Random attractors

  • Hans Crauel
  • Arnaud Debussche
  • Franco Flandoli


In this paper, we generalize the notion of an attractor for the stochastic dynamical system introduced in [7]. We prove that the stochastic attractor satisfies most of the properties satisfied by the usual attractor in the theory of deterministic dynamical systems. We also show that our results apply to the stochastic Navier-Stokes equation, the white noise-driven Burgers equation, and a nonlinear stochastic wave equation.

Key words

Random attractors stochastic dynamical systems deterministic nonautonomous systems stochastic Navier-Stokes equation stochastic Burgers equation stochastic nonlinear wave equation 

AMS subject classifications

58F11 58F12 60H15 34D45 35Q30 35Q35 35L05 


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  1. 1.
    A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes.J. Funct. Anal. 13, 195–222, 1973.CrossRefGoogle Scholar
  2. 2.
    Z. Brzezniak, M. Capinski, and F. Flandoli, Pathwise global attractors for stationary random dynamical systems.Prob. Th. Rel. Fields 95, 87–102, 1993.CrossRefGoogle Scholar
  3. 3.
    R. Carmona and D. Nualart, Random non-linear wave equations: Smoothness of the solutions.Prob. Th. Rel. Fields 79, 469–508, 1988.CrossRefGoogle Scholar
  4. 4.
    C. Castaing and M. Valadier,Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin, 1977.Google Scholar
  5. 5.
    D. H. Chambers, R. J. Adrian, P. Moin, D. S. Stewart, and H. J. Sung, Karhunen-Loeve expansion of Burger's model of turbulence.Phys. Fluids 31(9), 2573–2582, 1988.CrossRefGoogle Scholar
  6. 6.
    H. Choi, R. Temam, P. Moin, and J. Kim, Feedback control for unsteady flow and its application to Burgers equation.J. Fluid Mech. 253, 509–543, 1993.Google Scholar
  7. 7.
    H. Crauel and F. Flandoli, Attractors for random dynamical systems,Prob. Th. Rel. Fields 100, 365–393, 1994.CrossRefGoogle Scholar
  8. 8.
    H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems,J. Dynamics Differential Equations, 1994.Google Scholar
  9. 9.
    G. Da Prato and D. Gatarek, Stochastic Burgers equation with correlated noise, Preprint 4, Scuola Normale Superiore di Pisa, 1994.Google Scholar
  10. 10.
    G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions.Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.Google Scholar
  11. 11.
    G. Da Prato, A. Debussche, and R. Temam, Stochastic Burger's equation.Nonlin. Diff. Eq. Appl. 1, 389–402, 1994.CrossRefGoogle Scholar
  12. 12.
    F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations.Nonlin. Diff. Eq. Appl. 1, 403–423, 1994.CrossRefGoogle Scholar
  13. 13.
    F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Preprint 14, Scuola Normale Superiore di Pisa.Prob. Th. Rel. Fields 102(3), 367–391, 1995.CrossRefGoogle Scholar
  14. 14.
    F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Preprint 20, Scuola Normale Superiore di Pisa.Comm. Math. Phys. 172(1), 119–141, 1995.CrossRefGoogle Scholar
  15. 15.
    J. K. Hale,Asymptotic Behaviour of Dissipative Dynamical Systems, Mathematical Surveys and Monographs, Vol. 25, AMS, Providence, 1988.Google Scholar
  16. 16.
    A. Haraux, Attractors of asymptotically compact processes and applications to nonlinear partial differential equations.Comm. PDE 13(11), 1383–1414, 1988.Google Scholar
  17. 17.
    A. Haraux,Systèmes Dynamiques Dissipatifs et Applications. Collection RMA 17, Masson, Paris, 1991.Google Scholar
  18. 18.
    I. Hosokawa and K. Yamamoto, Turbulence in the randomly forced one dimensional Burgers flow.J. Stat. Phys. 13, 245, 1975.CrossRefGoogle Scholar
  19. 19.
    H. Morimoto, Attractors of probability measures for semilinear stochastic evolution equations.Stoch. Anal. Appl. 10, 205–212, 1992.Google Scholar
  20. 20.
    B. Schmalfu\, Measure Attractors of the Stochastic Navier-Stokes equation, Report 258, Institut für Dynamische Systeme, Bremen, 1991.Google Scholar
  21. 21.
    R. Temam,Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.Google Scholar
  22. 22.
    M. I. Vishik,Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.Google Scholar
  23. 23.
    M. I. Vishik and A. V. Fursikov,Mathematical Problems of Statistical Hydromechanics, Kluver, Dordrecht, 1980.Google Scholar
  24. 24.
    H. F. Yashima,Equations de Navier-Stokes Stochastiques Non Homogenes et Applications, Tesi di perfezionamento, Scuola Normale Superiore, Pisa, 1992.Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Hans Crauel
    • 1
  • Arnaud Debussche
    • 2
  • Franco Flandoli
    • 3
  1. 1.SaarbrückenGermany
  2. 2.Laboratoire d'Analyse Numérique et CNRSUniversité Paris-SudOrsay CedexFrance
  3. 3.Dipartimento di Matematica Applicata U. DiniPisaItaly

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