Uniform Attractor for the Fractional Nonautonomous Long-Short Wave Equations

Ge, Huanmin; Xin, Jie

doi:https://doi.org/10.1155/2014/712183

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 712183 | https://doi.org/10.1155/2014/712183

Uniform Attractor for the Fractional Nonautonomous Long-Short Wave Equations

Huanmin Ge¹and Jie Xin¹

Academic Editor: Prasanta K. Panigrahi

Received14 Mar 2014

Accepted03 Sept 2014

Published20 Oct 2014

Abstract

We firstly proved the existence and the uniqueness of the solution for the -periodic fractional nonautonomous long-short wave equations with translation compact force by using Galerkin method and then obtained the compact uniform attractor of the system.

1. Introduction

In this paper, we consider the following fractional nonautonomous long-short wave equations with translation compact forces: with initial and periodic boundary conditions: where is an unknown and complex-valued function, is an unknown real valued function, ; , and nonautonomous terms and are time-dependant external forces and translation compact (see Definition 1).

We all know that the long-short wave resonance equations play an important role in fluid mechanics and have rich physical and mathematical properties. There are more and more resent papers treating the long-short wave resonance equations. Guo studied the global solution for one class of the system of LS nonlinear wave interaction in [1] and the periodic initial value problems and initial value problems for one class of generalized long-short type equations in [2]. The papers [3–5] studied the existence of a global attractor of it. Cui et al. developed the weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces in [6].

The Schrödinger type equation has been of great importance describing nonrelativistic quantum mechanical behavior. It is well known that Feynman and Hibbs derive the standard (nonfractional) Schrödinger type equation by applying path integrals over Brownian paths in [7]. Recently Laskin generalized the Schrödinger equation to space fractional cases using path integrals over Lévy trajectories in [8, 9]. In [10], the authors discussed the models and numerical methods of the fractional calculus. The fractional Schrödinger type equation is used to describe better physical phenomenon and has attracted more and more attention of researchers. Guo and Xu studied some applications of the Schrödinger equation in physics (see [11]). In [12], the authors obtained the approximate analytical solutions of the fractional nonlinear Schrödinger equations by using the homotopy perturbation method. Eid et al. studied the -dimensional fractional Schrödinger equation and obtained its exact solutions in [13]. Guo et al. investigated the fractional nonlinear Schrödinger equation and showed the existence and uniqueness of its global smooth solution by using energy method in [14]. Goubet [15] studied regularity of the attractor for a weakly damped nonlinear Schrödinger equation in .

The rest of this paper is arranged as follows. In Section 2, we recall some basic definitions, introduce preparation results, and analyse some fractional calculation laws which depend heavily on -period. In Section 3, we introduce some preparation lemma and give the uniform a priori estimates (uniform in initial data and symbol in the symbol space and large time). In Section 4, we show the existence and uniqueness of the solution of the system. In Section 5, we prove the existence of strong compact uniform attractor of the system.

Through the paper, we denote the norm of with the usual inner product by . We denote the norm of for all by . For simplicity and convenience, the letter represents a constant, which may be different from one to others. represents the constant expressed by the parameters appearing in the parentheses.

2. Preliminaries

In this section, we introduce notations definition and preliminary facts. We firstly recall the following known definitions (see [6, 16–18]) and some main lemmas (see [16, 19, 20]).

Definition 1. Suppose is a Banach space, is a function, and is the translation operator. The hull of is defined by (i)is said to be translation bounded in if is bounded in which Then consists of all the translation bounded functions in .(ii) is called translation compact in if is compact in , where the convergence is taken in the sense of local quadratic mean convergence topology of . The collection of all the translation compact functions in is denoted by .

Let be a Banach space, and the following definitions are common.

Definition 2. Let be a parameter set. , is said to be a family of processes in , if, for each , is a process; that is, the two-parameter mapping from to satisfies(i), (ii), where is called the symbol space and is called the symbol.

A subset is said to be uniformly absorbing set for the family of processes , if, for any and subset denoting the set of all bounded subsets of , there exists such that for all . A set is called uniformly attracting for the family of process , if, for each fixed and every , it satisfies that

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

Definition 4. , a family of processes in , is said to be -continuous, if, for any fixed and , , projection is continuous from to .

Definition 5. The space denotes all measurable functions with the norm for , and

Lemma 6. Let be a compact metric space and suppose is a family of operators defined on , satisfying(i)(ii)translation identity: where is an arbitrary process in compact metric space . Note that 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 7. Assumption (11) holds if the system has a unique solution.

Lemma 8. Let be a uniform convex Banach space (particularly, a Hilbert space), and let be a sequence in . If and , then .

Lemma 9. Let be a sequence in space . If , then
Since the solution , if it exists, is a -periodic function, we have the Fourier expansion: where . Therefore, and is defined by Since the following definitions make sense. Let and let be a complete space of the set under the norm: Then we can easily check that is a Banach space and that, for , is space-periodic with the period and its order derivatives are in . And for , when . is a Hilbert space with the inner product
For brevity, we introduce and . We denote the space of functions by with norm Similarly, we denote the space of by with norm

Assumption 10. Suppose that the symbol belongs to 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 assume .

Remark 11. Due to the conception of translation compact/boundedness, we remark that(i), ;(ii), where is a translation operator.

3. A Uniform A Priori Estimates

In this section, we will get some uniform a priori estimates which hold uniformly independently of initial data, time, and symbols in symbol space (). In the following, we denote that and , which will not cause any confusions.

We first recall the Gagliardo-Nirenberg and the Young inequalities (see [21]).

Lemma 12. Let . Then for , there exists a constant such that where .

Lemma 13. Let . Then for each satisfying , it holds that

Lemma 14. Assume that(i) satisfy Assumption 10;(ii) and solves problem (1)–(4).Then there exist positive constants and such that

Proof. Taking the inner product of (1) with , we have Taking the imaginary part of (27), we get By (28) and Remark 11, we have By Gronwall inequality we get the lemma.

Lemma 15. Assume that(i), and satisfy Assumption 10;(ii) and solves problem (1)–(4).Then there exist positive constants and such that

Proof. Taking the inner product of (1) with , we have Taking the real part of (31), we have where So we have Taking the inner product of (1) with , we have Taking the real part of (35), we have By (34) and (36), we have By Lemmas 12 and 13 and the condition , we have So we reduce that Similarly, we also derive that Taking the inner product of (2) with , we have By (1), we have where By (41)(43), we get By Lemmas 12 14 and the condition , we have So we obtain that Similarly, we can also get that Set So by (39) and (46), we get By (40) and (47), we also get Setting , , then we deduce that By Gronwall inequality, we have Obviously for any , we have So by (52)~(54), there exists a such that for any .
By (48),(53), and (55), we get Then setting , we get By using Lemma 14, we conclude the lemma.

Lemma 16. Assume that(i), and satisfy Assumption 10;(ii) and solves problem (1)–(4).Then there exist positive constants and such that

Proof. Taking the inner product of (1) with , we have Taking the real part of (59), we have where, by (1) and (2), we have By Lemmas 12, 14, and 15 and , we see that By (60)~(62), Lemmas 12~ 15, and Hölder inequality, we can see that Taking the inner product of (2) with , we have where, by Lemmas 12~ 15 and , we can see that So by (64) and (65), we get Set and Then by (63) and (66), we can deduce that which has the same form with (51) in the proof of Lemma 15. Similar to the study of (51), there exist positive constants and such that which conclude the proof of Lemma 16.

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 above section, one can readily get the existence of the solution by ’s method (see [20, 22–24]). We show the theorem and prove it briefly for readers’ convenience.

Theorem 17. Set , and satisfy Assumption 10; for each , then system (1)–(4) has a unique global solution , .

Proof. We prove this theorem by two steps.
Step 1. The existence of solution.
By ’s method, we construct the approximate solution of the periodic initial value problem (1)(4). We apply the following approximate solution: to approach , the solution of the problem (1)–(4). And for satisfies We see that system (71) is an initial boundary value problem of ordinary differential equations (ODE). By the standard existence theory for ODE and uniform a priori estimates in Section 3, for any , there exists a unique solution of (71), such that There is a subsequence of and such that Due to the above proof and the continuous extension theorem, is the solution Of (1)(4).
Step 2. The uniqueness of solution.
Suppose are two solutions of problem (1)–(4). Let , and then satisfies Obviously, is uniformly bounded. Note that .
Taking the inner product of (74) with and taking the imaginary part, we can get Taking the inner product of (75) with , we can obtain Taking the inner product of (74) with and taking the real part, we can get Differentiating (74) with respect to , taking the inner product of with , and taking the imaginary part, we can get Taking the inner product of (75) with , we have Therefore by (78)(81), we conclude that From Gronwall inequality and (76), we have Therefore, we complete the proof of the theorem.

5. Uniform Absorbing Set and Uniform Attractor

In this section, we will prove the existence of the strong compact uniform attractor of problem (1)~(4) applying Ball et al.’s idea (see [19, 22]). Firstly, we construct a bounded uniformly absorbing set. Next, we show the weak uniform attractor of the system. Lastly, we derive that the weak uniform attractor is actually the strong one.

Theorem 18. Under assumptions of Theorem 17, admits a strong compact uniform attractor .

Proof. We prove this theorem by three steps.
Step 1. possess a bounded uniformly absorbing set in .
Let . By Theorem 17, is a bounded absorbing set of the process .
By Assumption 10, we know that, for each , holds. So the solution of (1)(4) satisfies Then we can get that the set is a bounded uniformly absorbing set of .
Step 2. we prove the existence of weakly compact uniform attractor in .
From Lemma 6, Theorem 17, and Step 1, we only need to prove that is -continuous. We denote weak convergence by and weak convergence by .
For any fixed , let If we can deduce that where , we will obtain that is -continuous. By (86) and Theorem 17, we can get that Then by Lemmas 12 16, we can see that Note that and . By (89) and (90), we find that and Because of Theorem 17 and (93), we easily see that there exist a subsequence of and , such that Besides, for any , by (89) there exists such that By (94) and compactness embedding theorem, we can get that
Next, we will obtain that is a solution of problem (1)(4).
For , by (91) we have that Since by (90), (94), and (97), Then we have And by (94), we have that By using the similar methods to the other terms of (98), we have So, we can get that which shows that ( satisfies (1).
For any with , by (91) we find that We know that Assumption (86) implies that Then from (105) and (106), we have while by (104) we know that So by (107) and (108), we have that By (104) and (110), we have For any , with , then we repeat the procedure of proofs of (105)(108) by (96) having From (96), (111), and (112), we have that Similarly, we can also derive that From (113) and (114), we deduce (87). We complete the proof of the step.
Step 3. We show the weakly compact uniform attractor is actually the strong one.
From the proof of Lemma 16, we know each solution trajectory for problem (1)–(4) satisfies where By the uniform boundedness and the compactness embedding, we have that , , and are all weakly continuous in .
From Step 2, we can see that the point if and only if there exist two sequences and such that for all , it uniformly satisfies that where as . If the weak convergence implies strong one, we obtain is actually the strong compact attractor. For each fixed , because of , we consider it as . By Lemma 16 and Theorem 17, is bounded in . Then there exists a subsequence of and a point , such that Let where is the translation operator on . Since is translation compact symbol, there exists a symbol such that Then by (118), (119), and the weak -continuity of , we can get that From (119), we can see that the solution trajectory is created by starting at . By (115), (119), and (122), we have that Let in (122). Since and are weakly continuous in , , and the Lebesgue dominated convergence theorem, we can obtain that
Since , we can see the solution as at corresponding to the initial data and the symbol . Similarly to (122), we have Deducting (125) from (124), we can get that As , we can get that On the other hand, the weak convergence implies that From the above two inequalities, we get that Similarly to the above arguments, by using (116) we can derive that Then we get that in . We complete the proof of the theorem.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the NSF of China (nos. 11371183 and 11271050) and the NSF of Shandong Province (no. ZR2013AM004).

References

B. Guo, “The global solution for one class of the system of LS nonlinear wave interaction,” Journal of Mathematical Research and Exposition, vol. 7, no. 1, pp. 69–76, 1987.
View at: Google Scholar | MathSciNet
B. Guo, “The periodic initial value problems and initial value problems for one class of generalized long-short type equations,” Journal of Engineering Mathematics, vol. 8, pp. 47–53, 1991.
View at: Google Scholar
X. Du and B. Guo, “The global attractor for LS type equation in $R^{1}$ ,” Acta Mathematicae Applicatae Sinica, vol. 28, pp. 723–734, 2005.
View at: Google Scholar
Y. Li, “Long time behavior for the weakly damped driven long-wave—short-wave resonance equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 261–289, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
R.-F. Zhang, “Existence of global attractor for LS type equations,” Journal of Mathematical Research and Exposition, vol. 26, no. 4, pp. 708–714, 2006.
View at: Google Scholar | MathSciNet
H. Cui, J. Xin, and A. Li, “Weakly compact uniform attractor for the nonautonomous long-short wave equations,” Abstract and Applied Analysis, vol. 2013, Article ID 601325, 13 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, NY, USA, 1965.
N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Physics Letters A, vol. 268, no. 4–6, pp. 298–305, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
N. Laskin, “Fractional quantum mechanics,” Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 62, no. 3, pp. 3135–3145, 2000.
View at: Publisher Site | Google Scholar
D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, 2012.
X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” Journal of Mathematical Physics, vol. 47, no. 8, Article ID 082104, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
D. Baleanu and A. K. Golmankhaneh, “Solving of the fractional non-linear and linear Schrödinger equations by homotopy perturbation method,” Romanian Journal of Physics, vol. 54, no. 10, pp. 823–832, 2009.
View at: Google Scholar | MathSciNet
R. Eid, S. I. Muslih, D. Baleanu, and E. Rabei, “On fractional Schrödinger equation in α-dimensional fractional space,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1299–1304, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
B. Guo, Y. Han, and J. Xin, “Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 468–477, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
O. Goubet, “Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^{2}$ ,” Advances in Differential Equations, vol. 3, no. 3, pp. 337–360, 1998.
View at: Google Scholar | MathSciNet
V. V. Chepyzhov and M. I. Vishik, “Attractors of nonautonomous dynamical systems and their dimension,” Journal de Mathématiques Pures et Appliquées, vol. 73, no. 3, pp. 279–333, 1994.
View at: Google Scholar | MathSciNet
J. Bergh and J. Laofstraom, Interpolation Spaces, Springer, Berlin, Germany, 1976.
V. Chepyzhov and M. Vishik, “Non-autonomous evolutionary equations with translation-compact symbols and their attractors,” Comptes Rendus de l'Académie des Sciences Série I: Mathématique, vol. 321, no. 2, pp. 153–158, 1995.
View at: Google Scholar | MathSciNet
J. M. Ball, “Global attractors for damped semilinear wave equations,” Discrete and Continuous Dynamical Systems Series A, vol. 10, no. 1-2, pp. 31–52, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
B. Guo, Infinite Dimensional Dynamical Systems, National Defense Industry Press, Beijing, China, 1st edition, 2000.
J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, Germany, 1976.
View at: MathSciNet
J. Xin, B. Guo, Y. Han, and D. Huang, “The global solution of the $(2 + 1)$ -dimensional long wave-short wave resonance interaction equation,” Journal of Mathematical Physics, vol. 49, no. 7, Article ID 073504, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
B. Guo and B. Wang, “The global solution and its long time behavior for a class of generalized LS type equations,” Progress in Natural Science, vol. 6, no. 5, pp. 533–546, 1996.
View at: Google Scholar | MathSciNet
R. Zhang and B. Guo, “Global solution and its long time behavior for the generalized long-short wave equations,” Journal of Partial Differential Equations, vol. 18, no. 3, pp. 206–218, 2005.
View at: Google Scholar | MathSciNet

Copyright

Copyright © 2014 Huanmin Ge 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.

PDF Download Citation

Download other formats

Order printed copies

Views

580

Downloads

697

Citations