# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [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
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.