Time-Periodic Solution of the Weakly Dissipative Camassa-Holm Equation
Chunyu Shen1,2
Academic Editor: Shangbin Cui
Received31 May 2011
Accepted31 Aug 2011
Published26 Oct 2011
Abstract
This paper is concerned with time-periodic solution of the weakly dissipative Camassa-Holm equation with a periodic boundary condition. The existence and uniqueness of a time periodic solution is presented.
1. Introduction
The Camassa-Holm equation
modeling the unidirectional propagation of shallow water waves over a flat bottom, where represents the fluidβs free surface above a flat bottom (or equivalently, the fluid velocity at time and in the spatial direction).
Since the equation was derived physically by Camassa and Holm [1, 2], many researchers have paid extensive attention to it. The Camassa-Holm equation is also a model for the propagation of axially symmetric waves in hyperelastic rods [3, 4]. It has a bi-Hamiltonian structure [5, 6] and is completely integrable [1, 2, 7β11]. It is a reexpression of geodesic flow on the diffeomorphism group of the circle [12] and on the Virasoro group [13]. Its solitary waves are peaked [7], and they are orbitally stable and interact like solitons [14β16]. The peakons capture a characteristic of the traveling waves of greatest height-exact traveling solutions of the governing equations for water waves with a peak at their crest [17β19].
The Cauchy problem of the Camassa-Holm equation has been extensively studied. It has been shown that this equation is locally well posed [20β25] for initial data with . Moreover, it has global strong solutions modeling permanent waves [20, 24β27] but also blow-up solutions modeling wave breaking [20β28]. On the other hand, it has global weak solutions with initial data [29β35]. Moreover, the initial-boundary value problem for the Camassa-Holm equation on the half-line and on a finite interval was discussed in [36, 37]. It is observed that if is the solution of the Camassa-Holm equation with the initial data in , we have for all ,
It is worth pointing out that the advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solitons and models wave breaking [2, 20, 21].
In general, it is difficult to avoid energy dissipation mechanisms in a real world. Ott and Sudan [38] investigated how the KdV equation was modified by the presence of dissipation and the effect of such dissipation on the solitary solution of the KdV equation. Ghidaglia [39] investigated the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system.
The Camassa-Holm equation with dissipative term is
where is the forcing term, is a dissipative term, can be a differential operator or a quasi-differential operator according to different physical situations.
With and , (1.3) becomes weakly dissipative Camassa-Holm equation
where is a constant.
The local well-posedness, global existence, and blow-up phenomena of the Cauchy problem of (1.4) on the line [40] and on the circle [41] were studied. A new global existence result and a new blow-up result for strong solutions to this equation with certain profiles are presented recently [42]. We found that the behaviors of (1.4) are similar to the Camassa-Holm equation in a finite interval of time, such as the local well-posedness and the blow-up phenomena, and that there are considerable differences between (1.4) and the Camassa-Holm equation in their long-time behaviors. The global solutions of (1.4) decay to zero as time goes to infinity provided the potential is of one sign (see [40, 41]). This long-time behavior is an important feature that the Camassa-Holm equation does not possess. It is well known that the Camassa-Holm equation has peaked traveling wave solutions. But the fact that any global solution of (1.4) decays to zero means that there are no traveling wave solutions of (1.4).
Another difference between (1.4) and the Camassa-Holm equation is that (1.4) does not have the following conservation laws
which play an important role in the study of the Camassa-Holm equation.
Equation (1.4) has the same blow-up rate as the Camassa-Holm equation does when the blow-up occurs [41]. This fact shows that the blow-up rate of the Camassa-Holm equation is not affected by the weakly dissipative term, but the occurrence of blow-up of (1.4) is affected by the dissipative parameter [40, 41].
In the paper, we would like to consider the following weakly dissipative Camassa-Holm equation
where is the weakly dissipative term, is a constant, and the forcing term is -periodic in time and -periodic in spatial . Without loss of generality, we assume further , where . When system is periodically dependent on time , we want to know whether there exists time-periodic solution with the same period for the system. In many nonlinear evolution equations, the study of time-periodic solution has attracted considerable interest (e.g., [43β45]). In this paper, we will prove that (1.6)β(1.8) have a solution by using the Galerkin method [46], and Leray-Schauder fixed point theorem [44].
Our paper is organized as follows. In Section 2, we give some notations and definition of some space used in this paper. In Section 3, we prove the existence of the approximate solution and give uniform a priori estimates needed where we prove the convergence of a sequence of the approximate solution. Section 4 is devoted to the study of the existence and uniqueness of time-periodic solution for (1.6)β(1.8).
2. Preliminaries
Before starting our work, it is appropriate to introduce some notations and inequalities that will be used in the paper.
Let be a Banach space, we denote by the set of -periodic -valued measurable functions on with continuous derivatives up to order . The norm in the space is .
For , the space is the set of -periodic -valued measurable functions on such that
The space denote the set of functions which belong to together with their derivatives up to order , and we write in particular when is a Hilbert space.
and are classical Sobolev spaces. For simplicity, we write by as and by .
The following inequalities (see [47]) will be used in the proofs later
where , as , .
3. A Priori Estimates
In this section, we first prove that (1.6)β(1.8) have a sequence of approximate solutions , then give a prior, estimates about .
We denote the unbounded linear operator on , then the set of all linearly independent eigenvectors of , that is, , with , is an orthonormal basis of . For any and a group of function , where , the function is called an approximate solution to (1.6)β(1.8) if it satisfies the equation as follows:
where and . By the classical theory of ordinary differential equations, for any fixed , the equation , has a unique -periodic solution and the mapping is continuous and compact in . Hence by Leray-Schauder fixed point theorem, we want to prove the existence of an approximate solution only to show for all possible solution of (3.1) replaced by instead of nonlinear term , where is a constant which only depends on , , , , and .
Lemma 3.1. If , then
where is a constant which only depends on , , , , , and , and .
Proof. Multiplying (3.1) by and summing up over from 1 to , we obtain
Then, we can get
Notice that , . From Youngβs inequality, we have , where is a constant. According to the above relations, we can derive from (3.4) that
where . Considering the time periodicity of and integrating (3.5) over , we get
Hence, there exists such that . From (3.5), we have . Integrating the above inequality with respect to from to , we deduce that
Hence, we infer
which concludes our proof.
From Lemma 3.1 and Leray-Schauder fixed point theorem, (3.1) has solution , which is also a sequence of approximate solutions of (1.6)β(1.8). In order to obtain the convergence of sequence , we need to give a priori estimates for the high-order derivers of .
Lemma 3.2. If , then
where is a constant which only depends on , , , , , , , , and , and .
Proof. Multiplying (3.1) by and summing up over from 1 to , we have
The above equation yields
From Youngβs inequality, we have
where is a constant. From (2.2), (3.8), and Youngβs inequality, we can deduce that
From (2.3), (3.8), Cauchy-Schwarz inequality, Youngβs inequality, and Lemma 3.1, we get
Taking (3.11)β(3.15) into account, we can infer that
where . Integrating (3.16) about from 0 to and considering the time periodicity of , we get
Hence, there exists such that
From (3.16), we have
Then we can obtain
which concludes our proof.
In the following, we continue to establish a priori estimates for high-order derivers of the approximate solution by an inductive argument.
Lemma 3.3. For any , if , then
where is a constant which only depends on , , , , , , , , , and .
Proof. By Lemma 3.2, we know the conclusion of Lemma 3.3 holds for . Assume that for the conclusion of Lemma 3.3 holds, we want to prove that the same statement holds for also. Multiplying (3.1) by and summing up over from 1 to , we have
Follow the same methods discussed in Lemma 3.2, we have
From the conclusion of Lemma 3.3 for , (2.2), (2.4) and Youngβs inequality, we can deduce that
Similarly, we can also deduce that
From the conclusion of Lemma 3.3 for , Youngβs inequality and (2.3), we have
Combining (3.25) and the above inequality, we can get
Similarly,
Taking (3.22)β(3.24) and (3.27)-(3.28) into account, we can deduce that
From the above relation, we can infer
where . Integrating (3.30) about from 0 to , there exists such that
From (3.30), we have
Integrating the above inequality from to and with (3.31), we can easily obtain
The proof is completed.
Lemma 3.4. For any , if , then
where is a constant which only depends on , , , , , , , , , and .
Proof. We first prove the conclusion of Lemma 3.4 holds for . Multiplying (3.1) by and summing up over from 1 to , we have
By Lemma 3.3, if , then we have . Hence,
Therefore, from (3.35) and (3.36), it is easy to know that
Assume that the conclusion of Lemma 3.4 holds for , we want to prove that the conclusion of Lemma 3.4 also holds for . Multiplying (3.1) by and summing up over from 1 to , we have
By Lemma 3.3, if , then for . Hence,
Taking (3.38) and (3.39) into account, it follows
This completes the proof of Lemma 3.4 by an inductive argument.
4. Existence and Uniqueness of Time-Periodic Solution
We have proved that (1.6)β(1.8) have a sequence of approximate solutions . In this section, we want to prove that the sequence converges and the limit is a solution of (1.6)β(1.8).
By Lemmas 3.1β3.4 and standard compactness arguments, we conclude that there is a subsequence which we denote also by such that for any , if , we have
From the above lemmas, we know that the nonlinear terms are well defined
as , uniformly in ,
as , uniformly in ,
as , uniformly in .
Consequently, it follows that
Thanks to the estimates obtained in the previous section, we have
a.e. on .
So we obtain that the existence of time periodic solution for (1.6)β(1.8), which is the following theorem.
Theorem 4.1. Given ,ββ, there exists a time periodic solution to (1.6)β(1.8), such that .
Under the assumption of Theorem 4.1, we are unable to prove the uniqueness of the solution for (1.6)β(1.8). But if we assume that is sufficiently small, then the result can be obtained.
Theorem 4.2. Suppose that the assumption in Theorem 4.1 holds. If is sufficiently small, then the time periodic solution of (1.6)β(1.8) in Theorem 4.1 is unique.
Proof. Let and be any two time periodic solutions of (1.6)β(1.8). With , we can get from (1.6) that
Taking the inner product of (4.7) with , we have
Since,
Hence, if is sufficient small such that , , then it follows from (4.8)β(4.11), we get
where is suitable constant. Applying Gronwallβs inequality, we derive that
Since is -periodic in , then for any positive integer we have
Then we can infer that
It follows from that , which completes the proof of Theorem 4.2.
References
R. Camassa and D. D. Holm, βAn integrable shallow water equation with peaked solitons,β Physical Review Letters, vol. 71, no. 11, pp. 1661β1664, 1993.
A. Constantin and W. A. Strauss, βStability of a class of solitary waves in compressible elastic rods,β Physics Letters A, vol. 270, no. 3-4, pp. 140β148, 2000.
B. Fuchssteiner and A. S. Fokas, βSymplectic structures, their Bäcklund transformations and hereditary symmetries,β Physica D, vol. 4, no. 1, pp. 47β66, 1981/82.
A. Constantin, βOn the scattering problem for the Camassa-Holm equation,β The Royal Society of London. Proceedings. Series A, vol. 457, no. 2008, pp. 953β970, 2001.
A. Constantin, βOn the inverse spectral problem for the Camassa-Holm equation,β Journal of Functional Analysis, vol. 155, no. 2, pp. 352β363, 1998.
A. Constantin, V. S. Gerdjikov, and R. I. Ivanov, βInverse scattering transform for the Camassa-Holm equation,β Inverse Problems, vol. 22, no. 6, pp. 2197β2207, 2006.
A. Boutet de Monvel and D. Shepelsky, βRiemann-Hilbert approach for the Camassa-Holm equation on the line,β Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 343, no. 10, pp. 627β632, 2006.
A. Constantin and H. P. McKean, βA shallow water equation on the circle,β Communications on Pure and Applied Mathematics, vol. 52, no. 8, pp. 949β982, 1999.
A. Constantin and B. Kolev, βGeodesic flow on the diffeomorphism group of the circle,β Commentarii Mathematici Helvetici, vol. 78, no. 4, pp. 787β804, 2003.
A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, βOn geodesic exponential maps of the Virasoro group,β Annals of Global Analysis and Geometry, vol. 31, no. 2, pp. 155β180, 2007.
A. Constantin, βExistence of permanent and breaking waves for a shallow water equation: a geometric approach,β Université de Grenoble. Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321β362, 2000.
A. Constantin and J. Escher, βWave breaking for nonlinear nonlocal shallow water equations,β Acta Mathematica, vol. 181, no. 2, pp. 229β243, 1998.
Y. A. Li and P. J. Olver, βWell-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,β Journal of Differential Equations, vol. 162, no. 1, pp. 27β63, 2000.
Z. Yin, βWell-posedness, blowup, and global existence for an integrable shallow water equation,β Discrete and Continuous Dynamical Systems. Series A, vol. 11, no. 2-3, pp. 393β411, 2004.
Z. Yin, βWell-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation,β Journal of Evolution Equations, vol. 4, no. 3, pp. 391β419, 2004.
A. Constantin and J. Escher, βGlobal existence and blow-up for a shallow water equation,β Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 26, no. 2, pp. 303β328, 1998.
A. Constantin and J. Escher, βWell-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,β Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475β504, 1998.
A. Constantin and J. Escher, βOn the blow-up rate and the blow-up set of breaking waves for a shallow water equation,β Mathematische Zeitschrift, vol. 233, no. 1, pp. 75β91, 2000.
A. Bressan and A. Constantin, βGlobal conservative solutions of the Camassa-Holm equation,β Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215β239, 2007.
A. Constantin and L. Molinet, βGlobal weak solutions for a shallow water equation,β Communications in Mathematical Physics, vol. 211, no. 1, pp. 45β61, 2000.
L. Molinet, βOn well-posedness results for Camassa-Holm equation on the line: a survey,β Journal of Nonlinear Mathematical Physics, vol. 11, no. 4, pp. 521β533, 2004.
E. Wahlén, βGlobal existence of weak solutions to the Camassa-Holm equation,β International Mathematics Research Notices, vol. 2006, Article ID 28976, 12 pages, 2006.
Z. Xin and P. Zhang, βOn the weak solutions to a shallow water equation,β Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411β1433, 2000.
A. Bressan and A. Constantin, βGlobal dissipative solutions of the Camassa-Holm equation,β Analysis and Applications, vol. 5, no. 1, pp. 1β27, 2007.
J. Escher and Z. Yin, βInitial boundary value problems of the Camassa-Holm equation,β Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 377β395, 2008.
J. Escher and Z. Yin, βInitial boundary value problems for nonlinear dispersive wave equations,β Journal of Functional Analysis, vol. 256, no. 2, pp. 479β508, 2009.
J.-M. Ghidaglia, βWeakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,β Journal of Differential Equations, vol. 74, no. 2, pp. 369β390, 1988.
S. Y. Wu and Z. Y. Yin, βBlowup and decay of solutions to the weakly dissipative Camassa-Holm equation,β Acta Mathematicae Applicatae Sinica, vol. 30, no. 6, pp. 996β1003, 2007.
S. Wu and Z. Yin, βBlow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation,β Journal of Mathematical Physics, vol. 47, no. 1, Article ID 013504, 12 pages, 2006.
S. Wu and Z. Yin, βGlobal existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,β Journal of Differential Equations, vol. 246, no. 11, pp. 4309β4321, 2009.
H. Kato, βExistence of periodic solutions of the Navier-Stokes equations,β Journal of Mathematical Analysis and Applications, vol. 208, no. 1, pp. 141β157, 1997.
B. Wang, βExistence of time periodic solutions for the Ginzburg-Landau equations of superconductivity,β Journal of Mathematical Analysis and Applications, vol. 232, no. 2, pp. 394β412, 1999.
Y. Fu and B. Guo, βTime periodic solution of the viscous Camassa-Holm equation,β Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 311β321, 2006.