Effective Dynamical Behavior Of Stochastic Partial Differential Equations With A High Oscillation

In the study of complex physical phenomena,the system needs to take the instability induced by the high oscillation related with dramatically varying temperature in random thermal environment into consideration.Since high oscillation problems are widely used in various fields,it is crucial important to explore the system with a high oscillation.However,the behavior of systems affected by the high oscillation becomes difficult to be predicted and presented.Not only in theory but also in practice,it is significant to eliminate the influence of the high oscillation and remain the domain part of the system.This thesis considers the effective dynamical behavior of the stochastic weakly damped sine-Gordon equation with a high oscillation and the stochastic quasi-geostrophic flow equation with a high oscillation.Note that the high oscillation not only appears in the system,but also on the boundary.Therefore,it also discusses the effective dynamical behavior of a stochastic partial differential equation with a highly oscillating dynamical boundary condition in this thesis.These systems are regarded as the slow and fast systems in which all components evolve at drastically different rates due to the singular perturbing parameter at a large time scale.When the slow component is frozen,it shows that the fast component is ergodic.Giving a full consideration to the different expression of the slow component in each system,it further obtains that the slow component admits tightness by the classical Prokhorov theorem.Applying the averaging principle,the nonlinear term coupled high oscillation is handled to simplify the original system into an effective system.In other words,the high oscillation in the original system is averaged out by using the unique invariant measure,and an effective system further is established.Then employing the classical Khaminskiis time discretization method,the auxiliary equation is constructed on the partitioning time interval.And the auxiliary system converges to the original system and the effective system respectively as the singular perturbing parameter tends to zero.Moreover,there exists the corresponding effective process converging to the original in some strong sense.It further provides a more accurate rate of the original solution converging to the effective.More precisely,it establishes the moderate deviation principle of a stochastic partial differential equation with a highly oscillating dynamical boundary condition under a certain deviation scale.