Hausdorff dimension of a random invariant set

Journal des Mathematiques Pures et Appliquees

Volumn 77, Issue 10, 1998, Pages 967-988

  Debussche, A ^a  

^a CNRS   (France)

Author keywords

Attractor; Hausdorff dimension; Lyapunov exponents; Stochastic partial differential equation

Indexed keywords

EID: 0032325270     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-7824(99)80001-4     Document Type: Article

References (21)

1. [1] A. V. Babin, M.I. Vishik, Attractors of Navier-Stokes systems and of parabolic equations and estimates for their dimensions, J. Soviet Math., 28, no 5, 1985, p. 619-627.
2. [2] A. V. Babin, M.I. Vishik, Attractors for Evolution Equations, North-Holland, Amsterdam, 1992.
3. [3] C. Castaing, M. Valadier, Convex analysis and mesurable multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin 1977.
4. [4] P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Memoirs of the Amer. Math. Soc., Vol. 53, no 314, 1985.
5. [5] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Prob. Theory and Related Fields, 100, 1994, p. 365-393.
6. [6] H. Crauel, F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, To appear in Journal of Dyn. and Diff. Eq...
7. [7] H. Crauel, A. Debussche, F. Flandoli, Random attractors, Journal of Dyn. and Diff. Eq., vol. 9, no 2, 1997, p. 307-341.
8. [8] G. Da Prato, J. Zabczyck, Ergodicity for infinite dimensional systems, London Mathematical Society, Lecture Note Series 229, Cambridge University Press, 1996.
9. [9] A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, vol. 15. no 4, 1997, p. 473-492.
10. [10] A. Douady, J. Oesterlé, Dimension de Hausdorff des attracteurs, C. R. Acad. Sci. Paris, Sér. A, 290, 1980, p .1135-1138.
11. [11] N. Dunford, J. T. Schwartz, Linear Operators, Interscience, New York, 1958.
12. [12] C. Foias, R. Temam, Some analytic and geometrical properties of the solutions of Navier-Stokes equations, J. Math. Pures Appl., 58, 1979, p. 339-368.
13. [13] J.K. Hale, Asymptotic behaviour of dissipative dynamical systems, Mathematical Surveys and Monographs, Vol. 25, AMS, Providence, 1988.
14. [14] O. A. Ladyzhenskaya, On the finitness of the dimension of bounded invariant sets for the Navier-Stokes equations and other related dissipative systems, J. Soviet Math., 28, no 5, 1985, p, 714-725.
15. [15] J.-L. Lions (1969) Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars.
16. [16] J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, Journal of Diff. Eq., 22, 1976, p. 331.
17. [17] M. Marion, Attractors for reaction-diffusion equations ; Existence and estimate of their dimension, Appl. Anal., 25, 1987, p. 101-147.
18. [18] M. Reed, B. Simon, Methods of modem mathematical physics, Academic Press, New York, 1978.
19. [19] B. Schmalfuß, Measure attractors and stochastic attractors, Technical Report 332, Universität Bremen, Institut für Dynamische Systeme, 1995. To appear.
20. [20] B. Schmalfuß, The stochastic attractor of the stochastic Lorenz system, To appear.
21. [21] R. Temam, Infinite dimensional dynamical systems in mechanics and physics, Springer-Verlag, New York, second edition, 1997.