Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 875749, 6 pages

http://dx.doi.org/10.1155/2013/875749

Research Article

Perturbation of Stochastic Boussinesq Equations with Multiplicative White Noise

Chunde Yang and Xin Zhao

Institute of System Theory and Application, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 28 December 2012; Accepted 28 March 2013

Academic Editor: Xiaofeng Liao

Copyright © 2013 Chunde Yang and Xin Zhao. 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 studies the Boussinesq equations perturbed by multiplicative white noise and shows the existence and uniqueness of the global solution. It also gets some regularity results for the unique solution.

1. Introduction

The Boussinesq equation is a mathematics model of thermohydraulics, which consists of equations of fluid and temperature in the Boussinesq approximation. The deterministic case has been studied systematically by many authors (e.g., see [1–3]). However, in many practical circumstances, small irregularity has to be taken into account. Thus, it is necessary to add to the equation a random force, which is in general a space-time white noise, as considered recently by many authors for other equations (see [4–11]). The random attractors of boussinesq equations with multiplicative noise have been investigated by [12]. In this paper, We will study the perturbation of stochastic boussinesq equations with multiplicative white noise.

We will consider the following stochastic two-dimension a Boussinesq equations perturbed by a multiplicative white noise of Stratonovich form:

The domain occupied by the fluid is , and , is the canonical basis of . The unknown , , and stand for the velocity vector, temperature, and pressure, respectively. is the temperature at the top, , while is the temperature at the boundary below, . The constant numbers , , and   are related to the usual Prandtl, Grashof, and Rayleigh numbers.

is two-sided Wiener processes on the probability space , where , is the Borel sigma-algebra induced by the compact-open topology of , and is a Wiener measure.

We supplement (1) with the following boundary condition:

When an initial-valued problem is considered, we supplement these equations with

The existence of a compact random attractor and its Hausdorff, fractal dimension estimates have been investigated by [12]. We will solve pathwise (1)–(3). By using the Faedo-Galerkin approximation and a priori estimates, we prove the existence and uniqueness of the global solution and show that the solution continuously depends on the initial value. We also get some regularity results of the solutions.

2. Mathematical Setting and Basic Estimates

Let

and change to ; then (1) can be rewritten as

Let the process be

Then , and if we let

we get the new equations (no stochastic differential appears here)

with the boundary conditions

and the initial value conditions

To solve (8)–(12), we consider the Hilbert space with the scalar products () and norms , where and

We also consider the subspace of , where is the space of functions in vanishing at and and periodic in the direction of . is a Hilbert space for the scalar product and the norm

and . We also denote by and the canonical scalar product and norm in and .

The bilinear form

determines a linear isomorphism from into and from into the dual　space , defined by

with ,  where

Four spaces , , , and satisfy

and all embedding injections are densely continuous. It is well known that is self-adjoint and positive and is a compact self-adjoint in .

We also consider the trilinear forms on defined by

The trilinear form is continuous on or even on. We associate with the form the bilinear continuous operator which map into and into , defined by

Finally, we define the continuous operators in

Now, we can set (8) in the operator form. If is the solution of (8) and is a test function in , we multiply (8) by and (9) by , integrate over , and add the resulting equation. The pressure term disappears and after simplification we find

which can be reinterpreted as

Note that this equation differs from the determined case, and in determined case, the family of operator is independent of the time . Initial condition (12) can be reinterpreted as

To solve (23)-(24), we also need some Sobolev norm estimates on the bilinear and the operators and .

Lemma 1. The bilinear operators and are continuous and satisfy

(i),  for all  ,(ii),  for all  ,(iii),  for all  , ,(iv),  for all  , ,

where  , ,   are appropriate constants and  ,  .

Proof. The proof is the same as the deterministic case (see [10]).

Lemma 2. The linear continuous operators and satisfy

Proof. By (21), we have

which implies by the Poincare inequality

that (25) holds true. Since , it follows from (25) that (26) holds true.

Lemma 3. The bilinear form on satisfies

Proof. By (15), we have

which imply (28).

3. Existence and Uniqueness

In this section, we will prove the existence and uniqueness of the global solution of (23)-(24), equivalently (8)–(12) or (1)–(3). We are working almost surely for .

Theorem 4. Assume that , then there exists a unique solution of (23)-(24), such that

and the mapping is continuous from H into D(A), for all .

Proof. Since is a self-adjoint compact operator in , it follows from a classical spectral theorem that there exists a sequence and a family of elements which is completely orthogonal in such that

For each , we look for an approximate solution of the following form:

satisfying

and initial condition

where is the projector in (or ) on the space spanned by. Since and commute, the above equation is also equivalent to

where

in view of the linearity of .

The existence of on any finite interval follows from standard results of the existence of solutions of ordinary differential equations that is a consequence of these results and of the following priori estimates:

Indeed, multiplying (34) by , summing these relations for , and noting that (by Lemma 1), we find

which implies by Lemma 2, (29), and the Young inequality that

that is,

where is defined in (29) and is a appropriate constant. Using the classical Gronwall lemma we find

Integrating (41) for from 0 to and using above estimates we have

where and are independent of . Thus, we have proved (38).

We also claim that

Indeed, it follows from Lemma 1 that with appropriate constant c, which, together with (38), implies that and thus remain bounded in . Since both operators and are continuous (Lemmas 2 and 3), it follows from (38) that and thus remain bounded in . Therefore, by (36),

remains bounded in , which proved (44).

By weak compactness, it follows from (38) and (44) that there exists a , for all subsequence still denoted by , such that

We pass to the limit in (34) and find that

which implies that satisfies (23). In particular,  . This implies by [10, Lemma II.3.1] that is almost everywhere equal to a continuous function from [] into . Therefore initial condition (24) follows by a passage to the limit in (35). follows from [10, Lemma II.3.2] and the facts that and . Furthermore, if we show that uniqueness, then the fact that , for all , implies that .

To prove the uniqueness and continuous dependence of on (in ), we let be a solution of (23)-(24) such that . Similar to (39),   must satisfy the energy equality

By using Lemmas 1–3 and Gronwall lemma, we get the following similar estimates:

which has proved the continuous dependence. For the uniqueness, we let be two solutions of (23)-(24) and . Then is also a solution with . Thus, (49) implies that , that is, .

4. Regularity Results

In this section, we will consider further regularity results for the unique solution. The main result is that , and thus provided the initial function . More precisely, we have the following.

Theorem 5. Assume that , and let be the unique solution of (23)-(24). Then,

Proof. Let be the approximate solution (33) in the proof of Theorem 4. We first claim that

Indeed, multiplying (34) by , summing these relations for , and using (32), we find

By Lemma 1(iv) and the Young inequality, we find

For , by Lemma 2, (25), and the Young inequality, we have

Noting also that

and , we find from (52) and all the above estimates that

By (38), is bounded in . This, together with Lemma 3, implies that (56) can be rewritten as

for some appropriate constant . By Gronwall lemma, it follows from (57) and (38) that

which implies by Lemma 3 again that remains bounded in . Integrating in (57) from to  , we have

which proved the second argument of (51), and thus (51) holds.

Taking the limit in (51) (by weak compactness), we then find that is in . We need also to prove that u is continuous from into . This is proved as follows.

Since and with densely continuous injection, it follows from [10, Lemma II.3.3] that is weakly continuous; that is,   is continuous for every . Similarly   is continuous for every . Thus, by taking the limit in (52) and applying [10, Lemma II.3.2], we obtain an equality similar to (52) for :

which holds in the distribution sense on . Since ,  it follows from Lemma 1 and Lemma 2 that

and thus , which implies by [10, Lemma ] that the function is continuous. Therefore, since is a norm on equivalent to (by Lemma 3), it follows that is continuous for the norm topology.

Acknowledgment

This work was supported by the Foundation of Science and Technology Project of Chongqing Education Commission (KJ100513).

References

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

B. Guo, “Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations,” Chinese Annals of Mathematics B, vol. 16, no. 3, pp. 379–390, 1995.

B. L. Guo, “Spectral method for solving two-dimensional Newton-Boussinesq equations,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 3, pp. 208–218, 1989.

Y. R. Li and B. L. Guo, “Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,” Journal of Differential Equations, vol. 245, no. 7, pp. 1775–1800, 2008.

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer, Berlin, Germany, 1998.

T. Caraballo, J. A. Langa, and J. C. Robinson, “Stability and random attractors for a reaction-diffusion equation with multiplicative noise,” Discrete and Continuous Dynamical Systems, vol. 6, no. 4, pp. 875–892, 2000.

T. Caraballo, J. A. Langa, and J. C. Robinson, “Upper semicontinuity of attractors for small random perturbations of dynamical systems,” Communications in Partial Differential Equations, vol. 23, no. 9-10, pp. 1557–1581, 1998.

H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 307–341, 1997.

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.

G. Da Prato, A. Debussche, and R. Temam, “Stochastic Burger's equation,” Nonlinear Differential Equations and Applications, vol. 1, no. 4, pp. 389–402, 1994.

J. C. Robinson, “Stability of random attractors under perturbation and approximation,” Journal of Differential Equations, vol. 186, no. 2, pp. 652–669, 2002.

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