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The Turbulent Fluid as a Dynamical System

  • David Ruelle
Conference paper

Abstract

This paper reviews the applications of the theory of differentiable dynamical systems to the understanding of chaos and turbulence in hydrodynamics. It is argued that this point of view will remain useful in the still elusive strongly turbulent regime.

Keywords

Rayleigh Number Strange Attractor Characteristic Exponent Ergodic Measure Sensitive Dependence 
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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Références

  1. [1]
    C. Foias and R. Temam. Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. pures et appl. 58, 339–368 (1979).MathSciNetzbMATHGoogle Scholar
  2. [2]
    D. Ruelle. Differentiable dynamical systems and the problem of turbulence. Bull. Amer. Math. Soc. 5, 29–42 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    O.A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. 2nd ed., Nauka, Moscow, 1970; 2nd English ed., Gordon Breach, New York, 1969.Google Scholar
  4. [4]
    J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969.zbMATHGoogle Scholar
  5. [5]
    R. Temam. Navier-Stokes equations. Revised ed., North Holland, Amsterdam, 1979.zbMATHGoogle Scholar
  6. [6]
    V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations. Springer, Berlin, 1986.zbMATHGoogle Scholar
  7. [7]
    L. Caffarelli, R. Kohn and L. Nirenberg. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. pure appl. Math. 35, 771–831 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    V. Scheffer. A self-focussing solution to the Navier-Stokes equations with a speed-reducing external force. pp 1110–1112 in Proceedings of the International Congress of Mathematicians 1986 Amer. Math. Soc., Providence R.I., 1987.Google Scholar
  9. [9]
    E. Hopf. A mathematical example displaying the features of turbulence. Commun. pure appl. Math. 1, 303–322 (1948).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    L.D. Landau. On the problem of turbulence. Dokl. Akad. Nauk SSSR 44 No 8, 339–342 (1944).Google Scholar
  11. [11]
    H. Poincaré. Science et méthode. Ernest Flammarion, Paris, 1908.Google Scholar
  12. [12]
    E.N. Lorenz. Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963).CrossRefGoogle Scholar
  13. [13]
    D. Ruelle and F. Takens. On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    G. Ahlers. Low temperature studies of the Rayleigh-Bénard instability and turbulence. Phys. Rev. Lett. 33, 1185–1188 (1974).CrossRefGoogle Scholar
  15. [15]
    J.P. Gollub and H.L. Swinney. Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927–930 (1975).CrossRefGoogle Scholar
  16. [16]
    Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980).MathSciNetCrossRefGoogle Scholar
  17. [17]
    M.F. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Statist. Phys. 19, 25–52 (1987).MathSciNetCrossRefGoogle Scholar
  18. [18]
    M.J. Feigenbaum. The universal metric properties of nonlinear transformation. J. Statis. Phys. 21, 669–706 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    P. Cvitanović. Universality in Chaos. Adam Hilger, Bristol, 1984.zbMATHGoogle Scholar
  20. [20]
    Hao Bai-Lin. Chaos. World Scientific, Singapore, 1984.zbMATHGoogle Scholar
  21. [21]
    J.-P. Eckmann. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    P. Bergé, Y. Pomeau and Chr. Vidal. Order within Chaos. J. Wiley, New York, 1987.Google Scholar
  23. [23]
    N.H. Packard, J.P. Crutchfield, J.D. Farmer and R.S. Shaw. Geometry from a time series. Phys. Rev. Letters 45, 712–716 (1980).CrossRefGoogle Scholar
  24. [24]
    J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985).MathSciNetCrossRefGoogle Scholar
  25. [25]
    L.-S. Young. Dimension, entropy and Lyapunov exponents. Ergod. Th. and Dynam. Syst. 2, 109–124 (1982).zbMATHCrossRefGoogle Scholar
  26. [26]
    P. Grassberger and I. Procaccia. Measuring the strangeness of strange attractors. Physica 9D, 189–208 (1983).Google Scholar
  27. [27]
    P. Frederickson, J.L. Kaplan, E.D. Yorke and J.A. Yorke. The Lyapunov dimension of strange attractors. J. Diff. Equ. 49, 185–207 (1983).MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    F. Ledrappier. Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    P. Collet, J.L. Lebowitz and A. Porzio. The dimension spectrum of some dynamical systems. J. Statist. Physics 47, 609–644 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    D. Ruelle. Resonances of chaotic dynamical systems. Phys. Rev. Letters 56, 405–407 (1986).MathSciNetCrossRefGoogle Scholar
  31. [31]
    B. Malraison, P. Atten, P. Bergé and M. Dubois. Dimension of strange attractors: an experimental determination for the chaotic regime of two convective systems. J. Physique-Lettres 44, L–897–L–902 (1983).Google Scholar
  32. [32]
    J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto. Lyapunov exponents from time series. Phys. Rev. A 34, 4971–4979 (1986).MathSciNetCrossRefGoogle Scholar
  33. [33]
    D. Ruelle. Large volume limit of the distribution of characteristic exponents in turbulence. Commun. Math. Phys. 87, 287–302 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    D. Ruelle. Characteristic exponents for a viscous fluid subjected to time dependent forces. Commun. Math. Phys. 93, 285–300 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    E. Lieb and W. Thirring. Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities; pp 269–303 in Essays in honor of Valentine Bargman (edited by E. Lieb, B. Simon, and A.S. Wightman) Princeton University Press, Princeton, NJ, 1976.Google Scholar
  36. [36]
    E. Lieb. On characteristic exponents in turbulence. Commun. Math. Phys. 92, 473–480 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    A.V. Babin and M.I. Vishik. Attractors for partial differential equations of evolution and estimation of their dimension. Usp. Mat. Nauk 38 No 4 (232), 133–182 (1983).MathSciNetGoogle Scholar
  38. [38]
    P. Constantin, C. Foias and R. Temam. Attractors representing turbulent flows. Amer. Math. Soc. Memoirs No 314. Providence, R.I., 1985.Google Scholar
  39. [39]
    J. Mallet-Paret. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Diff. Eq. 22, 331–348 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    J.-P. Eckmann and D. Ruelle. Two-dimensional Poiseuille flow. Physica Scripta. T9, 153–154 (1985).CrossRefGoogle Scholar
  41. [41]
    A. Lafon. Borne sur la dimension de Hausdorff de l’attracteur pour les équations de Navier-Stokes à deux dimensions. C.R. Acad. Sc. Paris 298, Sér. I, 453–456 (1984).MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • David Ruelle

There are no affiliations available

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