amer. J. Math. 124, 1153–1219zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Desjardins B., Grenier E., Lions P.-L., Masmoudi N. (1999). Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471CrossRefMathSciNetGoogle Scholar
  7. 7.
    DiPerna R.J., Lions P.-L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Ducomet B., Feireisl E. (2004). On the dynamics of gaseous stars. Arch. Ration. Mech. Anal. 174, 221–266zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ebin D.B. (1977). The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. of Math. 105(2): 141–200CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ebin, D.B.: Viscous fluids in a domain with frictionless boundary. Global Analysis - Analysis on Manifolds. (Ed. H. Kurke, J. Mecke, H. Triebel, R. Thiele) Teubner-Texte zur Mathematik 57, Teubner, Leipzig, 93–110 (1983)Google Scholar
  11. 11.
    Feireisl E. (2003). Dynamics of viscous compressible fluids. Oxford University Press, OxfordGoogle Scholar
  12. 12.
    Feireisl, E.: Mathematical theory of compressible, viscous, and heat conducting fluids. To appear in Comput. Appl. Math. (2007)Google Scholar
  13. 13.
    Feireisl E., Novotný (2005). On a simple model of reacting compressible flows arising in astrophysics. Proc. Roy. Sect. Soc. Edinburgh Sect. A 135, 1169–1194zbMATHCrossRefGoogle Scholar
  14. 14.
    Gallavotti G. (1999). Statistical mechanics: A short treatise. Springer-Verlag, HeidelbergzbMATHGoogle Scholar
  15. 15.
    Gallavotti G. (2002). Foundations of fluid dynamics. Springer-Verlag, New YorkGoogle Scholar
  16. 16.
    Hagstrom T., Lorenz J. (2002). On the stability of approximate solutions of hyperbolic-parabolic systems and all-time existence of smooth, slightly compressible flows. Indiana Univ. Math. J. 51: 1339–1387CrossRefMathSciNetGoogle Scholar
  17. 17.
    Hoff D. (1998). The zero Mach number limit of compressible flows. Comm. Math. Phys. 192, 543–554zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Hoff D. (2002). Dynamics of singularity surfaces for compressible viscous flows in two space dimensions. Comm. Pure Appl. Math. 55, 1365–1407zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Klainerman S., Majda A. (1981). Compressible and incompressible fluids. Comm. Pure Appl. Math. 34, 481–524zbMATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Klein R., Botta N., Schneider T., Munz C.D., Roller S., Meister A., Hoffmann L., Sonar T. (2001). Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Engrg. Math. 39, 261–343zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lin C.K. (1995). On the incompressible limit of the compressible Navier–Stokes equations. Comm. Partial Differential Equations 20, 677–707zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lions P.-L., Masmoudi N. (1998). Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627zbMATHMathSciNetGoogle Scholar
  23. 23.
    Métivier G., Schochet S. (2001) The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158, 61–90zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Métivier G., Schochet S. (2003). Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differential Equations 187, 106–183zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Müller, I., Ruggeri, T.: Rational extended thermodynamics. Springer Tracts in Natural Philosophy 37. Springer-Verlag, Heidelberg, 1998Google Scholar
  26. 26.
    Oxenius J. (1986). Kinetic theory of particles and photons. Springer-Verlag, BerlinGoogle Scholar
  27. 27.
    Rajagopal K.R., Shrinivasa A.R. (2004). On thermodynamical restrictions of continua. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460, 631–651zbMATHADSCrossRefMathSciNetGoogle Scholar
  28. 28.
    Schochet S. (1986). The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit. Comm. Math. Phys. 104, 49–75zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Schochet S. (1994). Fast singular limits of hyperbolic pde’s. J. Differential Equations 114, 476–512zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Schochet S. (2005). The mathematical theory of low Mach number flows. M2AN 39, 441–458zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematical Institute AS ČRPraha 1Czech Republic
  2. 2.Université du Sud Toulon-VarLa GardeFrance

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