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Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations. (English) Zbl 0582.35092

The fractal and Hausdorff dimension of the universal attractor for the Navier-Stokes equation are estimated in terms of the Grashof number, improving the previous estimations. The equations governing the transport of finite-dimensional volume elements in the phase space are established, and the existence of a critical dimension (such that every volume depassing this dimension decays exponentially in time) is proved. The authors show that upper bounds for the fractal and Hausdorff dimension of the attractor may be obtained in terms of the global Lyapunov exponents [replacing the local exponents used in the J. L. Kaplan-J. A. Yorke conjecture, Lect. Notes Math. 730, 204-227 (1979; Zbl 0448.58020)].
Reviewer: G.Pasa

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure

Citations:

Zbl 0448.58020
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References:

[1] Babin, Dokladi Akad. Nauk S. S. S. R. 264 pp 780– (1982)
[2] Babin, Uspehi Mat. Nauk 3 pp 225– (1982)
[3] Constantin, C. R. Acad. Sci. Paris 296 pp 23– (1983)
[4] Douady, C. R. Acad. Sci. Paris 290 pp 1135– (1980)
[5] Farmer, Physica D 4D pp 366– (1982)
[6] Solutions statistiques des équations de Navier-Stokes, Cours au College de France, 1974, mimeographed notes.
[7] Foias, Comm. in P. D. E. 6 pp 329– (1981)
[8] , , and , Asymptotic analysis of the Navier-Stokes equations, Physica D., 1983, pp. 157–188. · Zbl 0584.35007
[9] Foias, Rend. Sem. Padova 39 pp 1– (1967)
[10] Foias, J. Math. Pures et Appl. 58 pp 339– (1979)
[11] Foias, Physics Letters 93A pp 451– (1983) · doi:10.1016/0375-9601(83)90628-X
[12] Ordinary Differential Equations, Birkhäuser, Boston, 1982, p. 242.
[13] and , Chaotic behaviour of multidimensional difference equations, Functional Differential Equations and Approximation of Fixed Points, and , editors, Lecture Notes in Mathematics 730, Springer, Berlin, 1979, p. 219.
[14] Perturbation Theory for Linear Operators, Springer, Berlin, 1976. · doi:10.1007/978-3-642-66282-9
[15] Ladyzhenskaia, Dokladi Adad. Nauk S. S. S. R. 263 pp 802– (1982)
[16] Mallet-Paret, J. Diff. Eq. 22 pp 331– (1976)
[17] Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977.
[18] Métivier, J. Math. Pures et Appl. 57 pp 133– (1978)
[19] Minea, Revue Roumaine de Math. Pures et Appl. 21 pp 1071– (1976)
[20] Ergotic theory of differential dynamical systems, Publications Mathematiques IHES 50, 1979, pp. 275–306.
[21] Large volume limit of the distribution of charactertics exponents in turbulence, IHES preprint 82145.
[22] Singular Integrals and the Differentiability Properties of Functions, Princeton University Press, 1970.
[23] Navier-Stokes equations and nonlinear functional analysys, NSF/CBMS Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.
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