Abstract and Applied Analysis

Volume 2013 (2013), Article ID 830715, 17 pages

http://dx.doi.org/10.1155/2013/830715

Research Article

Uniform Attractor and Approximate Inertial Manifolds for Nonautonomous Long-Short Wave Equations

Hongyong Cui1,2 and Jie Xin¹

¹School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

²School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 28 August 2013; Revised 17 October 2013; Accepted 18 October 2013

Academic Editor: Carlo Bianca

Copyright © 2013 Hongyong Cui and Jie Xin. 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.

Nonautonomous long-short wave equations with quasiperiodic forces are studied. We prove the existence of the uniform attractor for the system by means of energy method, which is widely used to deal with problems who have no continuity (with respect to the initial data) property, as well as to those which Sobolev compact imbedding cannot be applied. Afterwards, we construct an approximate inertial manifold by means of extending phase space method and we estimated the size of the corresponding attracting neighborhood for this manifold.

1. Introduction

In this paper, we investigate the long time behavior of solutions for the following nonautonomous generalized dissipative LS equations with quasiperiodic forces:

with initial conditions and space-periodic boundary conditions as

where and and are positive constants.

The long-wave short-wave (LS) resonance equations arise in many kinds of physical models (see [1–4]). Due to their rich physical and mathematical properties, the LS equations have drawn much attention. The autonomous situations, including the existence of solutions, the solitary waves and their stability, and the long time behaviors of the solutions, have been deeply researched (see [5–14]).

Recently, the nonautonomous case of LS equations with translation compact forces was studied in [15]. Because of the nonlinear resonance of the equations, it is difficult to prove the continuity of the process generated by (1)–(4). Thus, it is hard to construct the uniform attractor directly by constructing a compact uniform absorbing set even if the forces are translation compact, and in [15] only the weakly compact uniform attractor for the system is obtained.

In this paper, we firstly investigate the compact uniform attractor for systems (1)–(4) by employing the energy equations and the energy method presented by Ball (see [16, 17]). The energy method can be concisely understood as the following two steps (e.g., in autonomous cases): construct a weakly compact attractor and prove the strong compactness of the weak attractor, that is, verify that the weak attractor is actually the strong one. To accomplish Step 1, one can construct a bounded (weakly compact) absorbing set and the weak continuity of the system. Step 2 is usually deduced by applying proper energy inequalities together with Lemmas 11 and 12. Obviously, this method is good at solving problems which are not continuous and those that lack Sobolev embeddings (such as systems defined in unbounded domains).

Besides, approximate inertial manifolds (AIM) for the system is studied afterwards. This manifold is a finite-dimensional smooth surface in a phase space, whose small vicinity attracts all the trajectories at a much higher speed than global attractors. To investigate AIM, by employing the extending phase space method we transfer the nonautonomous system to an autonomous one , and we get the AIM for by constructing the AIM for .

The main result of this paper contains Theorems 13 and 17. It is summarized by the following.

Main Theorem.  Assume that

(i), are quasiperiodic forces satisfying Assumption 1;(ii)generalized and are quasilinear functions satisfying (9) and (10).

Then systems (1)–(4) generate a family of processes in . Moreover, the family of processes admits a compact uniform attractor and an AIM in .

We would like to remark that the existence of the compact uniform attractor for the system does not depend heavily on the quasiperiodicity of the forces. It still holds when the forces are just translation bounded (see Remark 14), that is, it strengthens the result in [15].

This paper is organized as follows. In Section 2, we show the LS equations in details and we deeply introduce the quasiperiodicity conditions. In Section 3, we get the uniform a prior estimates for the solutions. In Section 4, we study the unique existence of the solution. In Section 5 the existence of the uniform attractor for (5)–(8) is obtained by applying weak convergence method. In Section 6, AIM for (5)–(8) is constructed by extending and splitting the phase space and making use of projection operators.

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 any Banach space . Besides, notations “”, “”, and “” denote weakly converges to, weakly star converges to and strongly converges to, respectively. And we denote different constants by the same letter , and represents that the constant relies only on the parameters that appear in the brackets.

2. Preliminaries

We show the nonautonomous dissipative generalized long-short wave equations with quasiperiodic forces in details as follows:

with initial value conditions

and periodic boundary value conditions

where is an unknown complex valued vector, is an unknown real valued function, are positive constants and nonautonomous terms , and are time-depended external forces satisfying quasiperiodicity conditions (see Assumption 1); non-linear terms and are given real-valued functions, satisfying

where are given positive constants for .

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

is called the hull of in , denoted by . is translation compact (resp., translation bounded) if is compact (resp., bounded) in .

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

Assumption 1. For , we suppose and it satisfies quasiperiodicity conditions; namely,

and for all ,

where , , and are rational and independent; is differentiable to each position and

If satisfy Assumption 1, we can consider the symbol space as

Since there is a continuous mapping , from [18] we know that the symbol space can be replaced by . And, for each , the translation operator acting on can be defined as

Therefore, is translation compact.

Proposition 2. Under Assumption 1, we can deduce the following properties:

(i)  is translation bounded in  ; that is,

where , , (ii)for all , ,(iii), which can be seen directly from (15) and the fact that

Moreover, is translation bounded in . Similarly to (18), by the continuity of , we can find a constant , which is independent of , such that

For brevity, we set , , and let with the norm

Similarly, we let and for each ,

Then systems (5)–(8) can be rewritten as

where the symbol or and the symbol space or .

3. Uniform a Prior Estimates of the Solutions

In this section, we derive uniform a priori estimates of the solutions both in time and in symbols which come from the symbol space . First we recall the following interpolation inequality.

Lemma 3. Let , , such that , , . Then we have

for , where and , .

Lemma 4. Let Assumption 1 hold. If and , then the solutions of problem (5)–(8) satisfy

where and , whenever .

Proof. Taking the inner product of (5) with in we get that

Taking the imaginary part of (26), we obtain that

By Young inequality and Proposition 2 we have

Then by Gronwall lemma we can complete the proof.

In the following, we denote by , which will not cause no confusions.

Lemma 5. Under assumptions of (9) and (10) and Assumption 1, if  , then solutions of problems (5)–(8) satisfy

where and , whenever .

Proof. Taking the inner product of (5) with in and taking the real part, we have

By (6) we know that

which shows that

where is introduced by

Taking the inner product of (5) with in and taking the real part, we get that

Multiply (34) by and add the resulting identity to (32) 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. For all , when is sufficiently large, by (9), Lemmas 3 and 4 we have

By (9) we deduce that

And then

By (36)–(41) we deduce that

Similarly we can also deduce that

Taking the inner product of (6) with in , we have

By (5) we get that

It comes from (44)–(46) that

Dealing with the right hand side of inequality (47), by Lemmas 3 and 4, we get

Therefore,

Analogously, we can also deduce that

Set , and

Then by (42) and (53) and (43) and (54) we can respectively deduce that

which shows that if we set , we can deduce that

where . By Gronwall lemma we have that

Similarly to (39), (40), (51), and (48), for we have

And then

where when . Then by (58) we infer that

where . By (55), (59) and (61) we infer that

Choosing , we have

which concludes the proof by using Lemma 4.

Lemma 6. Under assumptions of Lemma 5, if , then solutions of problems (5)–(8) satisfy

where and , whenever .

Proof. Taking the real part of the inner product of (5) with in , we have

By (5) and (6), we have

Since

we know that

Multiply (5) by and take the real part, we find that

Therefore,

Now we deal with (70) to get (78). Due to equalities

we deduce that

We take care of terms in (72) as follows

It follows from (72)–(75) that

And then

From (70) and (77) we have

By (65), (66), (68), and (78) we conclude that

where .

For later purpose, we let

Then from (79) we have

or

By Lemma 5 and Agmon inequality we have

In the following, we denote . By Lemma 3 and (84) we estimate the size of to get

Taking the inner product of (6) with in , we see that

Since

by (86) we can deduce that

From (5) we know that

Taking the real part of the inner product to (89) with in , we have

Because of

it holds that

By (92) and (88), we find that

That is,

For later use, we let

Then identity (94) as being equivalent to

Similarly to (85) we estimate each term in (94) and then we get

Let , and

By (85) and (98) we deduce that

which has the same form with (57) in the proof of Lemma 5. Similarly to the study of (57) we can derive that

where and when . By (80) we deduce that

and then by (99), (101), and (102), we deduce that

which concludes the proof by Lemma 5.

To study the AIM for the system, we construct the following higher order estimate.

Lemma 7. Under the assumptions of (9) and (10) and Assumption 1, for each , solutions of (5)–(8) satisfy

where .

Proof. Taking the partial derivative of (6) with respect to , by (9) and (10) we have

Taking the partial derivatives of (5) with respect to and , we have that

Taking the inner product of (106) with and taking the imaginary part, we get

Since

dealing with each term in (107) in the same way, we deduce that

By Gronwall lemma and (5) it follows that

where , which completes the proof.

4. Unique Existence of the Solution

In this section, we show the unique existence theorem of the solutions. Since uniform a priori estimates have been established in the former section, one can readily get the existence of the solution by Galërkin’s method (see [9, 14, 16, 19]) or operator semigroup method (see [6]). We show the theorem and prove it briefly for readers’ convenience.

Theorem 8. Under assumptions of Lemma 6, for each , systems (5)–(8) has a unique global solution , for all .

Proof. We prove this theorem briefly by two steps.

Step 1. Existence. By Galërkin’s method, we apply the following approximate solution

to approach and the solution of the problems (5)–(8), where is a orthogonal basis of satisfying . And satisfies, ,

It is easy to see that system (112) is an initial boundary value problem of ODE for the unknown coefficients ; the solution of which is known to be unique. Like [19], by the a priori estimates in Section 3 we know that converges (weakly star) to a which solves (5)–(8).

Step 2. Uniqueness. Suppose and are two solutions of the problems (5)–(8). Let ; then satisfies

which has nothing to do with symbols. Similarly to [19–21], we can deduce that , which concludes the proof.

Similarly to [16], by Lemma 7 we note that systems (5)–(8) has an unique global smooth solution in , and, moreover, it follows the following lemma.

Lemma 9. Under the assumptions of Lemma 7, for each , problems (5)–(8) have a unique global solution , for all , satisfying

where relies on the data and relies on the data and whenever .

5. Existence of the Compact Uniform Attractor

In this section, we derive the existence of the compact uniform attractor for the system applying Ball’s idea (see [16, 17]). That is, first we construct the weak uniform attractor (the convergences are taken in the sense of weak topology), and then we show that the weak uniform attractor is actually the strong one.

First we recall the following facts. Each solution trajectory for systems (5)–(8) satisfie

where ,  ,  , and are given by (80), (81), (95), and (96), respectively. Moreover, by the uniform boundedness and the compactness embedding (for all ) we have that , and , are all weakly continuous in .

Since we have uniformly estimated the size of solutions by Lemma 6 and shown the unique existence of the solution by Theorem 8, following the method of [8, 15] we have the following theorem.

Theorem 10. Under assumptions of Theorem 8, the family of processes generated by systems (5)–(8) is weakly -continuous and it admits a weakly compact uniform attractor in .

Proof. Since by Lemma 6 we know there exists a bounded uniform absorbing set, it suffices to prove that is weakly -continuous and the existence of the weak uniform attractor follows.

For any fixed , let

we will complete the proof if we deduce that

where , and .

By (117) and Theorem 8 we get the boundedness

By Agmon inequality we see that

Note that

and . By (120) and (121) we find that and

Due to Theorem 8 and (124) we know that there exists and subsequences of , which are still denoted by , such that

Besides, for all , by (120) we know that there exists such that

By (125) and a compactness embedding theorem, we claim that

In the following, we will show that is a solution of problems (5)–(8).

For all , , by (122) we find that

Due to

and by (121), (128), and (125)

we have

Taking care of other terms of (129) in similar methods and taking the limit, we have

Therefore, in the sense of distributions it holds that

which shows that satisfies (5).

For all and for all with , , by (122) we find that

Assumption (117) implies that

Then taking the limit of (135), by (136) we have

While from (134) we know that

It come from (137) and (138) that

and then

Equations (134) and (140) imply that

For all , , with , . Then repeating the procedure of the proofs of (135)–(138), by (127) we deduce that

It comes from (127), (141), and (142) that

Similarly, we can also deduce that

which together with (143) proves (118) and then the theorem.

To prove the strong compactness of the attractor , we recall the following two lemmas.

Lemma 11. Let be a uniform convex Banach space (particularly, a Hilbert space) and let be a sequence in . If and , then .

Lemma 12. Let be a sequence in space . If , then

Theorem 13. Under assumptions of Theorem 8, the weak uniform attractor in Theorem 10 is actually the strong one for the system in .

Proof. Since a point belongs to the weak uniform attractor if and only if there exist two sequences and such that for all , it uniformly holds that

where as . The theorem is concluded if the weak convergence is strong.

For each fixed, since , we can consider it as ,  . By Lemma 6 we know is bounded in , and then there exists a and a subsequence of , which is still denoted by , such that

Let

where is the translation operator on . Since is quasiperiodic, there exists a such that

Therefore, by (147) and (148) and the weak -continuity of we see that

and by taking ,

From the first equality of (148) we can consider as the solution trajectory, starting at , created by . Hence, by (115) and from the boundedness that we find that

Since and are weakly continuous in , by taking in (152), from (148), (147), (150), (151), and the Lebesgue dominated convergence theorem we get that

While , we can consider as the solution at corresponding to the initial data and the symbol . Similarly to (152) we have

Deducting (154) from (153), we see that

Since is fixed arbitrarily, let ; we conclude that

On the other hand, by Lemma 12, the weak convergence implies that

It follows from the previous two inequalities that

Similarly to the previous arguments, by using (116) we can derive that

By (146), (158), and (159) and Lemma 11, we conclude that in , which completes the proof.

Remark 14. We remark that up to this point the quasiperiodicity of the forces is not essentially necessary. We have actually used the uniform boundedness and the weak compactness of the symbol space in , which can be totally satisfied by translation bounded external forces. In other words, if are relaxed to be translation bounded: , then all the results here still hold.

6. Approximate Inertial Manifolds for (5)–(8)

6.1. Extending and Splitting the Phase Space

From Theorem 13 we know the systems (5)–(8) create a family of processes , which admit a compact uniform attractor in . Then from phase plane extension formula in [18], there is a semigroup , where

which is created by the following autonomous system:

Let be a bilinear operator: , a nonlinear operator: , and . Equations (161)–(163) can be transformed into the following abstract differential form

Since is an unbounded self-conjugate compact operator, there is a complete orthogonal set of eigenfunctions of such that , and

For all , we let be a projective operator and let . Taking the projection of (165) we get

where , , , and .

By Parseval’s formula we can get the following proposition.

Proposition 15. For all , there is the following expansion:

and if ,

Moreover, because of (166), it holds that

From Lemma 9, Proposition 15 and Agmon inequality , we can deduce the following lemma.

Lemma 16. If assumptions of Lemma 9 are satisfied, the solution satisfies

6.2. Constructing the AIM

Now we are in the position to show the AIM for (5)–(8).

Set ,

where satisfies

Let be orthogonal projection mappings.

The following theorem shows that , the graph of mapping , is just an AIM for the autonomous system (165) and is for problems (5)–(8), which concludes this paper.

Theorem 17. Under assumptions of Lemma 9, it holds that

where is the graph of and depends only on the data. Moreover,

which shows that is an approximate inertial manifold for problems (5)–(8).

Proof. From (176) and (168) we deduce that

which implies that

Since

from (181) and Lemma 16, we get

Because , there exists such that for all ,

From (177) and (170) we see that

While ,

where and . Then from (185), we see that

By Lemma 16 we can deduce that

While from Lemma 6 we know that

Therefore,

By (187), (188), (189), and (190), we get

Then from (184) and (191) we can conclude that

The estimate (179) follows from

and we complete the proof.

Acknowledgments

The authors thank an anonymous referee for his/her helpful comments on the paper. This work was supported by the NSF of China (nos. 11371183, 11271050, and 11071162) and the NSF of Shandong Province (no. ZR2013AM004).

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