Well Posedness And Asymptotics For 2-D Stochastic Generalized Ginzburg-landau Equation With Robin Boundary Data

Ginzburg-Landau equation caught attention of many experts and scholars due to much physical connotation. We were interested in the stochastic gen-eralized Ginzburg-Landau equation in 2 dimensional bounded domain, and this work studied the well posedness and asymptotic behavior for this model with Robin bound-ary data.The first chapter briefly introduced the physical background, the known studies of Ginzburg-Landau equation, well posedness of stochastic partial differential equa-tion and stochastic dynamical system theory. Meanwhile, chapter 1 presented some definitions, inequalities and lemmas which will be used in this work.Chapter 2 considered well posedness of the 2-D stochastic generalized Ginzburg-Landau equation with Robin boundary data. A truncated function was proposed to con-struct a truncated equation. By setting appropriate functional space and corresponding operator, with the help of Banach contraction principle for the truncated equation, we proved that the original equation has a unique local mild solution. Sobolev embed-ding, Holder inequality and properties of semigroup S(t) were used in the proof. Fur-thermore, the global existence was shown by setting energy functional and by taking advantage of Ito formula and Burkholder-Davis-Gundy inequality. It is necessary to overcome the difficulties caused by Robin boundary data by using Green formula and Trace theory.Chapter 3 proved the existence of random attractor for the stochastic dynami-cal system possessed by the 2-D stochastic generalized Ginzburg-Landau equation with Robin boundary data. Firstly, the stochastic dynamical system was equiva-lently changed to random dynamical system by introducing corresponding Ornstein- Uhlenbeck process to vanish the noise. Then by establishing absorbing sets in different spaces and by use of compact embedding theory, the existence of random attractor was obtained. Because the spacial dimension is 2, it will need more elaborate skills when employing Sobolev inequality, Gagliardo-Nirenburg inequality and Agmon inequality to estimate norms.The last chapter expressed the further plan and our next research.