Advertisement

Communications in Mathematical Physics

, Volume 90, Issue 2, pp 187–206 | Cite as

Self-similar universal homogeneous statistical solutions of the Navier-Stokes equations

  • C. Foias
  • R. Temam
Article

Abstract

In this note we consider a family of statistical solutions of the Navier-Stokes equations (i.e. time dependent solutions of the Hopf equation) which seem to constitute the rigorous mathematical framework for the theory of homogeneous turbulence [1], [13]. The main feature of these solutions is that they are the transforms under suitable scalings of thestationary statistical solutions of a new system of equations (the Eq. (2) below).

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Statistical 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Batchelor, G. K.: The theory of homogeneous turbulence. Cambridge: Cambridge University Press 1967Google Scholar
  2. 2.
    Bensoussan, A., Temam, R.: Equations stochastiques de type Navier-Stokes. J. Funct. Anal.13, 195–222 (1973)Google Scholar
  3. 3.
    Cafarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Preprint (to appear)Google Scholar
  4. 4.
    Chorin, A. J.: Lectures on Turbulence. Boston: Publish or Perish Inc. 1975Google Scholar
  5. 5.
    Foias, C.: Ergodic problems in functional spaces related to Navier-Stokes equations. Proc. Intern. Conf. Function. Anal. and related topics, Tokyo (1969) pp. 290–304Google Scholar
  6. 6.
    Foias, C.: Statistical study of the Navier-Stokes equations. I. Rend. Sem. Mat. Univ. Padova48, 219–348 (1972) II., idem,49, 9–123 (1973)Google Scholar
  7. 7.
    Foias, C., Manley, O. P., Temam, R.: A new representation of Navier-Stokes equations governing self-similar homogeneous turbulence. (in preparation)Google Scholar
  8. 8.
    Foias, C., Temam, R.: Some analytic and geometric properties of the solutions of the Navier-Stokes equations. J. Math. Pures Appl.58, 339–368 (1979)Google Scholar
  9. 9.
    Foias, C., Temam, R.: Homogeneous statistical solutions of the Navier-Stokes equations. Indiana Univ. Math. J.29, 913–957 (1980)Google Scholar
  10. 10.
    Foias, C., Temam, R.: Homogeneous statistical solutions of the Navier-Stokes equations, II. (in preparation)Google Scholar
  11. 11.
    Hopf, E.: Statistical hydrodynamic and functional calculus. J. Rat. Mech. Anal.16, 87–123 (1948)Google Scholar
  12. 12.
    Ladysenskaya, O. A., Vershik, A. M.: Sur l'évolution des mesures déterminées par les équations de Navier-Stokes et la résolution du probleme de Cauchy pour l'équation statistique de Hopf. Ann. Scuola Norm. Sup. Pisa, S. IV4, 209–230 (1977)Google Scholar
  13. 13.
    Landau, L. D., Lifshitz, E. M.: Fluid mechanics. New York: Pergamon 1959Google Scholar
  14. 14.
    Leray, J.: Sur le mouvement plan d'un liquide visqueux emplissant l'espace. Acta Math.63, 193–248 (1934)Google Scholar
  15. 15.
    Orszag, S. A.: Lectures on the statistical theory of turbulence Fluid Dynamics (Summer School, Les Houches, 1973). London: Gordon and Breach (1977) pp. 235–374Google Scholar
  16. 16.
    Ruelle, D.: Differential dynamical systems and the problem of turbulence. Bull. Am. Math. Soc.5, 29–42 (1981)Google Scholar
  17. 17.
    Scheffer, V.: Turbulence and Hausdorff dimension, In: Turbulence and Navier-Stokes equations, Temam R. (ed.), Lecture Notes in Mathematics, Vol. 565, Berlin, Heidelberg, New York: Springer (1976) pp. 94–112Google Scholar
  18. 18.
    Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pac. J. Math.66, 535–552 (1976)Google Scholar
  19. 19.
    Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys.55, 97–112 (1977)Google Scholar
  20. 20.
    Scheffer, V.: The Navier-Stokes equations in space dimension four. Commun. Math. Phys.61, 41–68 (1978)Google Scholar
  21. 21.
    Temam, R.: Navier-Stokes equations, 2nd edn. Amsterdam: North-Holland 1979Google Scholar
  22. 22.
    Vishik, M. L., Fursikov, A. V.: Solutions statistiques homogènes des systèmes paraboliques et du systéme de Navier-Stokes. Ann. Scuola Norm. Pisa, S. IV4, 531–576 (1977)Google Scholar
  23. 23.
    Vishik, M. L., Fursikov, A. V.: Translationally homogeneous statistical solutions and individual solutions with infinite energy of the system of Navier-Stokes equations. Sib. Math. J.19, 710–729 (1978)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • C. Foias
    • 1
  • R. Temam
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Laboratoire d'Analyse Numérique, Batiment 425Université Paris-SudOrsayFrance

Personalised recommendations