Revista Matemática Complutense

, Volume 26, Issue 2, pp 299–340 | Cite as

Mathematical analysis of fluids in motion: from well-posedness to model reduction

  • Eduard Feireisl


This paper reviews some recent results on the Navier-Stokes-Fourier system governing the evolution of a general compressible, viscous, and heat conducting fluid. We discuss several concepts of weak solutions, in particular, using the implications of the Second law of thermodynamics. We introduce the concept of relative entropy and dissipative solution and show the principle of weak-strong uniqueness. The second part of the paper is devoted to problems of model reduction and the related singular limits. Several examples of singular limits are presented: The incompressible limit, the inviscid limit, the low Rossby number limit and their combinations.


Navier-Stokes-Fourier system Compressible fluid Singular limits Relative entropy 

Mathematics Subject Classification

35Q30 76D05 



Eduard Feireisl acknowledges the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.


  1. 1.
    Babin, A., Mahalov, A., Nicolaenko, B.: Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48, 1133–1176 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Babin, A., Mahalov, A., Nicolaenko, B.: 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50 (Special Issue), 1–35 (2001)Google Scholar
  3. 3.
    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)zbMATHGoogle Scholar
  4. 4.
    Bechtel, S.E., Rooney, F.J., Forest, M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bechtel, S.E., Rooney, Q., Wang, F.J.: A thermodynamic definition of pressure for incompressible viscous fluids. Int. J. Eng. Sci. 42, 1987–1994 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bella, P., Feireisl, E., Pražák, D.: Long time behavior and stabilization to equilibria of solutions to the Navier-Stokes-Fourier system driven by highly oscillating unbounded external forces. J. Dyn. Differ. Equ. (2013) (to appear)Google Scholar
  7. 7.
    Bulíček, M., Málek, J., Rajagopal, K.R.: Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear- rate dependent viscosity. Indiana Univ. Math. J. 56, 51–86 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Callen, H.: Thermodynamics and an Introduction to Thermostatistics. Wiley, New York (1985)zbMATHGoogle Scholar
  10. 10.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics, volume 32 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, Oxford (2006)Google Scholar
  11. 11.
    Cheskidov, A., Friedlander, S., Shvydkoy, R.: On the energy equality for weak solutions of the 3D Navier-Stokes equations. In: Advances in Mathematical Fluid Mechanics, pp. 171–175. Springer, Berlin (2010)Google Scholar
  12. 12.
    Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, New York (1979)zbMATHCrossRefGoogle Scholar
  13. 13.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators: With Applications to Quantum Mechanics and Global Geometry. Texts and monographs in physics, Springer, Berlin (1987)Google Scholar
  14. 14.
    Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70, 167–179 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    De Lellis, C., Székelyhidi, L. Jr.: The \(h\)-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. (N.S.), 49(3), 347–375 (2012)Google Scholar
  16. 16.
    Desjardins, B., Grenier, E.: 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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Desjardins, B., Grenier, E., Lions, P.-L., Masmoudi, N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)MathSciNetGoogle Scholar
  18. 18.
    Duchon, J., Robert, R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13, 249–255 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ducomet, B., Feireisl, E.: On the dynamics of gaseous stars. Arch. Rational Mech. Anal. 174, 221–266 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Eliezer, S., Ghatak, A., Hora, H.: An Introduction to Equations of States, Theory and Applications. Cambridge University Press, Cambridge (1986)Google Scholar
  21. 21.
    Ericksen, J.L.: Introduction to the Thermodynamics of Solids. Applied Mathematical Sciences, revised edn, vol. 131, Springer, New York (1998)Google Scholar
  22. 22.
    Fefferman, C.L.: Existence and smoothness of the Navier-Stokes equation. In: The Millennium Prize Problems, pp. 57–67. Clay Math. Inst., Cambridge (2006)Google Scholar
  23. 23.
    Feireisl, E.: Local decay of acoustic waves in the low Mach number limits on general unbounded domains under slip boundary conditions. Commun. Partial Differ. Equ. (2010) (submitted)Google Scholar
  24. 24.
    Feireisl, E., Gallagher, I., Gerard-Varet, D., Novotný, A.: Multi-scale analysis of compressible viscous and rotating fluids. Commun. Math. Phys. 314, 641–670 (2012)zbMATHCrossRefGoogle Scholar
  25. 25.
    Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser-Verlag, Basel (2009)zbMATHCrossRefGoogle Scholar
  26. 26.
    Feireisl, E., Novotný, A.: Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Commun. Math. Phys. (2012) (to appear)Google Scholar
  27. 27.
    Feireisl, E., Novotný, A.: Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Feireisl, E., Novotný, A., Sun, Y.: A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. (2012) (submitted)Google Scholar
  29. 29.
    Feireisl, E., Pražák, D.: A stabilizing effect of a high frequency driving force on the motion of a viscous, compressible, and heat conducting fluid. Discr. Cont. Dyn. Syst. Ser. S. 2, 95–111 (2009)Google Scholar
  30. 30.
    Feireisl, E., Pražák, D.: Asymptotic Behavior of Dynamical Systems in Fluid Mechanics. AIMS, Springfield (2010)zbMATHGoogle Scholar
  31. 31.
    Feireisl, E., Schonbek, M.E.: On the Oberbeck-Boussinesq approximation on unbounded domains. In: Abel Symposium Lecture Notes. Springer, Berlin (2011)Google Scholar
  32. 32.
    Gallavotti, G.: Foundations of Fluid Dynamics. Springer, New York (2002)CrossRefGoogle Scholar
  33. 33.
    Golse, F.: The Boltzmann equation and its hydrodynamic limits. In: Evolutionary Equations. Vol. II. Handb. Differ. Equ., pp. 159–301. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  34. 34.
    Hesla, T.I.: Collision of Smooth Bodies in a Viscous Fluid: A Mathematical Investigation. PhD Thesis Minnesota (2005)Google Scholar
  35. 35.
    Hillairet, M.: Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Partial Differ. Equ. 32(7–9), 1345–1371 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hillairet, M., Takahashi, T.: Collisions in three-dimensional fluid structure interaction problems. SIAM J. Math. Anal. 40(6), 2451–2477 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Hoff, D.: Dynamics of singularity surfaces for compressible viscous flows in two space dimensions. Commun. Pure Appl. Math. 55, 1365–1407 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Hoff, D., Santos, M.M.: Lagrangean structure and propagation of singularities in multidimensional compressible flow. Arch. Ration. Mech. Anal. 188(3), 509–543 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Jesslé, D., Jin, B.J., Novotný, A.: Navier-Stokes-Fourier system on unbounded domains: weak solutions, relative entropies, weak-strong uniqueness (2012) (preprint)Google Scholar
  40. 40.
    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
  41. 41.
    Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Klein, R.: Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Z. Angw. Math. Mech. 80, 765–777 (2000)zbMATHCrossRefGoogle Scholar
  44. 44.
    Klein, R.: Multiple spatial scales in engineering and atmospheric low Mach number flows. ESAIM Math. Mod. Numer. Anal. 39, 537–559 (2005)Google Scholar
  45. 45.
    Klein, R., Botta, N., Schneider, T., Munz, C.D., Roller, S., Meister, A., Hoffmann, L., Sonar, T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Eng. Math. 39, 261–343 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Leis, R.: Initial-Boundary Value Problems in Mathematical Physics. B.G. Teubner, Stuttgart (1986)zbMATHCrossRefGoogle Scholar
  47. 47.
    Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Lighthill, J.: On sound generated aerodynamically I. General theory. Proc. R. Soc. Lond. A. 211, 564–587 (1952)Google Scholar
  49. 49.
    Lighthill, J.: On sound generated aerodynamically II. General theory. Proc. R. Soc. Lond. A. 222, 1–32 (1954)Google Scholar
  50. 50.
    Lighthill, J.: Waves in Fluids. Cambridge University Press, Cambridge (1978)zbMATHGoogle Scholar
  51. 51.
    Lions, P.-L.: Mathematical Topics in Fluid Dynamics, vol. 1. Incompressible Models. Oxford Science Publication, Oxford (1996)Google Scholar
  52. 52.
    Lions, P.-L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Liu, J.-G., Liu, J., Pego, R.L.: On incompressible Navier-Stokes dynamics: a new approach for analysis and computation. In: Hyperbolic Problems: Theory, Numerics and Applications, vol. I, pp. 29–44. Yokohama Publ., Yokohama (2006)Google Scholar
  54. 54.
    Liu, J.-G., Liu, J., Pego, R.L.: Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate. Commun. Pure Appl. Math. 60(10), 1443–1487 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Masmoudi, N.: The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary. Arch. Rational Mech. Anal. 142(4), 375–394 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Masmoudi, N.: Asymptotic problems and compressible and incompressible limits. In: Málek, J., Nečas, J., Rokyta, M. (eds.) Advances in Mathematical Fluid Mechanics, pp. 119–158. Springer, Berlin (2000)Google Scholar
  57. 57.
    Masmoudi, N.: Ekman layers of rotating fluids: the case of general initial data. Commun. Pure Appl. Math. 53(4), 432–483 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Masmoudi, N.: Incompressible inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré, Anal. non linéaire 18, 199–224 (2001)Google Scholar
  59. 59.
    Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Dafermos, C., Feireisl, E. (eds.) Handbook of Differential Equations, III. Elsevier, Amsterdam (2006)Google Scholar
  60. 60.
    Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible and heat conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37, Springer, Heidelberg (1998)Google Scholar
  63. 63.
    Priezjev, N.V., Darhuber A.A., Troian, S.M.: Slip behavior in liquid films on surfaces of patterned wettability: comparison between continuum and molecular dynamics simulations. Phys. Rev. E 71, 041608 (2005)Google Scholar
  64. 64.
    Priezjev, N.V., Troian, S.M.: Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions. J. Fluid Mech. 554, 25–46 (2006)zbMATHCrossRefGoogle Scholar
  65. 65.
    Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Reed, M., Simon, B.: Methods of modern mathematical physics, vol. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)Google Scholar
  67. 67.
    Scheffer, V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343–401 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Schochet, S.: The mathematical theory of low Mach number flows. M2ANMath. Model Numer. Anal. 39, 441–458 (2005)Google Scholar
  69. 69.
    Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Shnirelman, A.: Weak solutions of incompressible Euler equations. In: Handbook of Mathematical Fluid Dynamics, vol. II, pp. 87–116. North-Holland, Amsterdam (2003)Google Scholar
  71. 71.
    Shvydkoy, R.: Lectures on the Onsager conjecture. Discret. Contin. Dyn. Syst. Ser. S. 3(3), 473–496 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Sohr, H.: The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)zbMATHCrossRefGoogle Scholar
  73. 73.
    Valli, A.: A correction to the paper: An existence theorem for compressible viscous fluids (Ann. Mat. Pura Appl. 130(4), 197–213 (1982); MR 83h:35112). Ann. Mat. Pura Appl. 132 (4), 399–400 (1983)(1982)Google Scholar
  74. 74.
    Valli, A.: An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. 130(4), 197–213 (1982)Google Scholar
  75. 75.
    Valli, A., Zajaczkowski, M.: Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Wiedemann, E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(5), 727–730 (2011)Google Scholar
  77. 77.
    Xin, Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)zbMATHCrossRefGoogle Scholar

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© Universidad Complutense de Madrid 2013

Authors and Affiliations

  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrague 1Czech Republic
  2. 2.Faculty of Mathematics and Physics, Mathematical InstituteCharles University in PraguePrague 8Czech Republic

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