Abstract and Applied Analysis

Volume 2014 (2014), Article ID 428685, 12 pages

http://dx.doi.org/10.1155/2014/428685

Research Article

Random Attractors for Stochastic Ginzburg-Landau Equation on Unbounded Domains

Qiuying Lu,¹ Guifeng Deng,² and Weipeng Zhang³

¹Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China

²School of Mathematics and Information Science, Shanghai Lixin University of Commerce, Shanghai 201620, China

³School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

Received 18 February 2014; Revised 26 April 2014; Accepted 3 May 2014; Published 1 June 2014

Academic Editor: Milan Pokorny

Copyright © 2014 Qiuying Lu 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.

We prove the existence of a pullback attractor in for the stochastic Ginzburg-Landau equation with additive noise on the entire n-dimensional space . We show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system. We demonstrate that the system possesses a unique -random attractor, for which the asymptotic compactness is established by the method of uniform estimates on the tails of its solutions.

1. Introduction

In this paper, we study the following stochastic Ginzburg-Landau equation with additive noise defined in the entire space :

with the initial condition

where , , , , are real coefficients, with , , , and , , being time independent defined on and being independent two-sided real-valued Wiener processes on a complete probability space . Our aim is to study its long time behavior defined in the entire space .

Attractors are quite well investigated to describe the long time behavior of the deterministic equations (see, e.g., [1–7]). Recently, the concept of random attractors, which is in fact compact invariant set, was introduced to stochastic dynamical systems from the theory of attractors for deterministic equations in [8–10]. The existence of such random attractors for the Ginzburg-Landau equation perturbed by additive white noise and multiplicative white noise on bounded domains has been investigated, respectively, in [11, 12].

However, for unbounded domains, we cannot guarantee the compactness of solutions by the standard method since the Sobolev embeddings are no longer compact. Hence, to prove the existence of an attractor, we have to first overcome this difficulty. For deterministic equations, this difficulty has been overcome by employing the energy equation approach, introduced in [13, 14], and then used by others to prove the asymptotic compactness of deterministic equations in unbounded domains (see, e.g., [15–22]). In this paper, we prove the existence of a random attractor for the stochastic Ginzburg-Landau equation (1), defined on the unbounded domain with the help of tail estimates method, which was firstly established in [23] to the case of stochastic dissipative PDEs.

For the mathematical setting, we introduce complex Sobolev spaces. In general, we denote by the complexified space of a function space . For example, is the complexified space of . Denote by and the scalar product and the norm in either or . So, if , then , , , and

If , are in ,

We use letter to denote any positive constant which may change its value from line to line or even in the same line when necessary.

The whole paper is organized as follows. In Section 2, we first recall some definitions and propositions on random attractors for random dynamical systems (RDS). And then, by Ornstein-Uhlenbeck process, we obtain the continuous RDS associated with the stochastic Ginzburg-Landau equation (1). In Section 3, we concentrate to get the uniform estimate on the far-field values of the solution as and thus to further establish the asymptotic compactness of the solution operator . Then, we can exhibit our main result in the following theorem.

Theorem 1. The random dynamical system of stochastic Ginzburg-Landau equation with additive noise has a unique -random attractor in provided that .

2. RDS Associated with the Stochastic Ginzburg-Landau Equation on

2.1. Preliminaries on RDS

We first recall some definitions. For more details, one can refer to [8, 10, 24–26].

Definition 2. Let be a probability space and a family of measures preserving transformation such that is measurable, , and , for all , and then the flow together with the corresponding probability space is called a metric dynamical system.

For Wiener process in (1), we consider the probability space , where

is the Borel -algebra induced by the compact-open topology of , and is the corresponding Wiener measure on . The time shift is simply defined by

Then is a metric dynamical system.

Definition 3. A continuous random dynamical system (RDS) on over a metric dynamical system is a mapping:

which is -measurable such that, for -a.e. ,

(i) is the identity on ;(ii) for all ;(iii) is continuous for all .

Hereafter, we always assume that is a continuous RDS on over .

Definition 4. A random variable is called tempered with respect to the dynamical system if, for the associated stationary stochastic process , the invariant set for which

( applies only to two-sided time) has full -measure.

Definition 5. A random bounded set of is called tempered with respect to if, for -a.e. ,

where .

Definition 6. Let be a collection of random subsets of and . Then is called a random absorbing set for in if, for every and -a.e. , there exists such that

Definition 7. Let be a collection of random subsets of . Then is said to be -pullback asymptotically compact in if, for -a.e. , has a convergent subsequence in whenever , and with .

Definition 8. Let be a collection of random subsets of . Then a random set of is called a -random attractor (or -pullback attractor) for if the following conditions are satisfied: for -a.e. ,

(i) is compact, and is measurable for every ;(ii) is invariant; that is,

(iii) attracts every set in ; that is, for every ,

where is the Hausdorff semimetric given by for any and .

Proposition 9 (see [10, 25]). Let be an inclusion-closed collection of random subsets of and a continuous RDS on over . Suppose that is a closed random absorbing set for in and is -pullback asymptotically compact in . Then has a unique -random attractor which is given by

Remark 10. A collection of random subsets is called inclusion closed if, whenever is an arbitrary random set and is in with , for all , must belong to .

2.2. RDS Associated with the Stochastic Ginzburg-Landau Equation on

Denote , where

satisfies the one-dimensional Ornstein-Uhlenbeck equation

Since the random variable is tempered and is -a.e. continuous, there exists a tempered function such that

where satisfies, for -a.e. ,

thanks to Proposition  4.3.3 in [24]. From (16) to (17), we get, for -a.e. ,

Introduce the transformation

where is the solution of (1)-(2); then should satisfy

Similar to the procedure in [23], we can obtain that (20) has a unique solution with , which is continuous with respect to in . Let , and then is the solution of (1)-(2). Define by

for all . Then, we can claim that is a continuous random dynamical system associated with the stochastic Ginzburg-Landau equation on .

3. Existence of Random Attractors

In the following paper, we always assume that is the collection of all tempered subsets of with respect to . And then we are devoted to prove that has a random absorbing set in , and it is also -pullback asymptotically compact.

Proposition 11. There exists such that is a random absorbing set for in . Precisely, for any and -a.e. , there is such that

Proof. By multiplying (20) by , integrating over , and taking the real part, we get

Here

From (23) to (24),

We can see that the right-hand side of (25) can be bounded by

since , where .

So, for ,

which leads to

by multiplying (27) by and integrating from to .

By replacing by , we derive from (18) and (28) that, for all ,

By replacing by in (21), one has . Thereafter,

Recall that both the random variable and the random bounded set are tempered. Then, for any , there exists such that, for all ,

So far, for all ,

Select

then is a random absorbing set for in .

The proof is completed.

Lemma 12. Let and , and then, for any and -a.e. , that is, the two inequalities of (34) hold true for the solution of (1)-(2) and of (20) with , , such that

Proof. Fix , and then replace by and by in (28); we then obtain

With (18) and (26) in mind, by multiplying at both sides of the above equation, one can easily get

From (25) to (26),

Multiply (37) by and then integrate from to ; we then obtain

Keep the last two terms on the left-hand side of (38), and replace by ; we then have

However the second term on the right-hand side can be bounded by

due to (18) and (26). Together with (36), there is

The proof is completed.

Corollary 13. Let and , and then, for -a.e. , there exists such that the solutions of (1)-(2) and of (20), with , satisfy the following uniform estimates, for all :

Proof. Replace by and then replace by in (34); we then deduce

As both random variables and are tempered, there exists , such that, for all ,

which, together with (43), claims that, for all ,

With the same procedure as the above, we can also verify that, for all ,

The proof is completed.

Corollary 14. Let and , and then, for -a.e. , there exists such that the solution of (1)-(2) satisfies

Proof. Let just be the one in Corollary 13, and take and . Note that by (21) one has

Owing to (18), one has

Together with Corollary 13, we derive

by integrating (48) with respect to over .

The proof is completed.

Lemma 15. Suppose , and let and ; then, for -a.e. , there exists such that, for all ,

Proof. By multiplying (20) by , integrating over , and then taking the real part, we get

Since

while

we have

provided that .

Therefore, for the first term at the right-hand side of (52), we have

On the other hand, the second term at the right-hand side of (52) can be bounded by

By (52), (56)-(57), we can see that

That is,

where

Since , where , there exists a constant such that

Let , , where is the positive time taken in Corollary 13. By integrating (59) from to , we obtain

Integrate the above equation with respect to over to have

By replacing by , we derive

Thanks to Corollary 13, it follows from (61) and (64) that, for all ,

Then, together with (16), we obtain that, for all ,

The proof is completed.

Lemma 16. Suppose , and let and ; then, for every and -a.e. , there exist and such that the solution of (20) with satisfies, for all ,

Proof. Let be a smooth function defined on such that for all , and

Then there exists a constant such that , for all . Multiply (20) by , integrate over , and then take the real part to get

We now concentrate to estimate the terms in (69). Firstly,

Since

then we find that

Secondly,

Due to

we have

Thirdly,

Finally, from (69) to (76),

which implies

Proposition 11 together with Lemma 15 shows that there is such that, for all ,

Now, multiply (78) with , and then integrate over with respect to so that, for all ,

By replacing by in (80), we obtain that, for all ,

We now estimate the terms in (81) as follows.

Firstly, from (28), one deduces

Thus,

due to (18) and (26). Thus, for any given , there is such that, for all ,

For the second, replace by in (82); then we can find that the second term at the right-hand side of (81) satisfies

which implies that there exist and such that, for all and ,

For the third, from Lemma 12, we know that there is such that, for all , the third term at the right-hand side of (81) satisfies

Therefore, there is such that, for all and ,

Finally, note that the last two terms in (81) can be bounded by

and , where , and we can find such that, for all and ,

Accordingly, we have the following estimates for the last two terms in (81):

Let and . Then from (81), (84), (86), (88), and (91), we know that, for all and ,

That is, for any and ,

The proof is completed.

Lemma 17. Suppose , and let and ; then, for every and -a.e. , there exist and such that the solution of (1)-(2) satisfies, for all ,

Proof. Let and be the constants in Lemma 16. Then due to (16) and (90) we know that, for all and ,

Thus, together with Lemma 16, we derive, for all and ,

The proof is completed.

Up to now, we are ready to give the -pullback asymptotic compactness of , based on the former uniform estimates referring to the tails of solutions.

Proposition 18. Suppose that , and then the random dynamical system is -pullback asymptotically compact in . That is to say, for -a.e. , the sequence has a convergent subsequence in for , and .

Proof. Let , , and . By Proposition 11, we know that, for -a.e. ,

So, there is such that, up to a subsequence,

It only remains to prove that the weak convergence of (98) is indeed strong convergence. Let be small enough. Since , there exists , such that

From Lemma 17, there are and , for -a.e. , such that, for all ,

Since , let be large enough such that for every . Hence, it follows from (100) that, for all ,

On the other hand, from Proposition 11 and Lemma 15, there is such that, for all ,

Let such that for . Thus, from (102), we know that, for all ,

Denote for the set . Due to the compactness of embedding , we deduce from (103) that, up to a subsequence,

which tells us that, for the given , there exists such that, for all ,

By (99), (101), and (105), we conclude that, for all ,

Therefore, up to a subsequence,

is verified.

Up to now, we have proved that has a closed random absorbing set in by Proposition 11 and is -pullback asymptotically compact in , which is present in Proposition 18. So, the existence of unique -random attractor for stated in Theorem 1 immediately follows from Proposition 9.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the project sponsored by SRF for ROCS, SEM (no. 114329A4C11604), National Natural Science Foundation of China (nos. 11101370, 11101283, 11001041, and 11211130093), and the “521” Talent Program of Zhejiang Sci-Tech University (ZSTU) under Grant no. 11430132521304.

References

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, 1992.

X.-C. Fu and J. Duan, “Global attractors and invariant measures for non-invertible planar piecewise isometric maps,” Physics Letters A: General, Atomic and Solid State Physics, vol. 371, no. 4, pp. 285–290, 2007.

X.-C. Fu and J. Duan, “On global attractors for a class of nonhyperbolic piecewise affine maps,” Physica D: Nonlinear Phenomena, vol. 237, no. 24, pp. 3369–3376, 2008.

J. K. Hale, “Asymptotic behavior of dissipative systems,” American Mathematical Society, vol. 15, 1988.

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK, 2001.

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, NY, USA, 2002.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 2nd edition, 1997.

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

H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related Fields, vol. 100, no. 3, pp. 365–393, 1994.

F. Flandoli and B. Schmalfuss, “Random attractors for the 3 D stochastic navier-stokes equation with multiplicative white noise,” Stochastics and Stochastic Reports, vol. 59, pp. 21–45, 1996.

G. Wang, B. Guo, and Y. Li, “The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 849–857, 2008.

Y. Desheng, “The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise,” Journal of Mathematical Physics, vol. 45, no. 11, pp. 4064–4076, 2004.

J. M. Ball, “Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,” Journal of Nonlinear Science, vol. 7, no. 2, pp. 475–502, 1997.

J. M. Ball, “Global attractors for damped semilinear wave equations,” Discrete and Continuous Dynamical Systems, vol. 10, no. 1-2, pp. 31–52, 2004.

T. Caraballo, G. Łukaszewicz, and J. Real, “Pullback attractors for asymptotically compact non-autonomous dynamical systems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 64, no. 3, pp. 484–498, 2006.

J. M. Ghidaglia, “A note on the strong convergence towards attractors of damped forced KdV equations,” Journal of Differential Equations, vol. 110, no. 2, pp. 356–359, 1994.

O. Goubet and R. M. S. Rosa, “Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,” Journal of Differential Equations, vol. 185, no. 1, pp. 25–53, 2002.

J. Ning, “The H1-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains,” Nonlinearity, vol. 13, no. 4, pp. 1227–1238, 2000.

I. Moise and R. Rosa, “On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation,” Advances in Differential Equations, vol. 2, pp. 257–296, 1997.

I. Moise, R. Rosa, and X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity, vol. 11, no. 5, pp. 1369–1393, 1998.

R. Rosa, “The global attractor for the 2D navier-stokes flow on some unbounded domains,” Nonlinear Analysis: Theory, Methods and Applications, vol. 32, no. 1, pp. 71–85, 1998.

X. Wang, “An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors,” Physica D: Nonlinear Phenomena, vol. 88, no. 3-4, pp. 167–175, 1995.

P. W. Bates, K. Lu, and B. Wang, “Random attractors for stochastic reaction-diffusion equations on unbounded domains,” Journal of Differential Equations, vol. 246, no. 2, pp. 845–869, 2009.

L. Arnold, Random Dynamical Systems, Springer, 1998.

P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice dynamical systems,” Stochastics and Dynamics, vol. 6, no. 1, pp. 1–21, 2006.

H. Crauel, “Random point attractors versus random set attractors,” Journal of the London Mathematical Society, vol. 63, no. 2, pp. 413–427, 2001.