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The Turbulent Fluid as a Dynamical System

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New Perspectives in Turbulence

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.

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Références

  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).

    MathSciNet  MATH  Google Scholar 

  2. D. Ruelle. Differentiable dynamical systems and the problem of turbulence. Bull. Amer. Math. Soc. 5, 29–42 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  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. J.-L. Lions. Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969.

    MATH  Google Scholar 

  5. R. Temam. Navier-Stokes equations. Revised ed., North Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  6. V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations. Springer, Berlin, 1986.

    MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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. E. Hopf. A mathematical example displaying the features of turbulence. Commun. pure appl. Math. 1, 303–322 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  10. L.D. Landau. On the problem of turbulence. Dokl. Akad. Nauk SSSR 44 No 8, 339–342 (1944).

    Google Scholar 

  11. H. Poincaré. Science et méthode. Ernest Flammarion, Paris, 1908.

    Google Scholar 

  12. E.N. Lorenz. Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963).

    Article  Google Scholar 

  13. D. Ruelle and F. Takens. On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Ahlers. Low temperature studies of the Rayleigh-Bénard instability and turbulence. Phys. Rev. Lett. 33, 1185–1188 (1974).

    Article  Google Scholar 

  15. J.P. Gollub and H.L. Swinney. Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927–930 (1975).

    Article  Google Scholar 

  16. Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980).

    Article  MathSciNet  Google Scholar 

  17. M.F. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Statist. Phys. 19, 25–52 (1987).

    Article  MathSciNet  Google Scholar 

  18. M.J. Feigenbaum. The universal metric properties of nonlinear transformation. J. Statis. Phys. 21, 669–706 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Cvitanović. Universality in Chaos. Adam Hilger, Bristol, 1984.

    MATH  Google Scholar 

  20. Hao Bai-Lin. Chaos. World Scientific, Singapore, 1984.

    MATH  Google Scholar 

  21. J.-P. Eckmann. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Bergé, Y. Pomeau and Chr. Vidal. Order within Chaos. J. Wiley, New York, 1987.

    Google Scholar 

  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).

    Article  Google Scholar 

  24. J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985).

    Article  MathSciNet  Google Scholar 

  25. L.-S. Young. Dimension, entropy and Lyapunov exponents. Ergod. Th. and Dynam. Syst. 2, 109–124 (1982).

    Article  MATH  Google Scholar 

  26. P. Grassberger and I. Procaccia. Measuring the strangeness of strange attractors. Physica 9D, 189–208 (1983).

    Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Ledrappier. Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81, 229–238 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Collet, J.L. Lebowitz and A. Porzio. The dimension spectrum of some dynamical systems. J. Statist. Physics 47, 609–644 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Ruelle. Resonances of chaotic dynamical systems. Phys. Rev. Letters 56, 405–407 (1986).

    Article  MathSciNet  Google Scholar 

  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. J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto. Lyapunov exponents from time series. Phys. Rev. A 34, 4971–4979 (1986).

    Article  MathSciNet  Google Scholar 

  33. D. Ruelle. Large volume limit of the distribution of characteristic exponents in turbulence. Commun. Math. Phys. 87, 287–302 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  34. D. Ruelle. Characteristic exponents for a viscous fluid subjected to time dependent forces. Commun. Math. Phys. 93, 285–300 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  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. E. Lieb. On characteristic exponents in turbulence. Commun. Math. Phys. 92, 473–480 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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. J. Mallet-Paret. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Diff. Eq. 22, 331–348 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  40. J.-P. Eckmann and D. Ruelle. Two-dimensional Poiseuille flow. Physica Scripta. T9, 153–154 (1985).

    Article  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

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Authors
  1. David Ruelle
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Editor information

Editors and Affiliations

  1. Center for Fluid Mechanics, Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Lawrence Sirovich

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© 1991 Springer-Verlag New York Inc.

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Ruelle, D. (1991). The Turbulent Fluid as a Dynamical System. In: Sirovich, L. (eds) New Perspectives in Turbulence. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3156-1_4

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  • DOI: https://doi.org/10.1007/978-1-4612-3156-1_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7817-7

  • Online ISBN: 978-1-4612-3156-1

  • eBook Packages: Springer Book Archive

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