Journal of Function Spaces
Volume 2016 (2016), Article ID 1832840, 8 pages
http://dx.doi.org/10.1155/2016/1832840

Research Article

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

Yin Li,1,2 Ruiying Wei,1,2 and Donghong Cai³

¹School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China
²School of Mathematics and Statistics, Yunnan University, Yunnan 650091, China
³Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

Received 24 January 2016; Revised 27 April 2016; Accepted 28 April 2016

Academic Editor: Maria Alessandra Ragusa

Copyright © 2016 Yin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , where

We 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 obtain

Note that

On the other hand,

Then,

Therefore, (9) can also be written as

By 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 obtain

Using 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 , then

Using 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).

References

X. Pu and B. Guo, “Global well-posedness of the stochastic 2D Boussinesq equations with partial viscosity,” Acta Mathematica Scientia. Series B. English Edition, vol. 31, no. 5, pp. 1968–1984, 2011.

Y. R. Li and B. L. Guo, “Random attractors of Boussinesq equations with multiplicative noise,” Acta Mathematica Sinica, English Series, vol. 25, no. 3, pp. 481–490, 2009.

W. Q. Zhao and Y. R. Li, “Asymptotic behavior of solutions to stochastic Boussinesq equations with additive white noises,” Acta Mathematica Sinica. Chinese Series, vol. 56, no. 1, pp. 1–14, 2013.

H. Crauel and F. Flandoli, “Hausdorff dimension of invariant sets for random dynamical systems,” Journal of Dynamics and Differential Equations, vol. 10, no. 3, pp. 449–474, 1998.

A. Debussche, “Hausdorff dimension of a random invariant set,” Journal de Mathématiques Pures et Appliquées, vol. 77, no. 10, pp. 967–988, 1998.

X. Fan and H. Chen, “Attractors for the stochastic reaction-diffusion equation driven by linear multiplicative noise with a variable coefficient,” Journal of Mathematical Analysis and Applications, vol. 398, no. 2, pp. 715–728, 2013.

X. Fan, “Random attractor for a damped sine-Gordon equation with white noise,” Pacific Journal of Mathematics, vol. 216, no. 1, pp. 63–76, 2004.

C. Sun and H. Gao, “Hausdorff dimension of random attractor for stochastic Navier-Stokes-Voight equations and primitive equations,” Dynamics of Partial Differential Equations, vol. 7, no. 4, pp. 307–326, 2010.

S. Gala, “On the regularity criterion of strong solutions to the 3D Boussinesq equations,” Applicable Analysis, vol. 90, no. 12, pp. 1829–1835, 2011.

S. Gala and M. A. Ragusa, “Logarithmically improved regularity criterion for the Boussinesq equations in BESov spaces with negative indices,” Applicable Analysis, vol. 95, no. 6, pp. 1271–1279, 2016.

S. Gala, Z. Guo, and M. A. Ragusa, “A remark on the regularity criterion of Boussinesq equations with zero heat conductivity,” Applied Mathematics Letters, vol. 27, pp. 70–73, 2014.

S. Gala and Z. Zhang, “Osgood type regularity criterion for the 3D Newton-Boussinesq equation,” Electronic Journal of Differential Equations, vol. 2013, no. 223, 6 pages, 2013.

J. Fan and Y. Zhou, “A note on regularity criterion for the 3D Boussinesq system with partial viscosity,” Applied Mathematics Letters, vol. 22, no. 5, pp. 802–805, 2009.

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics And Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988.