Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids

  • Eduard Feireisl
Living reference work entry


The complete set of equations describing the motion of a general compressible, viscous, heat-conducting, and possibly rotating fluid arises as a mathematical model in a large variety of real world applications. The scale analysis aims at two different objectives: Rigorous derivation of a simplified asymptotic set of equations and understanding the passage from the original primitive system to the simplified target system. These issues are discussed in the context of compressible, viscous, heat conducting, and/or rotating fluids.


  1. 1.
    T. Alazard, Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (electronic) (2006)Google Scholar
  2. 2.
    T. Alazard, Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. 180, 1–73 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A.S. Almgren, J.B. Bell, C.A. Rendleman, M. Zingale, Low Mach number modeling of type Ia supernovae. I. hydrodynamics. Astrophys. J. 637, 922–936 (2006)CrossRefGoogle Scholar
  4. 4.
    S.N. Antontsev, A.V. Kazhikhov, V.N. Monakhov, Krajevyje zadaci mechaniki neodnorodnych zidkostej. Novosibirsk (1983)Google Scholar
  5. 5.
    A. Babin, A. Mahalov, B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48, 1133–1176 (1999)MathSciNetMATHGoogle Scholar
  6. 6.
    A. Babin, A. Mahalov, B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50(Special Issue), 1–35 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    C. Bardos, T.T. Nguyen, Remarks on the inviscid liit for the compressible flows. Arxive Preprint Series (2014). arXiv 1410.4952v1Google Scholar
  8. 8.
    S.E. Bechtel, F.J. Rooney, M.G. Forest, Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    D. Bresch, B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. 87, 57–90 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    M. Bulíček, J. Málek, K.R. Rajagopal, Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear- rate dependent viscosity. Indiana Univ. Math. J. 56, 51–86 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    L. Caffarelli, R.V. Kohn, L. Nirenberg, On the regularity of the solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    T. Chatelain, A. Henrot, Some results about Schiffer’s conjectures. Inverse Probl. 15(3), 647–658 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathematical Geophysics. Volume 32 of Oxford Lecture Series in Mathematics and Its Applications (The Clarendon Press/Oxford University Press, Oxford, 2006)Google Scholar
  14. 14.
    C. Cheverry, I. Gallagher, T. Paul, L. Saint-Raymond, Semiclassical and spectral analysis of oceanic waves. Duke Math. J. 161(5), 845–892 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    A.J. Chorin, J.E. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer, New York, 1979)CrossRefMATHGoogle Scholar
  16. 16.
    C.M. Dafermos, The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    A.-L. Dalibard, L. Saint-Raymond, Mathematical study of the β-plane model for rotating fluids in a thin layer. J. Math. Pures Appl. (9) 94(2), 131–169 (2010)Google Scholar
  18. 18.
    R. Danchin, Low Mach number limit for viscous compressible flows. M2AN Math. Model Numer. Anal. 39, 459–475 (2005)Google Scholar
  19. 19.
    R. Danchin, X. Liao, On the well-posedness of the full low Mach number limit system in general critical Besov spaces Commun. Contemp. Math. 14(3), 1250022-1–1250022-47 (2012)Google Scholar
  20. 20.
    B. Desjardins, E. Grenier, Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    D.B. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math. 105, 141–200 (1977)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    R. Farwig, H. Kozono, H. Sohr, An L q-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21–53 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    C.L. Fefferman, Existence and smoothness of the Navier-Stokes equation, in The Millennium Prize Problems (Clay Mathematics Institute, Cambridge, 2006), pp. 57–67MATHGoogle Scholar
  25. 25.
    E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford University Press, Oxford, 2004)MATHGoogle Scholar
  26. 26.
    E. Feireisl, Vanishing dissipation limit for the Navier-Stokes-Fourier system. Commun. Math. Sci., 14(6), pp. 1535–1551 (2015)CrossRefGoogle Scholar
  27. 27.
    E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids (Birkhäuser-Verlag, Basel, 2009)CrossRefMATHGoogle Scholar
  28. 28.
    E. Feireisl, A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    E. Feireisl, A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Commun. Math. Phys. 321, 605–628 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    E. Feireisl, A. Novotný, Inviscid incompressible limits under mild stratification: a rigorous derivation of the EulerBoussinesq system. Appl. Math. Optim. 70, 279–307 (2014)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    E. Feireisl, A. Novotný, Multiple scales and singular limits for compressible rotating fluids with general initial data. Commun. Partial Differ. Equ. 39, 1104–1127 (2014)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    E. Feireisl, M.E. Schonbek, On the Oberbeck-Boussinesq approximation on unbounded domains, in Abel Symposium Lecture Notes (Springer, Berlin, 2011)Google Scholar
  33. 33.
    E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392 (2001)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    E. Feireisl, A. Novotný, Y. Sun, Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–631 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    E. Feireisl, I. Gallagher, D. Gerard-Varet, A. Novotný Multi-scale analysis of compressible viscous and rotating fluids. Commun. Math. Phys. 314, 641–670 (2012)Google Scholar
  36. 36.
    E. Feireisl, B.J. Jin, A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14, 712–730 (2012)MathSciNetMATHGoogle Scholar
  37. 37.
    E. Feireisl, O. Kreml, Š Nečasová, J. Neustupa, J. Stebel, Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differ. Equ. 254, 125–140 (2013)Google Scholar
  38. 38.
    E. Feireisl, Š Nečasová, Y. Sun, Inviscid incompressible limits on expanding domains. Nonlinearity 27(10), 2465–2478 (2014)Google Scholar
  39. 39.
    E. Feireisl, R. Klein, A. Novotný, E. Kaminska, On singular limits arising in the scale analysis of stratified fluid flows. Math. Models Methods Appl. Sci. 26(3) 419–443 (2016)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    I. Gallagher, Résultats récents sur la limite incompressible. Astérisque (299). Exp. No. 926, vii, 29–57, 2005. Séminaire Bourbaki, vol. 2003/2004Google Scholar
  41. 41.
    I. Gallagher, L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations. C. R. Math. Acad. Sci. Paris 336(5), 401–406 (2003)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    I. Gallagher, L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results. Mém. Soc. Math. Fr. (N.S.) (107), v+116 (2006/2007)Google Scholar
  43. 43.
    G. Gallavotti, Statistical Mechanics: A Short Treatise (Springer, Heidelberg, 1999)CrossRefMATHGoogle Scholar
  44. 44.
    G. Gallavotti, Foundations of Fluid Dynamics (Springer, New York, 2002)CrossRefMATHGoogle Scholar
  45. 45.
    P.A. Gilman, G.A. Glatzmaier, Compressible convection in a rotating spherical shell. I. Anelastic equations. Astrophys. J. Suppl. 45(2), 335–349 (1981)MathSciNetCrossRefGoogle Scholar
  46. 46.
    G.A. Glatzmaier, P.A. Gilman, Compressible convection in a rotating spherical shell. II. A linear anelastic model. Astrophys. J. Suppl. 45(2), 351–380 (1981)MathSciNetGoogle Scholar
  47. 47.
    D. Gough, The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448–456 (1969)CrossRefGoogle Scholar
  48. 48.
    E. Grenier, Y. Guo, T.T. Nguyen, Spectral stability of Prandtl boundary layers: an overview. Analysis (Berlin) 35(4), 343–355 (2015)Google Scholar
  49. 49.
    D. Hoff, The zero-Mach limit of compressible flows. Commun. Math. Phys. 192(3), 543–554 (1998)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    D. Hoff, T.-P. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data. Indiana Univ. Math. J. 38(4), 861–915 (1989)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Ration. Mech. Anal. 25, 188–200 (1967)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    T. Kato, Nonstationary flows of viscous and ideal fluids in r 3. J. Funct. Anal. 9, 296–305 (1972)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    T. Kato, Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary, in Seminar on PDE’s, ed. by S.S. Chern (Springer, New York, 1984)Google Scholar
  55. 55.
    T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, in Seminar on Nonlinear Partial Differential Equations, Berkeley, 1983. Volume 2 of Mathematical Sciences Research Institute Publications (Springer, New York, 1984), pp. 85–98Google Scholar
  56. 56.
    T. Kato, C.Y. Lai, Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    S. Klainerman, A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    R. Klein, Scale-dependent models for atmospheric flows, in Annual Review of Fluid Mechanics. Annual Review of Fluid Mechanics, vol. 42 (Annual Reviews, Palo Alto, 2010), pp. 249–274Google Scholar
  59. 59.
    R. Klein, N. Botta, T. Schneider, C.D. Munz, S. Roller, A. Meister, L. Hoffmann, T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Eng. Math. 39, 261–343 (2001)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)MathSciNetCrossRefGoogle Scholar
  61. 61.
    J. Lighthill, On sound generated aerodynamically I. General theory. Proc. R. Soc. Lond. A 211, 564–587 (1952)Google Scholar
  62. 62.
    J. Lighthill, On sound generated aerodynamically II. General theory. Proc. R. Soc. Lond. A 222, 1–32 (1954)Google Scholar
  63. 63.
    P.-L. Lions, Mathematical Topics in Fluid Dynamics. Incompressible Models, vol. 1 (Oxford Science Publication, Oxford, 1996)Google Scholar
  64. 64.
    P.-L. Lions, Mathematical Topics in Fluid Dynamics. Compressible Models, vol. 2 (Oxford Science Publication, Oxford, 1998)Google Scholar
  65. 65.
    P.-L. Lions, N. Masmoudi, Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    P.-L. Lions, N. Masmoudi, Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999)MathSciNetCrossRefGoogle Scholar
  67. 67.
    F.B. Lipps, R.S. Hemler, A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 39, 2192–2210 (1982)CrossRefGoogle Scholar
  68. 68.
    A. Majda, High Mach number combustion. Lect. Notes Appl. Math. 24, 109–184 (1986)MathSciNetGoogle Scholar
  69. 69.
    A. Majda, Introduction to PDE’s and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, vol. 9 (Courant Institute, New York, 2003)Google Scholar
  70. 70.
    N. Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary. Arch. Ration. Mech. Anal. 142(4), 375–394 (1998)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data. Commun. Pure Appl. Math.. 53, 432–483 (2000)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. Henri Poincaré, Anal. non linéaire 18, 199–224 (2001)Google Scholar
  73. 73.
    N. Masmoudi, Examples of singular limits in hydrodynamics, in Handbook of Differential Equations, III, ed. by C. Dafermos, E. Feireisl (Elsevier, Amsterdam, 2006)Google Scholar
  74. 74.
    N. Masmoudi, Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230–240 (2007)MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    G. Métivier, S. Schochet, The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158, 61–90 (2001)MathSciNetCrossRefMATHGoogle Scholar
  76. 76.
    G. Métivier, S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differ. Equ. 187, 106–183 (2003)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    M. Michálek, Stability result for Navier-stokes equations with entropy transport. J. Math. Fluid Mech. 17(2), 279–285 (2015)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Y. Ogura, M. Phillips, Scale analysis for deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179 (1962)CrossRefGoogle Scholar
  79. 79.
    M. Oliver, Classical solutions for a generalized Euler equation in two dimensions. J. Math. Anal. Appl. 215, 471–484 (1997)MathSciNetCrossRefMATHGoogle Scholar
  80. 80.
    J. Pedlosky, Geophysical Fluid Dynamics (Springer, New York, 1987)CrossRefMATHGoogle Scholar
  81. 81.
    L. Saint-Raymond, Hydrodynamic limits: some improvements of the relative entropy method. Annal. I.H.Poincaré AN 26, 705–744 (2009)Google Scholar
  82. 82.
    S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit. Commun. Math. Phys. 104, 49–75 (1986)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    S. Schochet, Fast singular limits of hyperbolic PDE’s. J. Differ. Equ. 114, 476–512 (1994)MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    S. Schochet, The mathematical theory of low Mach number flows. M2ANMath. Model Numer. Anal. 39, 441–458 (2005)Google Scholar
  85. 85.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1967)MATHGoogle Scholar
  86. 86.
    F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain. J. Math. Fluid Mech. 16(1), 163–178 (2014)MathSciNetCrossRefMATHGoogle Scholar
  87. 87.
    V.A. Vaigant, A.V. Kazhikhov, On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid (in Russian). Sibirskij Mat. Z. 36(6), 1283–1316 (1995)MathSciNetMATHGoogle Scholar
  88. 88.
    G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, Cambridge, 2006)CrossRefGoogle Scholar
  89. 89.
    S. Wang, S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 31(4–6), 571–591 (2006)MathSciNetCrossRefMATHGoogle Scholar
  90. 90.
    E. Weinan, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. Acta Math. Sin. Engl. Ser. 16, 207–218 (2000)MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    R.Kh. Zeytounian. Asymptotic Modeling of Atmospheric Flows (Springer, Berlin, 1990)CrossRefMATHGoogle Scholar
  92. 92.
    R.Kh. Zeytounian, Asymptotic Modelling of Fluid Flow Phenomena. Volume 64 of Fluid Mechanics and Its Applications (Kluwer Academic, Dordrecht, 2002)Google Scholar
  93. 93.
    R.Kh. Zeytounian, Joseph Boussinesq and his approximation: a contemporary view. C. R. Mecanique 331, 575–586 (2003)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Evolution Differential Equations (EDE)Institute of Mathematics, Academy of Sciences of the Czech RepublicPragueCzech Republic

Personalised recommendations