A Note on the Generalized Camassa-Holm Equation

Wu, Yun; Zhao, Ping

doi:https://doi.org/10.1155/2014/975925

Journal of Function Spaces

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Research Article | Open Access

Volume 2014 | Article ID 975925 | https://doi.org/10.1155/2014/975925

A Note on the Generalized Camassa-Holm Equation

Yun Wu¹and Ping Zhao¹

Academic Editor: Naseer Shahzad

Received15 Oct 2014

Accepted24 Nov 2014

Published25 Dec 2014

Abstract

We study the generalized Camassa-Holm equation which contains the Camassa-Holm (CH) equation and Novikov equation as special cases with the periodic boundary condition. We get a blow-up scenario and obtain the global existence of strong and weak solutions under suitable assumptions, respectively. Then, we construct the periodic peaked solutions and apply them to prove the ill-posedness in with .

1. Introduction

In this paper we study the global strong and weak solutions to the generalized Camassa-Holm equation with periodic boundary condition: where , , and .

When , (1) reduces to the the well-known CH equation: The CH equation was derived independently by Fokas and Fuchssteiner in [1] and by Camassa and Holm in [2]. Fokas and Fuchssteiner derived (3) in studying completely integrable generalizations of the KdV equation with bi-Hamiltonian structures, while Camassa and Holm proposed (3) to describe the unidirectional propagation of shallow water waves over a flat bottom.

As shown in [2], the CH equation is completely integrable and possesses an infinite number of conservation laws. Moreover, the CH equation is such an equation that exhibits both phenomena of soliton interaction (peaked soliton solutions) and wave breaking (the solution remains bounded while its slope becomes unbounded in finite time [3]), while the KdV equation does not model breaking waves [4]. In fact, wave breaking is one of the most intriguing long-standing problems of water wave theory [5]. The essential feature of CH should be pointed out: the fact that the traveling waves have a peak at their crest is exactly like for the waves of greatest height solutions of the governing equations for water waves (see [6–8] for the details).

From a mathematical point of view the Camassa-Holm equation is well studied and a series of achievements had been made. Constantin [9] and Misiołek [10] investigated the Cauchy problem for the periodic Camassa-Holm equation. Constantin et al. [3, 11–14] studied the wave breaking of the Cauchy problem for the CH equation. Recently, Jiang et al. gave a new and direct proof for McKean’s theorem in [15]. Xin and Zhang [16] proved that (3) has global weak solutions for initial data in . Bressan and Constantin developed a new approach to the analysis of the CH equation and proved the existence of the global conservative and dissipative solutions in [17, 18]. Holden and Raynaud [19, 20] also obtained the global conservative and dissipative solutions. The large time behavior of the CH equation was firstly established in [21]. In [22], Himonas et al. studied the persistence properties and infinite propagation speed for the CH equation.

In 2009, Novikov [23] found a new integrable equation: It is derived that (4) possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities and admits exact peaked solutions with [24, 25], as well as the explicit formulas for multipeakon solutions [25, 26].

By using the Littlewood-Paley decomposition and Kato’s theory, the well-posedness of the Novikov equation has been studied in Besov spaces and in the Sobolev space (see [27, 28]). Wu and Yin [29] established some results on the existence and uniqueness of global weak solutions to the Novikov equation. Jiang and Ni [30] established some results about blow-up phenomena of the strong solution to the Cauchy problem for (4). For the periodic boundary condition case, Tılay [31] proved that for the periodic Novikov equation is locally well-posed in . Later the range of regularity index of local well-posedness was extended to in [32]; furthermore, it is shown that the solution maps for both periodic boundary value problem and Cauchy problem of the Novikov equation are not uniformly continuous from any bounded subset in into . When , Grayshan [33] proved that the properties of the solution map for (4) are not (globally) uniformly continuous in Sobolev spaces . For the nonuniform dependence and ill-posedness results in Besov spaces, we refer to [34–37].

In this paper we consider the generalized CH equation (1) with . Let , then (1) takes the form of a quasi-linear evolution equation of hyperbolic type: Applying the operator , (1) can be expressed as the following nonlocal form: From the above equation, we can view (1) as a nonlocal perturbation of the Burgers-type equation:

We recall the following results in [38] for (1) and (2).

Proposition 1. If and , then there exists a such that (1) and (2) have a unique solution which depends continuously on the initial data . Moreover, satisfies the solution estimate where is a constant depending on .

In this paper we use the following notations. We use to denote estimates that hold up to some universal constant which may change from line to line but whose meaning is clear from the context. stands for and . All function spaces are over and we drop in all function spaces if there is no ambiguity. For linear operators and , we denote . The Fourier transform of the function is defined by , . The inverse Fourier transform is given by .

The operator for any real number is defined by . is the standard Soblev space on whose norm is defined by

For each , stands for the Friedrichs mollifier defined by where stands for the convolution. Here and is a Schwartz function satisfying for all the and for any .

We will also use another mollifier. Define where the constant is chosen so that . For , we set . To define the mollifier , we first let be the characteristic function on and It follows that and when is smooth, as and . Then we can define by Obviously, if , , then and (as ). Moreover, if , then .

Now we present our results.

Theorem 2. Let , , and let be the maximal existence time of the corresponding solution to (1) and (2). Then blows up at if and only if

Theorem 3. Let , . If does not change sign, then the corresponding solution to (1) and (2) exists globally.

Theorem 4. Suppose with such that does not change sign. Then (1) and (2) have a unique global weak solution in the sense of distribution. Moreover, .

When , the solution map is not uniformly continuous, and we establish the ill-posedness as follows. We refer to [33, 36, 39] for the ill-posedness results for the CH equation, Novikov equation, and the b-family equation.

Theorem 5. If , then (1) and (2) are ill-posed in in the sense that the solution map is not uniformly continuous from into for any . More precisely, there exist two sequences of weak solutions and in of (1) and (2) such that, for any , there hold the following estimates: where only depend on .

We outline the rest of the paper. In the next section, we give some preliminaries. We deal with the blow-up criterion and prove Theorems 2 and 3 in Section 3. In Section 4, we study the global weak solution and prove Theorem 4. We demonstrate Theorem 5 in Section 5.

2. Preliminaries

Rewrite (1) and (2) as follows: where , with

The following estimates are useful in our work.

Lemma 6 (Kato-Ponce commutator estimate [40]). If , , and , then

Lemma 7 (see [40] or see the Moser estimate in [41]). If , then is an algebra, and

From the construction of the mollifier , we know that and, for any and ,

In addition, we have

Lemma 8 (see [42]). Let be a function such that . Then, there is a such that, for any ,

The following Calderon-Coifman-Meyer type commutator estimate is also useful (see Proposition 4.2, [43]).

Lemma 9. If and , then there is a such that

3. Blow-Up Criterion and Global Existence of Strong Solutions

The aim of this section is to prove Theorems 2 and 3 which show that the solution blows up only when the slope of the wave blows up and the solution exists globally if does not change sign. Rewrite (1) as the following form: First, we have the following lemma.

Lemma 10. Let , , and is a solution to (1) and (2) with . Then there exists a constant such that

Proof. Since the term is only in , we cannot apply to either side of (26) when . So we apply the operator to (26), multiply both sides of the resulting equation by , and integrate over to obtain where we used and We now estimate , , respectively. For , we first note that is self-adjoint, then commute the operator with , and use (21) and (22) to get
By using the Cauchy-Schwarz inequality, Lemmas 6, 7, and 8, integration by parts, and (23), we have Therefore
In the same way, can be estimated as
For , we use the Cauchy-Schwarz inequality, Lemma 7, and (23) to obtain For the estimate for , we have Similar to (34), , while Hence we obtain that Combining (32), (33), (34), and (37) yields Integrating both sides with respect to results in Let tend to ; we get (27) and therefore complete the proof of this lemma.

Remark 11. Take in (27); then for , we have This estimate will be also used in the proofs of Lemma 20.
Let be the solution to problem (1) and (2). We define to be its lifespan, Then the following alternative property holds:

Proof of Theorem 2. From (1) we can deduce that which implies that Taking in Lemma 10 and using , we obtain that Hence we know that if is finite, then is bounded and the case (ii) in (42) would not occur, which implies that can be extended beyond . On the other hand, if does not blow up, then is bounded on for any . Since , is also bounded on . Thus we complete the proof.

Remark 12. Actually, we can prove a more precise blow-up criterion for sufficiently regular solutions to (1); that is, if , with , then the solution blows up if and only if In fact, multiplying both sides of (5) by and integrating over , we have If is bounded from below on , that is, for some , for , then where . By Gronwall’s inequality, we obtain that the norm of is bounded on which is equivalent to the boundedness of since . On the other hand, since and is an algebra, we know that .

Remark 13. The new blow-up criterion (46) is better than the one obtained in Theorem 2, which is quite common for nonlinear hyperbolic PDE (see [5, 44]). For the Camassa-Holm and related equations, the blow-up criterion is often written as which is different from (46). In fact, if blows up and , then .
Let be the particle curve evolved by the solution; that is, it satisfies Since , , we see that and belong to for any . Therefore, for a fixed , (50) has a unique solution . Moreover, we have the following lemma.

Lemma 14. Let , . The map is an increasing diffeomorphism of , and for .

Proof. Differentiating (50) with respect to yields that Solving the above equation, we obtain Thus is positive and is an increasing diffeomorphism of before the blow-up time.

The following property for the strong solution is important in the proof of global existence.

Lemma 15. Let , , and is the solution to (1) and (2). One has the following identity: where , , and . Moreover, if , then

Proof. Differentiating with respect to , we have Solving the above equation with , we obtain (53). Furthermore, under the condition in the lemma, we have which implies (54).

Remark 16. If has a compact support in an interval , so does . Because of (53), we know is compactly supported in within its lifespan.
We note that the Green’s function of is for , where stands for the integer part of . For , we have A direct computation gives rise to Therefore, we have

Lemma 17. If , , and does not change sign, then the solution to (1) and (2) satisfies

Proof. We discuss the following results for the case ; the lemma follows by using a simple density argument. By (53) and the positivity of , we know keeps the sign of , and hence keeps the sign of . Therefore, employing (59), we obtain that, for , , That is, if does not change sign, then . By , we obtain the desired estimate.

Proof of Theorem 3. Combining Lemma 17 and Theorem 2, we have Theorem 3.

4. Global Weak Solutions

In this section, we prove that (1) and (2) have a unique global weak solution in lower-order Sobolev space , . First, we establish some estimates for the strong solutions to (1) with .

Lemma 18. Let , , and is the solution to (1) and (2) with initial value . Then there is a constant such that, for , one has

Proof. By using the operator , we can rewrite (26) as For , taking the norm of both sides of (63), it follows that The estimate for is straightforward. By using Lemma 7, we have Using , we have Using Lemma 7, it follows that Inserting (65), (66), and (67) into (64) yields the inequality (62).

To show the existence of weak solution to (1) and (2) in lower-order Sobolev space with , we will consider the following problem first: where is given in (17) and is the mollifier introduced in (11)–(13). It follows from Proposition 1 that, for each , there exists a such that the above problem has a unique solution .

Lemma 19. Let , . If does not change sign, then and .

Proof. We first note that, by the construction of , if (or ), then (or ). Thus, if does not change sign, so does . Using the notation in (61) implies Hence Theorem 3 yields that is a global solution.

Lemma 20. If , , and does not change sign, then for any , , with , and with .

Proof. By using (40), Lemma 19, and the Gronwall’s inequality, we obtain that Therefore , where only depends on . Similarly, by using (62) and Lemma 19 we have Thus for , we know is uniformly bounded in , and is uniformly bounded in . Besides, is uniformly bounded in with .

4.1. Existence of Global Weak Solution

Now we prove the existence of a global weak solution to problem (16).

Proposition 21. Suppose with such that does not change sign. Then (1) and (2) have a global weak solution in the sense of distribution. Moreover, .

Proof. For an arbitrary fixed , from Lemma 19, and are bounded in . From Lemma 20, when and , we have Let , . By the Aubin compactness theorem [45], there is a subsequence such that converges to some weakly in and , weakly in respectively. Moreover, Thus, the sequences , , converge to , , strongly in for any , respectively. Therefore, is a solution to (16) in the sense of distribution. By Lemmas 19 and 20, and .

4.2. Uniqueness of the Global Weak Solution

Proposition 22. Suppose with . If does not change sign, then the weak solution to (1) and (2) is unique.

Proof. Let be two solutions to (16) with the same initial data ; then satisfies where . Calculating the energy of yields the equation Since does not change sign, Lemma 19 implies that and . Employing the Cauchy-Schwarz inequality and integrating by parts yield For the estimate of the norm of , we have where . Hence Combining these inequalities, we obtain that Using , we have , which implies .

Proof of Theorem 4. With the aid of Propositions 21 and 22, we complete the proof of Theorem 4.

5. Ill-Posedness

In this section we establish the ill-posedness of (1) and (2) in the sense that the solution map is not uniformly continuous from into with either. Firstly, we show that (1) and (2) possess periodic peaked solutions.

5.1. Existence of Periodic Peaked Solutions

Proposition 23. Let and . Then is a periodic peaked solution to (1) if and only if . Moreover, with .

Proof. Let , , , and . Then (1) can be expressed as . Assuming that is the periodic solution to (1), we have where is the periodic Dirac delta function at (mod ). Thus Direct computation shows that Therefore, we have Using (80), we can compute that Similarly, Putting these results together, we see that which implies .
Obviously, if , we have Therefore, when , we have Hence with .

5.2. Proof of Theorem 5

By Proposition 23, we know that (1) has two sequences of periodic peakon (weak) solutions: where are constants velocity which will be specified later. By (88), we know

Note ; when , we have where .

When , it follows that Note that . For any fixed , we choose large enough such that and take , , . Let , ; then we have and where . Therefore, from (95), we have Since we have Hence we can infer from (93) that Combining (97) and (100), we complete the proof of Theorem 5.

Remark 24. From Theorems 2 and 5, we see is the critical index of regularity for well-posedness in Sobolev space for (1) and (2).

Remark 25. From the dynamical system point of view, when , we can deduce that none of the periodic peakons is Lyapunov stable. In fact, for all , for any periodic peakon with , following (93) and (95), we see that, for , If we let and such that then we have Hence is indeed unstable in the sense of Lyapunov.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11461014).

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Copyright © 2014 Yun Wu and Ping Zhao. 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.

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