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 thatMoreover, 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).