The Existence And The Hausdorff Dimensions Of Attractor For Stochastic Boussinesq Equations With Double Multiplicative White Noises

This article mainly investigates the stochastic Boussinesq equations driven by multiplica-te white noises both in the velocity and temperature equations.We prove the existence of ran-dom attractor for them on bounded and unbounded domains. Then we depict the complexity of randon attractor by the Hausdorff dimension, and give an upper bound.This paper is divided into five chapters:The first chapter introduces the research background of Boussinesq equations and resear-ch status both at home and abroad.Then we raise our main work in this paper.In the second chapter,we list the required basic knowledge on random dynamic systems and some frequently-used inequality.In the third chapter,we mainly study the long time behavior of the Boussinesq equations with double multiplicative white noises on the bounded domains.We introduce a stochastic process, which enables us to transform the stochastic equations with white noise into a stochastic equations without white noise.Then we prove the existence and the uniqueness of global solution for the new equations and we get a random dynamic system determined by the global solution.Moreover,we prove the random dynamic system possesses a random absorbing set and it is asymptotically compact.We get the existence of a random attractor finally.In order to describe the complexity of attractor,we will estimate the Hausdorff dimension of the random attractor in the fourth chapter. Firstly,we prove the RDS is differentiable on the random attractor according to the definition of differentiability. After that, we prove the other conditions of Hausdorff dimension.Finally,we prove the Hausdorff dimension of random attractor is limited, and an upper bound estimate is presented,In the fifth chapter,we are going to study the existence of random attractor for the stochastic Boussinesq equations with double multiplicative white noise on unbounded domains. Because the unboundedness of the domains, the Sobolev embeddings are not compact.It would be difficul for us to get the asymptotic compactness of the RDS. In order to overcome this difficulty, we use the "tail-estimates" technology. Roughly speaking, the idea is to divide the whole space into two parts:And prove that:(1) The RDS is asymptotically compact in H(Br).(2) The RDS is uniformly small in H(R2\Br). Here, H(·) is a function space defined later.