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(?,F,P,(?t)t?R),where = {w?C(R,R):w(0)= 0},F is the Borel?-algebra induced by the compact-open topology of ?,P is a Wiener measure,for???,and t?R,? satisfies ?tw(·)=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. |