Abstract and Applied Analysis

Volume 2013 (2013), Article ID 601325, 13 pages

http://dx.doi.org/10.1155/2013/601325

Research Article

Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations

Hongyong Cui, Jie Xin, and Anran Li

School of Mathematics and Information, Ludong University, Yantai, 264025, China

Received 8 November 2012; Revised 25 January 2013; Accepted 27 January 2013

Academic Editor: Lucas Jódar

Copyright © 2013 Hongyong Cui 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.

Solutions and weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces were studied in a bounded domain. We first established the existence and the uniqueness of the solution to the system by using Galerkin method and then obtained the uniform absorbing set and the weakly compact uniform attractor of the problem by applying techniques of constructing skew product flow in the extended phase space.

1. Introduction

The long wave-short wave (LS) resonance equations arise in the study of the interaction of the surface waves with both gravity and capillary modes presence and also in the analysis of internal waves, as well as Rossby wave [1]. In the plasma physics they describe the resonance of the high-frequency electron plasma oscillation and the associated low-frequency ion density perturbation [2]. Benney [3] presents a general theory for the interaction between the short wave and the long wave.

Due to their rich physical and mathematical properties the long wave-short wave resonance equations have drawn much attention of many physicists and mathematicians. For one-dimensional propagation of waves, there are many studies on this interaction. Guo [4, 5] obtains the existence of global solution for long-short wave equations and generalized long-short wave equations, respectively. The existence of global attractor was studied in [6–8]. The orbital stability of solitary waves for this system has been studied in [9]. In [10], Guo investigated the asymptotic behavior of solutions for the long-short wave equations with zero order dissipation in . The approximation inertial manifolds for LS type equations have been studied in [11]. The well posedness of the Cauchy problem for the long wave-short wave resonance equations was studied in [8, 12–17].

In this paper, nonautonomous LS equations with translation compact forces were studied. The essential difference between nonautonomous systems and autonomous ones is that the former get much influenced by the time-depended external forces, which breaks semigroup property of the flow or semiflow created by autonomous systems. Also, attractors of nonautonomous systems are no longer invariable; they change with the changing of the initial time. This makes it impossible for us to consider nonautonomous systems completely in the same way of autonomous ones. Fortunately, Chepyzhov and Vishik [18, 19] developed techniques by which skills in the study of autonomous systems can be used in dealing with nonautonomous problems. Their central idea is that constructing skew product flow in extended phase space is obtained by

where is a family of processes, is a translation semigroup, and the flow can be proved to be a semigroup under some preconditions, such as the translation identity and -continuity of , and more importantly, the compactness of the symbol space . By this means, we can get the uniform attractor by projecting the global attractor of to the phase space if the latter exists. We consider the following nonautonomous dissipative generalized long-short wave equations:

with the initial conditions

and the boundary value conditions

where . . Nonautonomous terms and are time-depended external forces, which are supposed to be translation compact (cf. [18] or Assumption 1). Nonlinear terms and are given smooth and real, satisfying

where , , , and constants , , and are given in for .

Our aim here is, firstly, to get the unique existence of solutions for problem (2)~(5) and then to derive the existence of weakly compact uniform attractor for it with the above-mentioned method. Here, and throughout this paper, uniform means uniform about symbols () in symbol space () unless there is special explanation. In fact, it is the same if we say uniform about the initial time, since the translation identity and the -continuity of hold in our case (cf. [20]).

Throughout this paper, we denote by the norm of with usual inner product , denote by the norm of for all , and denote by the norm of a usual Sobolev space for all . And we denote different constants by a same letter , and represents that the constant relies only on the parameters appearing in the brackets.

This paper is organized as follows. In Section 2, we recall some facts about the nonautonomous system. In Section 3, we provide the uniform a priori estimates in time. In Section 4, we obtain the unique existence of the solutions for problem (2)~(5) by Galerkin method. Section 5 contains the weakly compact uniform attractor for the nonautonomous system (2)~(5), and in the proof of Theorem 13, the -continuity of is proved.

2. Preliminary Results

Let be a topological space, and let be a function. The set

is called the hull of in , denoted by . is translation compact if is compact in .

We denote all the translation compact functions in by , where is a Banach space. Apparently, implies that is translation bounded; that is,

Let be a Banach space, and let a family of two-parameter mappings act in . We also need the following definitions and lemma (cf. [19, 20]).

Definition 1. Let be a parameter set. , is said to be a family of processes in Banach space , if for each , from to satisfies

Definition 2. , a family of processes in Banach space , is called -continuous, if for all fixed and , ,  projection  is continuous from to .

A set is said to be uniformly absorbing set for the family of processes , if for any and which denotes the set of all bounded subsets of , there exists , such that for all . A set is said to be uniformly attracting for the family of process , if for any fixed and every ,

Definition 3. A closed set is called the uniform attractor of the family of process , if it is uniformly attracting (attracting property), and it is contained in any closed uniformly attracting set of the family of process , (minimality property).

Lemma 4. Let be a compact metric space, and suppose is a family of operators acting on , satisfying the following:

(i)

(ii) translation identity:

where is arbitrarily a process in compact metric space . Moreover, if the family of processes is continuous, and it has a uniform compact attracting set, then the skew product flow corresponding to it has a global attractor on , and the projection of on , is the compact uniform attractor of .

Remark 5. Assumption (13) holds if the system has a unique solution.

For brevity, we rewrite system (2)~(5) in the vector form by introducing and . We denote by the space of vector functions with norm

Similarly, we denote by the space of with norm

Then system (2)~(5) can be considered as

where is the symbol of (16).

Assumption 1. Assume that the symbol comes from the symbol space defined by

where and the closure is taken in the sense of local quadratic mean convergence topology in the topological space . Moreover, we suppose that .

Remark 6. By the conception of translation compact/boundedness we remark that

(i),  ;(ii), where is an translation operator.

3. Uniform a Priori Estimates in Time

In this section, we derive uniform a priori estimates in time which enable us to show the existence of solutions and the uniform attractor. First we recall the following interpolation inequality (cf. [21]).

Lemma 7. Let , , such that , , . Then one has

for , where , , and .

Lemma 8. If and , then for the solutions of problem (2)~(5), one has

where , .

Proof. Taking the inner product of (2) with in we get that

Taking the imaginary part of (20), we obtain that

By Young inequality and Remark 6 we have

And then by Gronwall lemma we can complete the proof.

In the following, we denote that , which will not cause confusions.

Lemma 9. Under assumptions of (6), (7) and Assumption 1, if , solutions of problem (2)~(5) satisfy

where and .

Proof. Taking the inner product of (2) with in and taking the real part, we get that

By (3) we know that

which shows that

where is introduced by

Taking the inner product of (2) with in and taking the real part, we get that

Multiply (28) by , and add the resulting identity to (26) to get

That is,

In the following, we denote by any constants depending only on the data , and means it depends not only on but also on parameters in the brackets. , when is sufficiently large, by (6) and Lemmas 7 and 8, we have

By (6) we deduce that

And then

By (30)~(35) we get that

Similarly we can also deduce that

Taking the inner product of (3) with in , we see that

By (2) we get that

It comes from (38)~(40) that

Deal with the right hand side of inequality (41), by Lemmas 7 and 8,

So

Analogously, we can also deduce that

Set , and

Then by (36), (47) and (37), (48) we can, respectively, get

which shows that if we set , we can deduce that

where . By Gronwall lemma we see that

Similar to (33), (34), (45), and (42), for we have

And then

where when . Then by (52) we infer that

where and . By (49), (53), and (55) we infer that

Choose ; then we have

which concludes the proof by using Lemma 8.

Lemma 10. Under assumptions of Lemma 9, if , solutions of problem (2)~(5) satisfy

where and .

Proof. Taking the real part of the inner product of (2) with in , we have

By (2) and (3), we have

Since

we see that

Multiplying (2) by and taking the real part, we find that

therefore,

Now we deal with (64) to get (70). Due to equalities

we deduce that

We take care of terms in (66) as follows:

It follows from (66)~(67) that

And then

From (64) and (69) we have

By (59), (60), (62), and (70) we conclude that

where .

For later purpose, we let

Then from (71) we have

or

By Lemma 9 and Agmon inequality we have

In the following, we denote by . By Lemma 7 and (76) we estimate the size of to get

Taking the inner product of (3) with in , we see that

Since

by (78) we can deduce that

From (2) we know that

Taking the real part of the inner product to (81) with in , we have

Because of

it holds that

By (84) and (80), we find that

That is,

Similar to (77), we estimate each term in (86), and then we get

Let , and

By (77) and (87) we deduce that

which has the same form with (51) in the proof of Lemma 9. Similar to the study of (51), we can derive that

where and when . By (72) we deduce that

and then by (88), (90), and (91) we deduce that

which concludes the proof by Lemma 9.

4. Solutions for (2)~(5)

Theorem 11. Under assumptions of Lemma 10, for each , system (2)~(5) has a unique global solution , .

Proof. We prove this theorem briefly by two steps.

Step  1. The existence of the solution.

By Galërkin's method, we apply the following approximate solution:

to approach the solution of the problem (2)~(5), where is a orthogonal basis of satisfying . And satisfies

where . Then (94) becomes an initial boundary value problem of ordinary differential equations. According to the standard existence theory for the ordinary differential equations, there exists a unique solution of (94). Similar to [4, 22], by the a priori estimates in Section 3 we know that converges (weakly star) to a which solves (2)~(5).

Step  2. The uniqueness of the solution.

Suppose are two solutions of the problem (2)~(5). Let , then satisfies

Similar to [4, 5, 22], we can deduce that .

5. Uniform Absorbing Set and Uniform Attractor

From Theorem 11 we know that , the family of processes corresponding to (2)~(5), is well defined. And assumption (13) is satisfied.

Theorem 12. Under assumptions of Theorem 11, possesses a bounded uniformly absorbing set in .

Proof. Let . From Theorem 11 we know that is a bounded absorbing set of the process .

On the other hand, from Assumption 1 we know that for each , holds. Thus, the solution of our system satisfies

So the set is a bounded uniformly absorbing set of .

Theorem 13. Under assumptions of Theorem 12, admits a weakly compact uniform attractor .

Proof. To prove the existence of weakly compact uniform attractor in , from Lemma 4 and Theorems 11 and 12, the only thing we should do is to verify that is -continuous. Through the following proof, means weak converges, and means *weak converges.

For any fixed , let

We will complete the proof if we deduce that

where , , and .

From (97) and Theorem 11 we know that

By Agmon inequality,

We see that

Note that

and . By (100) and (102), we find that and

Due to Theorem 11 and (105), we know that there exist , and subsequences of , which are still denoted by , such that

Besides, for , by (100) we know that there exists , such that

By (106) and a compactness embedding theorem, we claim that

In the following, we shall show that is a solution of the problem (2)~(5).

For , , by (103) we find that

Since

by (102), (109), and (106),

Then we have

By using the similar methods to the other terms of (110), we have

Therefore, we obtain

which shows that satisfies (2).

For , with , by (103) we find that

Assumption (97) implies that

Then by (116) and (117), we have

While from (115) we know that

It come from (118) and (119) that

And then

By (115) and (121), we have

For , , with , , then repeating the procedure of proofs of (116)~(119), by (108) we deduce that

It comes from (108), (122), and (123) that

Similarly, we can also deduce that

By (124) and (125), we derive (98). We complete the proof of Theorem 13.

Acknowledgments

The authors were supported by the NSF of China (nos. 11071162, 11271050), the Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA09).

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