A Note on the Generalized Camassa-Holm Equation

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 [superscript] H s [/superscript] with s < 3 / 2 .

Academic Editor:Naseer Shahzad

Department of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, China

Received 15 October 2014; Accepted 24 November 2014; 25 December 2014

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.

1. Introduction

In this paper we study the global strong and weak solutions to the generalized Camassa-Holm equation with periodic boundary condition: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

When [figure omitted; refer to PDF] , (1) reduces to the the well-known CH equation: [figure omitted; refer to PDF] 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 Misiolek [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 [figure omitted; refer to PDF] . 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: [figure omitted; refer to PDF] It is derived that (4) possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities and admits exact peaked solutions [figure omitted; refer to PDF] with [figure omitted; refer to PDF] [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 [figure omitted; refer to PDF] and in the Sobolev space [figure omitted; refer to PDF] (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, Ti [figure omitted; refer to PDF] lay [31] proved that for [figure omitted; refer to PDF] the periodic Novikov equation is locally well-posed in [figure omitted; refer to PDF] . Later the range of regularity index of local well-posedness was extended to [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] into [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , Grayshan [33] proved that the properties of the solution map for (4) are not (globally) uniformly continuous in Sobolev spaces [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , then (1) takes the form of a quasi-linear evolution equation of hyperbolic type: [figure omitted; refer to PDF] Applying the operator [figure omitted; refer to PDF] , (1) can be expressed as the following nonlocal form: [figure omitted; refer to PDF] From the above equation, we can view (1) as a nonlocal perturbation of the Burgers-type equation: [figure omitted; refer to PDF]

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

Proposition 1.

If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then there exists a [figure omitted; refer to PDF] such that (1) and (2) have a unique solution [figure omitted; refer to PDF] which depends continuously on the initial data [figure omitted; refer to PDF] . Moreover, [figure omitted; refer to PDF] satisfies the solution estimate [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a constant depending on [figure omitted; refer to PDF] .

In this paper we use the following notations. We use [figure omitted; refer to PDF] 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. [figure omitted; refer to PDF] stands for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . All function spaces are over [figure omitted; refer to PDF] and we drop [figure omitted; refer to PDF] in all function spaces if there is no ambiguity. For linear operators [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we denote [figure omitted; refer to PDF] . The Fourier transform of the function [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The inverse Fourier transform is given by [figure omitted; refer to PDF] .

The operator [figure omitted; refer to PDF] for any real number [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the standard Soblev space on [figure omitted; refer to PDF] whose norm is defined by [figure omitted; refer to PDF]

For each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] stands for the Friedrichs mollifier defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] stands for the convolution. Here [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a Schwartz function satisfying [figure omitted; refer to PDF] for all the [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] .

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

Now we present our results.

Theorem 2.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] be the maximal existence time of the corresponding solution [figure omitted; refer to PDF] to (1) and (2). Then [figure omitted; refer to PDF] blows up at [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF]

Theorem 3.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] does not change sign, then the corresponding solution to (1) and (2) exists globally.

Theorem 4.

Suppose [figure omitted; refer to PDF] with [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] does not change sign. Then (1) and (2) have a unique global weak solution [figure omitted; refer to PDF] in the sense of distribution. Moreover, [figure omitted; refer to PDF] .

When [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] , then (1) and (2) are ill-posed in [figure omitted; refer to PDF] in the sense that the solution map is not uniformly continuous from [figure omitted; refer to PDF] into [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . More precisely, there exist two sequences of weak solutions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] of (1) and (2) such that, for any [figure omitted; refer to PDF] , there hold the following estimates: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] only depend on [figure omitted; refer to PDF] .

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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , with [figure omitted; refer to PDF]

The following estimates are useful in our work.

Lemma 6 (Kato-Ponce commutator estimate [40]).

If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Lemma 7 (see [40] or see the Moser estimate in [41]).

If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is an algebra, and [figure omitted; refer to PDF]

From the construction of the mollifier [figure omitted; refer to PDF] , we know that [figure omitted; refer to PDF] and, for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

In addition, we have [figure omitted; refer to PDF]

Lemma 8 (see [42]).

Let [figure omitted; refer to PDF] be a function such that [figure omitted; refer to PDF] . Then, there is a [figure omitted; refer to PDF] such that, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

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

Lemma 9.

If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then there is a [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] does not change sign. Rewrite (1) as the following form: [figure omitted; refer to PDF] First, we have the following lemma.

Lemma 10.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is a solution to (1) and (2) with [figure omitted; refer to PDF] . Then there exists a constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Proof.

Since the term [figure omitted; refer to PDF] is only in [figure omitted; refer to PDF] , we cannot apply [figure omitted; refer to PDF] to either side of (26) when [figure omitted; refer to PDF] . So we apply the operator [figure omitted; refer to PDF] to (26), multiply both sides of the resulting equation by [figure omitted; refer to PDF] , and integrate over [figure omitted; refer to PDF] to obtain [figure omitted; refer to PDF] where we used [figure omitted; refer to PDF] and [figure omitted; refer to PDF] We now estimate [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , respectively. For [figure omitted; refer to PDF] , we first note that [figure omitted; refer to PDF] is self-adjoint, then commute the operator [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , and use (21) and (22) to get [figure omitted; refer to PDF]

By using the Cauchy-Schwarz inequality, Lemmas 6, 7, and 8, integration by parts, and (23), we have [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF]

In the same way, [figure omitted; refer to PDF] can be estimated as [figure omitted; refer to PDF]

For [figure omitted; refer to PDF] , we use the Cauchy-Schwarz inequality, Lemma 7, and (23) to obtain [figure omitted; refer to PDF] For the estimate for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Similar to (34), [figure omitted; refer to PDF] , while [figure omitted; refer to PDF] Hence we obtain that [figure omitted; refer to PDF] Combining (32), (33), (34), and (37) yields [figure omitted; refer to PDF] Integrating both sides with respect to [figure omitted; refer to PDF] results in [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] tend to [figure omitted; refer to PDF] ; we get (27) and therefore complete the proof of this lemma.

Remark 11.

Take [figure omitted; refer to PDF] in (27); then for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] This estimate will be also used in the proofs of Lemma 20.

Let [figure omitted; refer to PDF] be the solution to problem (1) and (2). We define [figure omitted; refer to PDF] to be its lifespan, [figure omitted; refer to PDF] Then the following alternative property holds: [figure omitted; refer to PDF]

Proof of Theorem 2.

From (1) we can deduce that [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] Taking [figure omitted; refer to PDF] in Lemma 10 and using [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] Hence we know that if [figure omitted; refer to PDF] is finite, then [figure omitted; refer to PDF] is bounded and the case (ii) in (42) would not occur, which implies that [figure omitted; refer to PDF] can be extended beyond [figure omitted; refer to PDF] . On the other hand, if [figure omitted; refer to PDF] does not blow up, then [figure omitted; refer to PDF] is bounded on [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is also bounded on [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] , then the solution [figure omitted; refer to PDF] blows up if and only if [figure omitted; refer to PDF] In fact, multiplying both sides of (5) by [figure omitted; refer to PDF] and integrating over [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] If [figure omitted; refer to PDF] is bounded from below on [figure omitted; refer to PDF] , that is, for some [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . By Gronwall's inequality, we obtain that the [figure omitted; refer to PDF] norm of [figure omitted; refer to PDF] is bounded on [figure omitted; refer to PDF] which is equivalent to the boundedness of [figure omitted; refer to PDF] since [figure omitted; refer to PDF] . On the other hand, since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an algebra, we know that [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] which is different from (46). In fact, if [figure omitted; refer to PDF] blows up and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the particle curve evolved by the solution; that is, it satisfies [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] belong to [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . Therefore, for a fixed [figure omitted; refer to PDF] , (50) has a unique solution [figure omitted; refer to PDF] . Moreover, we have the following lemma.

Lemma 14.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The map [figure omitted; refer to PDF] is an increasing diffeomorphism of [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Proof.

Differentiating (50) with respect to [figure omitted; refer to PDF] yields that [figure omitted; refer to PDF] Solving the above equation, we obtain [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] is positive and [figure omitted; refer to PDF] is an increasing diffeomorphism of [figure omitted; refer to PDF] before the blow-up time.

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

Lemma 15.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the solution to (1) and (2). One has the following identity: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Moreover, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Proof.

Differentiating [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Solving the above equation with [figure omitted; refer to PDF] , we obtain (53). Furthermore, under the condition in the lemma, we have [figure omitted; refer to PDF] which implies (54).

Remark 16.

If [figure omitted; refer to PDF] has a compact support in an interval [figure omitted; refer to PDF] , so does [figure omitted; refer to PDF] . Because of (53), we know [figure omitted; refer to PDF] is compactly supported in [figure omitted; refer to PDF] within its lifespan.

We note that the Green's function of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] stands for the integer part of [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] A direct computation gives rise to [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF]

Lemma 17.

If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] does not change sign, then the solution [figure omitted; refer to PDF] to (1) and (2) satisfies [figure omitted; refer to PDF]

Proof.

We discuss the following results for the case [figure omitted; refer to PDF] ; the lemma follows by using a simple density argument. By (53) and the positivity of [figure omitted; refer to PDF] , we know [figure omitted; refer to PDF] keeps the sign of [figure omitted; refer to PDF] , and hence [figure omitted; refer to PDF] keeps the sign of [figure omitted; refer to PDF] . Therefore, employing (59), we obtain that, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] That is, if [figure omitted; refer to PDF] does not change sign, then [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . First, we establish some estimates for the strong solutions to (1) with [figure omitted; refer to PDF] .

Lemma 18.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the solution to (1) and (2) with initial value [figure omitted; refer to PDF] . Then there is a constant [figure omitted; refer to PDF] such that, for [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF]

Proof.

By using the operator [figure omitted; refer to PDF] , we can rewrite (26) as [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , taking the [figure omitted; refer to PDF] norm of both sides of (63), it follows that [figure omitted; refer to PDF] The estimate for [figure omitted; refer to PDF] is straightforward. By using Lemma 7, we have [figure omitted; refer to PDF] Using [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Using Lemma 7, it follows that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , we will consider the following problem first: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is given in (17) and [figure omitted; refer to PDF] is the mollifier introduced in (11)-(13). It follows from Proposition 1 that, for each [figure omitted; refer to PDF] , there exists a [figure omitted; refer to PDF] such that the above problem has a unique solution [figure omitted; refer to PDF] .

Lemma 19.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] does not change sign, then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Proof.

We first note that, by the construction of [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] (or [figure omitted; refer to PDF] ), then [figure omitted; refer to PDF] (or [figure omitted; refer to PDF] ). Thus, if [figure omitted; refer to PDF] does not change sign, so does [figure omitted; refer to PDF] . Using the notation [figure omitted; refer to PDF] in (61) implies [figure omitted; refer to PDF] Hence Theorem 3 yields that [figure omitted; refer to PDF] is a global solution.

Lemma 20.

If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] does not change sign, then for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Proof.

By using (40), Lemma 19, and the Gronwall's inequality, we obtain that [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] only depends on [figure omitted; refer to PDF] . Similarly, by using (62) and Lemma 19 we have [figure omitted; refer to PDF] Thus for [figure omitted; refer to PDF] , we know [figure omitted; refer to PDF] is uniformly bounded in [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is uniformly bounded in [figure omitted; refer to PDF] . Besides, [figure omitted; refer to PDF] is uniformly bounded in [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

4.1. Existence of Global Weak Solution

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

Proposition 21.

Suppose [figure omitted; refer to PDF] with [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] does not change sign. Then (1) and (2) have a global weak solution [figure omitted; refer to PDF] in the sense of distribution. Moreover, [figure omitted; refer to PDF] .

Proof.

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

4.2. Uniqueness of the Global Weak Solution

Proposition 22.

Suppose [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] does not change sign, then the weak solution to (1) and (2) is unique.

Proof.

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

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 [figure omitted; refer to PDF] into [figure omitted; refer to PDF] with [figure omitted; refer to PDF] either. Firstly, we show that (1) and (2) possess periodic peaked solutions.

5.1. Existence of Periodic Peaked Solutions

Proposition 23.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a periodic peaked solution to (1) if and only if [figure omitted; refer to PDF] . Moreover, [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then (1) can be expressed as [figure omitted; refer to PDF] . Assuming that [figure omitted; refer to PDF] is the periodic solution to (1), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the periodic Dirac delta function at [figure omitted; refer to PDF] (mod [figure omitted; refer to PDF] ). Thus [figure omitted; refer to PDF] Direct computation shows that [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] Using (80), we can compute that [figure omitted; refer to PDF] Similarly, [figure omitted; refer to PDF] Putting these results together, we see that [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] .

Obviously, if [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Therefore, when [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

5.2. Proof of Theorem 5

By Proposition 23, we know that (1) has two sequences of periodic peakon (weak) solutions: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are constants velocity which will be specified later. By (88), we know [figure omitted; refer to PDF]

Note [figure omitted; refer to PDF] ; when [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

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

Remark 24.

From Theorems 2 and 5, we see [figure omitted; refer to PDF] is the critical index of regularity for well-posedness in Sobolev space [figure omitted; refer to PDF] for (1) and (2).

Remark 25.

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

Acknowledgment

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

Conflict of Interests

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

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