Research Article | Open Access
Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations
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.
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 . In the plasma physics they describe the resonance of the high-frequency electron plasma oscillation and the associated low-frequency ion density perturbation . Benney  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 . In , 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 . 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.  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. ).
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,
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:
(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. ).
Lemma 7. Let , , such that , , . Then one has for , where , , and .
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.
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.
Proof. Taking the real part of the inner product of (2) with in , we have
By (2) and (3), we have
we see that
Multiplying (2) by and taking the real part, we find that
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
From (64) and (69) we have
By (59), (60), (62), and (70) we conclude that
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