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

Abstract

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.

1. Introduction

In this paper, we consider the following 2D stochastic Boussinesq equation on :where denotes the velocity of the fluid; is the temperature; and function represents the pressure. We set , and is the canonical basis of . Let us consider the Grashof and Prandtl numbers. is two-sided real-valued Wiener process on a probability space , here The Borel -algebra on is generated by the compact open topology, and is the corresponding Wiener measure; denotes the Stratonovich form white noise.

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 [6–13] 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 Hausdorff dimension of the random attractor in Section 4.

2. 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: , with the scalar product and norm , whereWe also consider the subspace: . Consider and its scalar product and norms are and , respectively, for any :

Now we give some definitions of common operators.

is the bilinear on : is isomorphism from into and from into the dual space , defined by where , being We consider trilinear forms on defined by is continuous on and .

We also define the bilinear continuous operator , which maps into and into , by Finally, we consider a family of linear continuous operators on :Assume that is a solution of (4)–(7) and is a test function in . Multiplying (4) by and (5) by , integrating over , and adding the resulting equation, we obtainNote that On the other hand, Then, Therefore, (9) can also be written asBy the Galerkin method (see [14, Theorem 3.1]), we can prove that (21) has a unique solution , which satisfies the following:

(i) For all , if , then

(ii) Denoting such solution by , for any , the mapping is continuous. Then, we define a stochastic flow by where and are projection operators.

We can prove that is a continuous RDS determined by (1) easily.

3. 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 .

Lemma 1. Let be a solution of (4)–(7), for any , and one has

Proof. Taking the scalar product of (4) with on By the incompressibility condition (6), we have From (27), we obtainUsing Poincaré inequality, ( is constant), we have Then, By the Gronwall inequality, for any , we get that Taking the scalar product of (5) with on as well Since , thenUsing Poincaré inequality , we have By Gronwall inequality, for any , we obtain that

Lemma 2. There exist two positive random variables and , such that, for all , and with , the solution of equations (4)–(7) satisfies

Proof. For , , from (26), we have taking , such that .
Then, for any , From (26), we find Then, , as .
Thus, there exists , for : By (37), we infer that , , is Since then . Therefor, Put Then, exists and it is tempered.
Getting , for any ,Now we are going to prove that is asymptotically compact.

Lemma 3. There exist two positive random variables and , such that, for all , and with , the solution of equations (4)–(7) satisfies

Proof. Integrating (34) on , we have From Lemma 1, we know that We get from (29),then Integrating the above formula over , then Let then

Lemma 4. There exist two positive random variables and , such that, for all , and with , the solution of (4)–(7) satisfies

Proof. Taking the scalar product of (5) with −Δ on , then Since then we have Hence, Using the Gronwall inequality over an arbitrary interval , we obtain Here, .
Integrating the above formula over the interval for , we obtain Let Thus, exists and it is tempered and we have Taking the scalar product of (4) with on Since , then Namely,where and .
By the Gronwall inequality over an arbitrary interval , we obtain Integrating the above formula over the interval for , we have Since let Thus, exists and it is tempered. Then, Let be a random closed ball centered at 0 with radius .
Then, is a random absorbing set for in . Since embedding is compact, then RDS is asymptotically compact in . Thus, the RDS determined by (1) possesses a random attractor.

4. Hausdorff Dimensions of Random Attractors

In this section, we will estimate the Hausdorff dimensions of . Now, we assume that are measurable and prove that the RDS is differentiable on .

Theorem 5. The RDS determined by (21) is almost surely uniformly differentiable on , , and its derivative is , , which is the solution of the following variational equation: where , , .

Proof. Let , , and , and we have We also set , and then Taking the scalar product of (75) with on ,Here, is the function about . By the Gronwall inequality, we obtain Taking the scalar product of (74) with on ,Then, By the Gronwall inequality, we get We get from (80) that From (80) to (81), we have The proof is completed.

Theorem 6. Set Then the Hausdorff dimension of random attractor satisfies

Proof. Taking the scalar product of (74) with on ,Then, By the Gronwall inequality, we get which implies that Since then Taking , then we have Assume that , here , and . is the orthogonal projection in onto the space spanned by . Let , , be an orthonormal basis in . Then, Set Then, We can get that when In this case, the Hausdorff dimension of random attractor satisfies

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant no. 11501373) and the Natural Science Foundation of Guangdong Province (no. 2016A030310019 and no. 2016A03030742), Guangdong Provincial culture of seedling of China (no. 2013LYM0081), the Education Research Platform Project of Guangdong Province (no. 2014KQNCX208), the Education Reform Project of Guangdong Province (no. 2015558), the Shaoguan Science and Technology Foundation (no. 20157201 and no. 20167201), and Education Reform Project of Shaoguan University (no. SYJY20121361 and no. SYJY20141576).