Hausdorff Dimension of a Random Attractor for Stochastic Boussinesq Equations with Double Multiplicative White Noises

This paper investigates the existence of random attractor for stochastic Boussinesq equations driven by multiplicative white noises in both the velocity and temperature equations and estimates the Hausdorff dimension of the random attractor.

Pu and Guo in [1] studied the global well-posedness of stochastic 2D Boussinesq equations with partial viscosity; Li and Guo [2] consider the stochastic Boussinesq equations, which are influenced by multiplicative white noise in the velocity equations.Recently, Zhao and Li [3] established the existence of a random attractor for stochastic Boussinesq equations with double additive white noises.
In many cases, the Hausdorff dimension of a random attractor is finite.Crauel and Flandoli [4] developed a technique to estimate the Hausdorff dimension of attractors for certain dynamical systems.But they required the noise to be bounded.Debussche [5] applied a random squeezing property to estimate the Hausdorff dimension without the assumption of bounded noise.A number of authors [6][7][8][9][10][11][12][13] used this method to estimate the Hausdorff dimension of attractors for many stochastic equations.
As far as we know, there are no results on stochastic 2D Boussinesq equations with double multiplicative white noises on bounded domains.In this paper, we prove the existence of random attractor for the corresponding RDS associated with problem (1).And we give an estimate of Hausdorff dimension for the random attractor.
The paper is organized as follows.In the first section, we recall some latest results and what we want to do for the 2D stochastic Boussinesq equations.We study the RDS determined by (1) in Section 2. In Section 3, we prove the existence of a random attractor.Finally, we estimate the

Random Dynamical System
The random dynamical system generated by stochastic Boussinesq equations with double multiplicative noise will be studied in this section.Thus, we introduce a process, which enables us to transform the stochastic equations into a deterministic equation with a random parameter. Let From ( 1), we get the stochastic Boussinesq equations without white noises: In the present paper, we consider the Hilbert space: H = H 1 × H 2 , with the scalar product (⋅, ⋅) and norm ‖ ⋅ ‖, where We also consider the subspace: and its scalar product and norms are ((⋅, ⋅)) and ‖ ⋅ ‖ 1 , respectively, for any  1 ,  2 ∈  2 : Now we give some definitions of common operators. is the bilinear on : is isomorphism from () into H and from  into the dual space   , defined by where () = ( 1 ) × ( 2 ), being We consider trilinear forms  on  defined by is continuous on  and (H  ()) 2 × H  ().
We also define the bilinear continuous operator , which maps  ×  into   and () × () into H, by Finally, we consider a family of linear continuous operators () = (, ) on H: Assume that  = (, )  is a solution of (4)- (7) (ii) Denoting such solution by (, ;  0 ,  0 ), for any  ≥  0 , the mapping is continuous.Then, we define a stochastic flow (, ) by where are projection operators.We can prove that (, ) is a continuous RDS determined by (1) easily.

Random Attractor
In this section, we will establish the existence of nonempty compact tempered random attractor for the RDS (, ) determined by (1).First we prove that (, ) possesses absorbing set in H.
Then, B() is a random absorbing set for (, ) in .Since  embedding H is compact, then RDS (, ) is asymptotically compact in H. Thus, the RDS determined by (1) possesses a random attractor.

Hausdorff Dimensions of Random Attractors
In this section, we will estimate the Hausdorff dimensions of A().Now, we assume that  0 ,  1 (),  2 (),  are F  measurable and prove that the RDS (, ) is differentiable on A().