Finite-dimensional limiting dynamics of random dynamical systems

摘要

One of the main tools in the description of the long-term dynamics of some stochastic partial differential equations (PDEs) is the random attractor introduced by Crauel and Flandoli (1994; Probability Theory and Related Fields, 100: 365-393) (see also Schmalfuss 1992; International Seminar on Applied Mathematics, pp. 185-192). This is an invariant random compact set attracting every bounded set in the pullback sense. The finite dimensionality of this set is related to the dependence on a finite number of degrees of freedom of the asymptotic pullback dynamics of some stochastic systems. In this paper we prove some results on determining modes for this pullback behaviour, from which we conclude finite-dimensional pull-back asymptotic dynamics for some stochastic PDEs, including the two-dimensional (2D) Navier-Stokes equations with additive noise. In particular, we also prove the finite fractal dimensionality of the random attractor associated to this model from fluid dynamics.