zbMATH — the first resource for mathematics

Fractal dimension of a random invariant set. (English) Zbl 1134.37364
Summary: In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche’s techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in A. Debussche’s paper [J. Math. Pures Appl. (9) 77, No. 10, 967–988 (1998;Zbl 0919.58044)] and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2D Navier-Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
28A80 Fractals
35B41 Attractors
37C45 Dimension theory of smooth dynamical systems
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
37L55 Infinite-dimensional random dynamical systems; stochastic equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Arnold, L., Random dynamical systems, (1998), Springer New York
[2] Babin, A.V.; Vishik, M.I., Attractors of evolution equations, (1992), North-Holland Amsterdam · Zbl 0778.58002
[3] Caraballo, T.; Langa, J.A.; Robinson, J.C., Stability and random attractors for a reaction – diffusion equation with multiplicative noise, Discrete contin. dynam. systems, 6, 875-892, (2000) · Zbl 1011.37031
[4] Caraballo, T.; Langa, J.A.; Robinson, J.C., A stochastic pitchfork bifurcation in a reaction – diffusion equation, Proc. R. soc. London A, 457, 2041-2061, (2001) · Zbl 0996.60070
[5] T. Caraballo, H. Crauel, J.A. Langa, J.C. Robinson, The effect of noise on the Chafee-Infante equation: a nonlinear case study, submitted for publication · Zbl 1173.60022
[6] Chepyzhov, V.V.; Ilyin, A.A., A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear anal., 44, 811-819, (2001) · Zbl 1153.37438
[7] Chepyzhov, V.V.; Vishik, M.I., Attractors for equations of mathematical physics, (2002), Amer. Math. Soc. Colloquium Publications Providence, RI · Zbl 0986.35001
[8] Chueshov, I.D., Gevrey regularity of random attractors for stochastic reaction – diffusion equations, Random oper. stochastic equations, 8, 143-162, (2000) · Zbl 0959.60045
[9] Chueshov, I.D.; Duan, J.; Schmalfuss, B., Determining functionals for random partial differential equations, Nodea: nonlinear differential equations appl., 10, 431-454, (2003) · Zbl 1041.60053
[10] Chueshov, I.D.; Scheutzow, M., On the structure of attractors and invariant measures for a class of monotone random systems, Dynam. systems, 19, 127-144, (2004) · Zbl 1061.37032
[11] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. amer. math. soc., 53, (1985) · Zbl 0567.35070
[12] Constantin, P.; Foias, C., Navier – stokes equations, (1988), Univ. of Chicago Press Chicago · Zbl 0687.35071
[13] Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probab. theory related fields, 100, 365-393, (1994) · Zbl 0819.58023
[14] Crauel, H.; Flandoli, F., Hausdorff dimension of invariant sets for random dynamical systems, J. dynam. differential equations, 10, 449-474, (1998) · Zbl 0927.37031
[15] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. dynam. differential equations, 9, 307-341, (1997) · Zbl 0884.58064
[16] Debussche, A., On the finite dimensionality of random attractors, Stochastic anal. appl., 15, 473-491, (1997) · Zbl 0888.60051
[17] Debussche, A., Hausdorff dimension of a random invariant set, J. math. pures appl., 77, 967-988, (1998) · Zbl 0919.58044
[18] Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R., Exponential attractors for dissipative evolution equations, Res. appl. math. ser., (1994), Wiley New York · Zbl 0842.58056
[19] Edmunds, D.E.; Triebel, H., Function spaces, entropy numbers, differential operators, (1996), Cambridge Univ. Press Cambridge · Zbl 0778.46022
[20] Falconer, K.J., Fractal geometry, (1990), Wiley Chichester · Zbl 0587.28004
[21] Flandoli, F.; Langa, J.A., Determining modes for dissipative random dynamical systems, Stoch. and stoch. rep., 66, 1-25, (1998) · Zbl 0923.35213
[22] Foias, C.; Olson, E.J., Finite fractal dimension and Hölder – lipschitz parametrization, Indiana univ. math. J., 45, 603-616, (1996) · Zbl 0887.54034
[23] Hale, J.K., Asymptotic behavior of dissipative systems, Math. surveys monogr., vol. 25, (1988), Amer. Math. Soc. Providence, RI · Zbl 0642.58013
[24] Hunt, B.; Kaloshin, V.Y., Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12, 1263-1275, (1999) · Zbl 0932.28006
[25] Hunt, B.R.; Sauer, T.; Yorke, J.A.; Hunt, B.R.; Sauer, T.; Yorke, J.A., Prevalence: an addendum, Bull. amer. math. soc., Bull. amer. math. soc., 28, 306-307, (1993) · Zbl 0782.28007
[26] Kukavica, I.; Robinson, J.C., Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem, Physica D, 196, 45-66, (2004) · Zbl 1058.35045
[27] Ladyzhenskaya, O.A., Attractors for semigroups and evolution equations, (1991), Cambridge Univ. Press Cambridge · Zbl 0729.35066
[28] Langa, J.A., Finite-dimensional limiting dynamics of random dynamical systems, Dynam. systems, 18, 57-68, (2003) · Zbl 1038.37041
[29] Langa, J.A.; Robinson, J.C., A finite number of point observations which determine a non-autonomous fluid flow, Nonlinearity, 14, 673-682, (2001) · Zbl 1193.37103
[30] Mañé, R., On the dimension of the compact invariant sets of certain nonlinear maps, (), 230-242
[31] Petersen, K., Ergodic theory, (1983), Cambridge Univ. Press Cambridge · Zbl 0507.28010
[32] Robinson, C., Dynamical systems: stability, symbolic dynamics, and chaos, (1995), CRC Press London · Zbl 0853.58001
[33] Robinson, J.C., Global attractors: topology and finite dimensional dynamics, J. dynam. differential equations, 11, 557-581, (1999) · Zbl 0953.34049
[34] Robinson, J.C., Infinite-dimensional dynamical systems, (2001), Cambridge University Press Cambridge · Zbl 1026.37500
[35] Romanov, A.V., Finite-dimensional limiting dynamics for dissipative parabolic equations, Mat. sbornik, 191, 415-429, (2000) · Zbl 0963.37074
[36] Sauer, T.; Yorke, J.A.; Casdagli, M., Embedology, J. stat. phys., 71, 529-547, (1993) · Zbl 0943.37506
[37] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, Springer appl. math. sci., vol. 68, (1988), Springer-Verlag Berlin · Zbl 0662.35001
[38] Walters, P., Introduction to ergodic theory, (2000), Springer-Verlag New York · Zbl 0958.28011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.