Limit Behaviours Of The Stationary Measure For Two Classes Of Stochastic Evolution Equations In A Thin Domain

This thesis firstly investigates limit behaviours of the stationary measure for the stochastic Ginzburg-Landau equation in a thin domain.The tightness of measures is derived by constructing fractional Sobolev space and splitting solutions.Analyzing weak convergence of the nonlinear term and applying the averaging projection with respect to domains,it obtains the weak convergence of stationary statistical solutions to study the convergence of stationary measures.It finally proves that the stationary measure of the stochastic Ginzburg-Landau equation converges weakly to that of the Schr ¨odinger equation in a two-dimensional bounded domain as the thickness of the thin domain and the viscosity tend to zero.In addition,this thesis studies limit behaviours of the stationary measure for the stochastic magnetohydrodynamic in a thin domain.It deduces from the α-model of the three-dimensional stochastic magnetohydrodynamic that the system possesses stationary measures.By the averaging projection with respect to domains,it proves that the stationary measure of the threedimensional α-model converges weakly to that of the two-dimensional stochastic magnetohydrodynamic as the regularization parameter and the thickness of the thin domain tend to zero.Employing the independence of the regularization parameter with respect to the thickness of the thin domain in the convergence of stationary measures,it finally shows that the stationary measure of the three-dimensional stochastic magnetohydrodynamic converges weakly to that of the stochastic magnetohydrodynamic in a two-dimensional torus as the thickness of the thin domain approaches to zero.