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Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise. (English) Zbl 0840.60054
Summary: A class of semilinear nonautonomous parabolic equations subjected to additive white noise is considered. The existence of a family of random \(N\)-dimensional approximate inertial manifolds (AIMs) whose neighborhoods of thickness of order \(\exp (- \delta \lambda^\beta_{N + 1})\) attract exponentially in the mean all the trajectories is proved for \(N\) large enough. Here \(\lambda_{N + 1}\) is the \((N + 1)\)th eigenvalue of the corresponding linear problem, and \(\delta\) and \(\beta\) are positive constants. We also construct a sequence of AIMs which converges to the exact inertial manifold, when a spectral gap condition is satisfied. These results remain true for deterministic autonomous and nonautonomous cases.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] Belopolskaya, Ya. I., and Daletcky, Yu. L. (1990).Stochastic Equations and Differential Geometry, Kluwer, Dordrecht.
[2] Bensousan, A., and Flandoli, F. (1994). Stochastic inertial manifold. Scuola Normale Superiore, Pisa, Preprint No. 07.
[3] Chow, S.-N., and Lu, K. (1988). Invariant manifolds for flows in Banach spaces.J. Diff. Eq. 74, 285–317. · Zbl 0691.58034 · doi:10.1016/0022-0396(88)90007-1
[4] Chueshov, I. D. (1993). Global attractors for nonlinear problems of mathematical physics.Russ. Math. Surv. 48(3), 133–161. · Zbl 0805.58042 · doi:10.1070/RM1993v048n03ABEH001033
[5] Chueshov, I. D. (1995). On approximate inertial manifolds for stochastic Navier-Stokes equations,J. Math. Anal. Appl. (in press). · Zbl 0849.60065
[6] Chueshov, I. D., and Girya, T. V. (1994a). Inertial manifolds for stochastic dissipative dynamical systems.Doklady Acad. Sci. Ukraine 7, 42–45. · Zbl 0851.60036
[7] Chueshov, I. D., and Girya, T. V. (1994b). Inertial manifolds and invariant measures for semilinear parabolic equations subjected to additive white noise. University of Bielefeld, BiBoS Preprint 657/7/94. · Zbl 0831.58022
[8] Constantin, P., and Foias, C. (1988).Navier-Stokes Equations, University of Chicago Press, Chicago. · Zbl 0687.35071
[9] Debussche, A., and Dubois, T. (1994). Approximation of exponential order of the attractor of a turbulent flow.Physica D 72, 372–389. · Zbl 0814.76030 · doi:10.1016/0167-2789(94)90239-9
[10] Debussche, A., and Temam, R. (1994). Convergent families of approximate inertial manifolds.J. Math. Pure Appl. 73, 489–522. · Zbl 0836.35063
[11] Foias, C., Manley, O., and Temam, R. (1988a). Modeling of the interaction of small and large eddies in two dimensional turbulent flows.Math. Mod. Num. Anal. 22, 93–114. · Zbl 0663.76054
[12] Foias, C., Sell, G. R., and Temam, R. (1988b). Inertial manifolds for nonlinear evolutionary equations.J. Diff. Eq. 73, 309–353. · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
[13] Foias, C., Sell, G. R., and Titi, E. (1989). Exponential tracking and approximation of inertial manifolds for dissipative equations.J. Dyn. Diff. Eq. 1, 199–224. · Zbl 0692.35053 · doi:10.1007/BF01047831
[14] Girya, T. V., and Chueshov, I. D. (1995). Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems.Matem. Sbornik 186(1), 29–45. · Zbl 0851.60036 · doi:10.1070/SM1995v186n01ABEH000002
[15] Henry, D. (1981).Geometric Theory of Semilinear Parabolic Equations, Springer, New York. · Zbl 0456.35001
[16] Jones, D. A., and Titi, E. S. (1994). A remark on quasi-stationary approximate inertial manifolds for Navier-Stokes equations.SIAM J. Math. Anal. 25, 894–914. · Zbl 0808.35102 · doi:10.1137/S0036141092230428
[17] Kuo, H.-H. (1975).Gaussian Measures in Banach Spaces, Springer, New York. · Zbl 0306.28010
[18] Temam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York. · Zbl 0662.35001
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