Wong-Zakai Approximations Of Attractors In Stochastic Ginzburg-Landau Equation

In this paper,we consider the existence and the upper semi-continuity of pull-back attractor in non-autonomous stochastic dynamical system.And we focus on the 1D-Ginzburg-Landau equations and 2D-Ginzburg-Landau equations which is difference about the existence and the upper semi-continuity of pullback attractor.Firstly,we introduce the background and the development prospect of non-autonomous stochastic dynamical systems and non-autonomous random attractors,and what has been done and what has been found so far.The proposal and devel-opment of the Wong-Zakai process are introduced,and the present research on the Ginzburg-Landau equations is also described.Secondly,we introduce a parametric dynamical system,and define a dynamical process ? with parametric equations.By an abstract combined results on both existence and upper semi-continuity of random attractors.In this paper,we have to verify three aspects:(a)the convergence of the solution operators,(b)the equi-absorption of the systems for all small size ? of difference noise and(c)the equi-asymptotic compactness in small size.Applying to the actual model,we consider the following Ginzburg-Landau equations:where O=(0,1)(?)R,?,?,?>0,?,??Cb(R,R)and f?Lloc2(R,L2(O)),consid-ering the existence of pullback attractor A? driven by linear multiplicative noise in terms of Wong-Zakai process.And we consider the following equations in Wiener-like process:(?)-(?+i?(t))?u=?u-(?+i?(t))|u|2u+f(t,x)+uodW/dt that the existence of pullback attractor A0.Moreover,we establish the upper semi-continuity of random attractors as the size of difference noise tend to zero.Finally,we study the non-autonomous 2D-Ginzburg-Landau equations per-turbed by white noise and Wong-Zakai noise,respectively.A basic objective is to obtain a pullback random attractors,which is a family of random sets indexed by two parameters.The main objective is to establish the double robustness of the attractors when both parameters are simultaneously convergent.We find a phenomenon in 2D-Ginzburg-Landau equations that the convergence of solutions is available when the initial data are restricted on a proper subspace only.So,we es-tablish the double robustness theorem by part joint-convergence,regularity,eventual local-compactness and recurrence.Our results relax the convergence conditions of cocycles in all known robustness theorems and can be applied to weakly dissipative equations,e.g.the 2D-GL equation in this paper.