Stochastic Partial Differential Equations With Two Time-scales

Some complex systems in science and engineering are described by stochastic ordinary differential equations (SODEs) or stochastic partial differential equations (SPDEs) with multiple time scales. The qualitative analysis for such stochastic sys-tems has been drawing more and more attention. This thesis mainly studies the asymp-totic behavior of SPDEs with two characteristic time-scales. It consists of two topics: the strong convergence in stochastic averaging principle for two time-scales SDPEs and the inertial manifolds reduction for two time-scales stochastic evolutionary sys-tems.This thesis is organized as follows:In Chapter 1, some primary definition, notion and the well-known It o formula in theory of stochastic processes and random dynamical systems is presented.In Chapter 2, the stochastic FitzHugh-Nagumo system subjected with additive noise is explored. Under the dissipative conditions, it can been shown that the "frozen" equation with respect to the fast motion has a unique invariant measure with exponen-tial mixing property. As a result, an effective equation for the slow motion can be de-rived by averaging its drift coefficient. And then, the strong convergence in stochastic averaging principle is proved by estimating the difference between the solution pro-cess of slow equation and that of the effective equation in suitable space.In Chapter 3, a stochastic parabolic equations with two time-scales on a bounded open interval is studied. The present model arise from some physical systems with noise perturbations to the state space. In this case, the noise is included in both fast motion and slow motion and it is of multiplicative type. The strong convergence in stochastic averaging principle for such stochastic parabolic equations is established by techniques similar to Chapter 2. But since more regular multiplicative noise is con-sider, the proof is thus modified and is of complexity. Under suitable conditions, the existence of an exponential mixing invariant measure for the fast equation with frozen slow variable is proved. As a consequence, the averaged equation which captures the dynamic of the slow motion can be obtained. Finally, with aid of the Burkholder- Davis-Gundy inequality, it is proved that the solution process of the slow equation strong convergence to the solution of the effective equation when the time scale pa-rameter tends to zero.In Chapter 4, invariant manifolds for infinite dimensional random dynamical sys-tems with two time-scales is studied. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations, which could be coupled SPDEs, or coupled SPDEs-SODEs. Under suitable conditions, it is proved that an exponen-tially tracking random invariant manifold exists, eliminating the fast motions for this coupled system. It is further shown that if the time scale parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differen-tial equations and coupled hyperbolic-hyperbolic partial differential equations.In the final chapter, a summary of this thesis is made and some questions that need to be improved and perfected in the future study are raised.