Upper Semicontinuity Of Random Attractors For Boussinesq Equations With Multiplicative Noise

In this paper,we study the asymptotic behavior of solutions for the Boussinesq equation with mult,iplicative noise,and prove that the random dynamical system generated by the equation exists a unique random attractor in a square integrable space,and it is upper semi-continuous.The paper investigates the Boussinesq equa-tion with multiplicative noise,the domian occupied by the fluid is D =(0,1)×(0,1):(?)This equation is supplemented with the boundary as follows:Among above,e ∈(0,1],e1,e2 are the canonical basis of R2.The unknown v =(v1,v2),P,T stand for the velocity vector,pressure and temperature respec-tively.The constant numbers v>0,k>0 are related to the usual Crashof and Prandtl.T1 is the temperature at the top,x2=1,while T0 = T1 + 1 is the tem-perature at the boundary below,x2 = 0.The multiplicative noise described by a process(t),which is a two-sided real-valued Wiener-process on a probabili-ty space(Ω,F,P,(θt)t∈R),where = {w∈C(R,R):w(0)= 0},F is the Borelσ-algebra induced by the compact-open topology of Ω,P is a Wiener measure,forω∈Ω,and t∈R,θ satisfies θtw(·)=w(·+t)-w(t).The paper contains four chapters:(?)In the first chapter:we introduce the background and the research status of RD-S,random attractor and the meaning of researching the Boussinesq equations.And we also provides some relevant basic knowledge.In the second chapter:there is no stochastic differential for introducing O-U transformation.The unique solution of the equation by using the Galerkin approx-imate method is obtained,which generates a continue random dynamical system.In the third chapter:we get the unique random attractor in L2(D)2×L2(D)space by uniform estimates of solutions and proving that there exist random ab-sorbing sets both in L2(D)2×L2(D)space,which combines with Sobolev compact embedding theory.In the fourth chapter:by the convergence of random dynamical system in L2(D)2×L2(D)space,the upper semi-continuity of random attractor for the Semi-linear Degenerate Parabolic equations is obtained.