Mathematical Analysis of Fluids in Motion

  • Eduard Feireisl
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 153)


Continuum fluid mechanics is a phenomenological theory based on macroscopic observable state variables, the time evolution of which is described by means of systems of partial differential equations. The resulting mathematical problems are highly non-linear and rather complex, even in the simplest physically relevant situations. We discuss several recent results and newly developed methods based on the concept of weak solution. The class of weak solutions is happily large enough in order to guarantee the existence of global-in-time solutions without any essential restrictions on the size of the relevant data. On the other hand, the underlying structural hypotheses impose quite severe restrictions on the specific form of constitutive relations. The best known open problems - hypothetical presence of vacuum zones, propagation of density oscillations, sequential stability of the temperature field, among others - are discussed. The final part of the study addresses the simplified problems, in particular, the incompressible Navier-Stokes system.


Weak Solution Entropy Production Rate Incompressible Limit Renormalize Solution Total Energy Balance 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Alazard. Low Mach number flows and combustion. SIAM J. Math. Anal., 38(4):1186–1213 (electronic), 2006.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    T. Alazard. Low Mach number limit of the full Navier-Stokes equations. Arch. Rational Mech. Anal., 180:1–73, 2006.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    A.V. Babin and M.I. Vishik. Attractors of evolution equations. North-Holland, Amsterdam, 1992.MATHGoogle Scholar
  4. 4.
    C. Bardos, F. Golse, and C.D. Levermore. Fluid dynamical limits of kinetic equations, I : Formal derivation. J. Statist. Phys., 63:323–344, 1991.CrossRefMathSciNetGoogle Scholar
  5. 5.
    C. Bardos, F. Golse, and C.D. Levermore. Fluid dynamical limits of kinetic equations, II : Convergence proofs for the Boltzman equation. Comm. Pure Appl. Math., 46:667–753, 1993.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    C. Bardos, C.D. Levermore, S. Ukai, and T. Yang. Kinetic equations: fluid dynamical limits and viscous heating. Bull. Inst. Math. Acad. Sin. (N.S.), 3(1):1–49, 2008.MATHMathSciNetGoogle Scholar
  7. 7.
    G.K. Batchelor. An introduction to fluid dynamics. Cambridge University Press, Cambridge, 1967.MATHGoogle Scholar
  8. 8.
    J.T. Beale, T. Kato, and A. Majda. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 94(1):61–66, 1984.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    D. Bresch and 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.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    D. Bresch, B. Desjardins, and D. Gérard-Varet. On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl., 87:227–235, 2007.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    J. Březina and A. Novotńy. On weak solutions of steady navier-stokes equations for monatomic gas. Preprint Nečas Center for Mathematical Modeling, 2007.Google Scholar
  12. 12.
    H. Callen. Thermodynamics and an Introduction to Thermostatistics. Wiley, New Yoerk, 1985.MATHGoogle Scholar
  13. 13.
    S. Chapman and T.G. Cowling. Mathematical theory of non-uniform gases. Cambridge Univ. Press, Cambridge, 1990.Google Scholar
  14. 14.
    G.-Q. Chen and H. Frid. On the theory of divergence-measure fields and its applications. Bol. Soc. Brasil. Mat. (N.S.), 32(3):401–433, 2001. Dedicated to Constantine Dafermos on his 60th birthday.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    G.-Q. Chen and M. Torres. Divergence-measure fields, sets of finite perimeter, and conservation laws. Arch. Ration. Mech. Anal., 175(2):245–267, 2005.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    G.-Q. Chen, M. Torres, and W.P. Ziemer. Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Comm. Pure Appl. Math., 62(2):242–304, 2009.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    P. Constantin and C. Fefferman. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J., 42(3):775–789, 1993.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    P. Constantin and C. Foias. Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.MATHGoogle Scholar
  19. 19.
    P. Constantin, C. Foias, and R. an Temam. Attractors representing turbulent flows. Mem. Amer. Math. Soc., 53, Providence, 1985.Google Scholar
  20. 20.
    P. Constantin, C. Foias, B. Nicolaenko, and R. Temam. Integral and inertial manifolds for dissipative partial differential equations. Springer-Verlag, New York, 1988.Google Scholar
  21. 21.
    R. Danchin. Low Mach number limit for viscous compressible flows. M2AN Math. Model Numer. Anal., 39:459–475, 2005.CrossRefMathSciNetGoogle Scholar
  22. 22.
    P.A. Davidson. Turbulence:An introduction for scientists and engineers. Oxford University Press, Oxford, 2004.MATHGoogle Scholar
  23. 23.
    R.J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98:511–547, 1989.CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev. Modern Phys., 57(3, Part 1):617–656, 1985.CrossRefMathSciNetGoogle Scholar
  25. 25.
    S. Eliezer, A. Ghatak, and H. Hora. An introduction to equations of states, theory and applications. Cambridge University Press, Cambridge, 1986.Google Scholar
  26. 26.
    C.L. Fefferman. Existence and smoothness of the Navier-Stokes equation. In The millennium prize problems, pp. 57–67. Clay Math. Inst., Cambridge, MA, 2006.Google Scholar
  27. 27.
    E. Feireisl. Dynamics of viscous compressible fluids. Oxford University Press, Oxford, 2004.MATHGoogle Scholar
  28. 28.
    E. Feireisl and A. Novotńy. Singular limits in thermodynamics of viscous fluids. Birkhäuser-Verlag, Basel, 2009.CrossRefMATHGoogle Scholar
  29. 29.
    E. Feireisl and D. Prăzák. Asymptotic behavior of dynamical systems in fluid mechanics. AIMS on Applied Mathematics, Vol. 4, Springfield 2010.Google Scholar
  30. 30.
    M Feistauer, J. Felcman, and I. Străskraba. Mathematical and compunational methods for compressible flow. Oxford Science Publications, Oxford, 2003.Google Scholar
  31. 31.
    C. Foias, O. Manley, R. Rosa, and R. Temam. Navier-Stokes equations and turbulence, volume 83 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2001.Google Scholar
  32. 32.
    J. Frehse, S. Goj, and M. Steinhauer. Lp - estimates for the Navier-Stokes equations for steady compressible flow. Manuscripta Math., 116:265–275, 2005.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    G. Gallavotti. Foundations of fluid dynamics. Springer-Verlag, New York, 2002.Google Scholar
  34. 34.
    F. Golse and C.D. Levermore. The Stokes-Fourier and acoustic limits for the Boltzmann equation. Commun. on Pure and Appl. Math., 55, 2002.Google Scholar
  35. 35.
    F. Golse and L. Saint-Raymond. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math., 155:81–161, 2004.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    F. Golse and L. Saint-Raymond. The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. (9), 91(5):508–552, 2009.CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    E. Hopf. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213–231, 1951.MATHMathSciNetGoogle Scholar
  38. 38.
    S. Klainerman and A. Majda. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math., 34:481–524, 1981.CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    S.N. Kruzhkov. First order quasilinear equations in several space variables (in Russian). Math. Sbornik, 81:217–243, 1970.CrossRefGoogle Scholar
  40. 40.
    O.A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 1969.MATHGoogle Scholar
  41. 41.
    J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63:193–248, 1934.CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    P.-L. Lions. Mathematical topics in fluid dynamics, Vol.1, Incompressible models. Oxford Science Publication, Oxford, 1996.Google Scholar
  43. 43.
    P.-L. Lions. Mathematical topics in fluid dynamics, Vol. 2, Compressible models. Oxford Science Publication, Oxford, 1998.Google Scholar
  44. 44.
    P.-L. Lions and N. Masmoudi. Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl., 77:585–627, 1998.CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    A.J. Majda and A.L. Bertozzi. Vorticity and incompressible flow, Vol. 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.Google Scholar
  46. 46.
    N. Masmoudi. Examples of singular limits in hydrodynamics. In Handbook of Dif-ferential Equations, III, C. Dafermos, E. Feireisl Eds., Elsevier, Amsterdam, 2006.Google Scholar
  47. 47.
    N. Masmoudi and L. Saint-Raymond. From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Comm. Pure Appl. Math., 56(9):1263–1293, 2003.CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    A. Mellet and A. Vasseur. On the barotropic compressible Navier-Stokes equations. Comm. Partial Differential Equations, 32(1-3):431–452, 2007.CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    A. Mellet and A. Vasseur. Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal., 39(4):1344–1365, 2007/08.CrossRefMathSciNetGoogle Scholar
  50. 50.
    F. Murat. Compacitépar compensation. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. Ser. 5, IV:489–507, 1978.MathSciNetGoogle Scholar
  51. 51.
    P. Pedregal. Parametrized measures and variational principles. Birkhäuser, Basel, 1997.MATHGoogle Scholar
  52. 52.
    P.I. Plotnikov and J. Sokolowski. Concentrations of stationary solutions to compressible Navier-Stokes equations. Comm. Math. Phys., 258, 2005.Google Scholar
  53. 53.
    P.I. Plotnikov and J. Sokolowski. Stationary solutions of Navier-Stokes equations for diatomic gases. Russian Math. Surveys, 62:3, 2007.CrossRefMathSciNetGoogle Scholar
  54. 54.
    P.I. Plotnikov and J. Sokolowski. On compactness domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations. 2002. Preprint.Google Scholar
  55. 55.
    G. Prodi. Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl., 48:173–182, 1959.CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    K.R. Rajagopal and A.R. Srinivasa. On thermodynamical restrictions of continua. Proc. Royal Soc. London, A 460:631–651, 2004.MathSciNetGoogle Scholar
  57. 57.
    S. Schochet. The mathematical theory of low Mach number flows. M2ANMath. Model Numer. anal., 39:441–458, 2005.CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    G.R. Sell and Y. You. Dynamics of evolutionary equations. Springer-Verlag, Berlin, Heidelberg, 2002.MATHGoogle Scholar
  59. 59.
    J. Serrin. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal., 9:187–195, 1962.CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    J. Simon. Compact sets and the space Lp(0, T;B). Ann. Mat. Pura Appl., 146:65–96, 1987.CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    L. Tartar. Compensated compactness and applications to partial differential equations. Nonlinear Anal. and Mech., Heriot-Watt Sympos., L.J. Knopps editor, Research Notes in Math 39, Pitman, Boston, pp. 136–211, 1975.Google Scholar
  62. 62.
    R. Temam. Navier-Stokes equations. North-Holland, Amsterdam, 1977.MATHGoogle Scholar
  63. 63.
    R. Temam. Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, 1988.MATHGoogle Scholar
  64. 64.
    C. Truesdell and K.R. Rajagopal. An introduction to the mechanics of fluids. Birkhäuser, Boston, 2000.CrossRefMATHGoogle Scholar
  65. 65.
    L. Truesdell. Notes on the history of the general equations of hydrodynamics. Amer. Math. Monthly, 60:445–458, 1953.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Eduard Feireisl
    • 1
  1. 1.Mathematical Institute of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic

Personalised recommendations