Abstract and Applied Analysis

Volume 2013 (2013), Article ID 437473, 12 pages

http://dx.doi.org/10.1155/2013/437473

Research Article

Periodic Conservative Solutions for a Modified Two-Component Camassa-Holm System with Peakons

Yujuan Wang and Yongduan Song

School of Automation, Chongqing University, Chongqing 400044, China

Received 11 August 2013; Accepted 24 October 2013

Academic Editor: Sining Zheng

Copyright © 2013 Yujuan Wang and Yongduan Song. 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.

Abstract.

This paper investigates the periodically weak conservative solutions for a modified periodic two-component Camassa-Holm system arisen from shallow water waves moving. The underlying approach is based on a system reformulation by introducing a skillfully defined characteristic and a set of newly defined variables. Such treatment makes it possible to overcome the difficulties inherent in the multicomponent system, leading to the establishment of the periodic conservative solutions for the original system.

1. Introduction

We consider in this paper a modified periodic two-component Camassa-Holm system with peakons [1], namely,

with the initial and periodic conditions

where , are periodic on the -variable. System (1) is a modified version of the coupled two-component Camassa-Holm system in [2],

which has been established by Fu and Qu to allow for peakon solitons in the form of a superposition of multipeakons. By parameterizing for system (3), it then takes the form of (1), which can be rewritten as a Hamiltonian system,

with the Hamiltonian , where , , and . Furthermore, it has the following conserved quantities:

A remarkable fact for the two types of coupled two-component Camassa-Holm system is that they have an infinite number of peakons, which take the following form in the periodic case that

where satisfy the explicit system of ordinary differential equations

In particular, when (or ), system (1) has the same peakon solitons as the Camassa-Holm (CH) equation.

Moreover, when , system (1) is reduced to the scalar Camassa-Holm equation,

The CH equation which models the unidirectional propagation of shallow water waves over a flat bottom has attracted considerable attention because it has a bi-Hamiltonian structure [3] and is completely integrable [4, 5]. Another interesting property of the CH equation is that the solutions experience wave breaking [4, 6, 7]. The presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [7, 8]. After wave breaking the solutions of the CH equation can be continued uniquely as either global conservative [9–12] or global dissipative solutions [13].

The important issue to address is the Cauchy problem for system (1) in the periodic case on . It has been shown that system (1) is locally well posed with periodic initial value and there exist global strong solutions which blow up in finite time for certain class of initial data [1]. Moreover, an existence result for a class of lower regularity of solutions [1] and the global attractor with the periodic boundary condition [14] were also given.

It should be noted that extending the solutions beyond wave breaking imposes significant challenge, as reflected in the case of multipeakons given by (6). Peakons interact in a way similar to that of solitons of the CH equation, and wave breaking may appear when at least two of the coincide. However, the solutions (6) are not smooth even with continuous functions . One possible way to interpret (6) as a weak solution of system (1) is to rewrite (6) as

In our recent work [15, 16], we investigate the continuity of the solutions for system (1) beyond collision time on the full line, leading to uniquely to either global conservative solutions where the energy is conserved for almost all times or global dissipative solutions where energy loss occurs only through wave breaking.

Motivated by our recent work [15–17], we study in this paper the continuation of the solutions for system (1) beyond wave breaking in the periodic case and present a novel approach from a Lagrangian point of view. The approach we applied here is mainly based on the reformulation by introducing a newly defined characteristic and a set of variables, where the associated energy is introduced as an additional variable to obtain a well-posed initial-value problem. The argument is inspired mainly by [11] in the study of the periodic conservative solution of the CH equation. However, one problematic issue is that we here deal with a coupled system where the mutual effect between two components makes the analysis more complicated than a single one. The newly introduced characteristic and the skillfully defined variables are crucial to overcome the above-mentioned problems. By overcoming the problems, we obtain the global periodic conservative solutions of (1).

The outline of this paper is as follows. The original system is transformed into a Lagrangian equivalent semilinear system by introducing a set of newly defined variables in Section 2 and the global solutions of the equivalent system are obtained in Section 3. Section 4 is devoted to establishing the periodic conservative solutions for the original system, which construct a global continuous semigroup. The paper is just closed in Section 5.

1.1. Notation

Throughout this paper, we identify all spaces of periodic functions with function spaces over the unit circle in ; that is, . We denote by the spatial convolution.

2. The Original System and the Lagrangian Equivalent System

2.1. Part A: The Original System

Let us introduce an operator , associated with its Green function , where stands for the integer part of , such that

Thus we can rewrite (1) as a form of a quasilinear evolution equation,

If we define , , , and as follows:

then (1) can be rewritten as

For smooth solutions, we have that the total energy

is constant in time. Indeed, differentiating the two equations in (13), respectively, with respect to and using the identity , we have

Multiplying the first equation in (13) by and the second by and multiplying the first equation in (15) by and the second by , we obtain the following four equalities:

Using the four identities in (16a), (16b), (16c), and (16d), we can get that

Thus (13) possesses the -norm conservation law defined as

where denotes the solution of system (13). Since , Young’s inequality ensures .

2.2. Part B: The Lagrangian Equivalent System

We reformulate system (13) as follows. We first introduce the corresponding characteristic , as the solution of

where the space is defined by . We define the cumulative energy distribution as

It is not hard to check that

Then it follows from (19) and (21) that

Using the fact that and the periodicity of and , it follows from (20) and (22) that

for all . Let us introduce the vector space given by

endowed with the norm , which is a Banach space. Then we have that .

Throughout the following, we use the notation

After the change of variables and , we obtain the following expressions for , , , , , , , and , namely,

where the variable is omitted for simplification and in the above we used the fact that and took for granted that is an increasing function for any fixed time (the validity will be proved later). In view of the periodicity of , , and , these expressions can be rewritten as

From the definition of the characteristic, the evolution equations for and in the new variables take the form

With these new variables, we now derive an equivalent system of (13), namely,

with , , , , , , , and given in (26). Differentiating (29) with respect to yields

which is semilinear with respect to the variables , , , and .

3. Global Solutions of the Lagrangian Equivalent System

We consider the Cauchy problem for (29) in the following form:

in the Banach space given by

where

with the norm (here we denote by for simplification and the same for other norms in the following), , , and ( denotes identity mapping; i.e., ). Note that there is a bijection between and . The advantage of the modification in (31) is that is a Banach space, which is important for the contraction argument, while is not Banach space. In the following, we use the Banach space and the set of variables .

Lemma 1. For any given , one has that , , , , , , , and defined by (26) are locally Lipschitz continuous from to . Moreover, one has

Proof. We firstly prove that given in (26) is locally Lipschitz continuous from to , and the others follow in the same way. Let and be two elements of where . We have

and . Since is locally Lipschitz on , we have

where is a generic constant depending only on , for all . It follows that

where we have used the fact that in . We handle the other terms of in the same way and then prove that is Lipschitz from to . Similarly, one proves that is Lipschitz from to . Direct computation by the chain rule gives us the expressions (34) for the derivatives of , , , , , , , and . Using the derivative of , we have

Thus we have proved that given in (26) is Lipschitz from to . Similarly, , , , , , , and defined in (26) are locally Lipschitz continuous from to .

Theorem 2. Let any initial data be given. The system (31) admits a unique local solution defined on some time interval , where depends only on .

Proof. To establish the local existence of solutions, one proceeds as in Lemma 1 and obtain that given by

with , is Lipchitz continuous on any bounded set of . The solutions of system (31) then can be rewritten as

Since is a Banach space, the theorem then follows from the standard contraction argument.

It remains to show that the local solution can be extended globally in time. Note that global solutions of (31) may not exist for all initial data in . However, they exist when the initial data belongs to the set which is defined as follows.

Definition 3. The set is composed of all such that

with , , where , which is a Banach space with the norm .

Theorem 2 gives us the existence of local solutions of (31) for initial data in . In the following, we will only consider initial data that belongs to . To obtain that the solution of (31) belongs to , we have to specify the initial condition for (30). Define

It is clear that . We take for , while for .

The global existence of the solution for initial data in relies essentially on the fact that the set is preserved by the flow as the next lemma shows.

Lemma 4. For initial data and some , one considers the local solution of system (31). Then

(i)  for all  ,(ii)  for almost every   and almost every  .

Proof. (i) Let be defined as before. For any given initial data , we have that the local solution of (31) belongs to , which satisfies (41a) for all . We now show that (41c) holds for any and therefore almost everywhere. Consider a fixed and drop it in the notation. On the one hand, it follows from (30) that

and on the other hand,

Thus . Notice that , and then for all and (41c) has been proved. It remains to prove that the inequalities in (41b) hold. Define . Assume that . Since is continuous with respect to , we have . It follows from (41c) that . Furthermore, (30) implies and . If , then which implies for all by the uniqueness of the solution of system (30). This contradicts the fact that for all . If , then . Since , there exists a neighborhood of such that for all . This contradicts the definition of . Hence, . We now have , which conversely implies for all , which contradicts the fact that . Thus we have proved for all . We now prove that for all . This follows from (41c) when . If , then from (41c). As we have seen, would imply that for some in a punctured neighborhood of , which is impossible. Hence, for all . Now we have that for all . If for some , it then follows that which implies for all , which contradicts the fact that for all . Thus .

(ii) Define the set . It follows from Fubini’s theorem that

where and .  From the above proof, we know that, for all , consists of isolated points that are countable. This means that . It follows from (45) and since that

This implies for almost all and therefore is strictly increasing and invertible w. r. t. .

Theorem 5. For any initial data , system (31) admits a unique global solution . Furthermore, for all , one has , which constructs a continuous semigroup.

Proof. To ensure that the local solution of system (31) can be extended to a global solution, it suffices to show that

We now consider a fixed and drop it for simplification. We have

where we have used the fact that and is an increasing function as . It is clear that for all . Since , we have and therefore . For , we have as is increasing. Since , we have

which implies that

for some constant and then . Similarly, one can prove that the bounds hold for , , , , , , and . Since and , we get

and therefore as . Let

After taking the -norms on both sides of (29) and (30), we have

Gronwall’s lemma implies that , and therefore (47) holds as and . It then follows that the map given by generates a continuous semigroup from standard ODE theory.

4. Global Conservative Solutions of the Original System

In this section, we show that the global solution of the equivalent system (31) yields a global conservative solution of the original system (13), which constructs a continuous semigroup.

It suffices to establish the correspondence between the Lagrangian equivalent system and the original system.

We first introduce the subsets and of defined by

where , and the set of relabeling functions is given by

For any , we denote the subsets of by as

with a useful characterization: if (), then almost everywhere. Conversely, if and there exists such that almost everywhere, then for some depending only on . With this useful characterization of , it is not hard to prove that the space is preserved by the governing equation (29).

We denote by the product group , where the group operation is given by for any . We define the map as

The map defines a group action of on . We then consider the quotient space of with respect to the group action. We denote the projection by . Let us introduce the subset of defined as

It turns out that there exists a bijection between and . To prove this, we introduce the composition map such as , where is given by

with and , and is given by

with . It is not hard to check that ,   that is, , and . Thus belongs to the same equivalence class as , and there corresponds to a map given by , where denotes the equivalence class of . It is not hard to check that . Thus, and therefore is a bijection from to . Note that any topology defined on is naturally transported into by the bijection . That is, if we equip with the metric induced by the -norm, that is for all ,   which is complete, then the topology on is defined by a complete metric given by for any .

For any initial data , we denote by the continuous semigroup with the solution of (29). Equation (13) is invariant with respect to relabeling. That is, for any and , we have that . Therefore, the map given by is well defined, which generates a continuous semigroup.

To obtain a semigroup of solution for (13), we have to consider the space , which characterizes the solutions in the original system:

where and is a positive periodic Radon measure with being its absolute continuous part.

In the following, we establish a bijection between and to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in ) into the original system (functions in ).

Let us first introduce the mapping , which transforms the Lagrangian equivalent system into the original system. In the other direction, we obtain the energy density in the original system, by pushing forward by the energy density in Lagrangian equivalent system, where the pushforward of a measure by a measurable function is defined as

for all Borel set . Let be defined as

with and . We have that , which does not depend on the representative of that we choose. We denote by the mapping to any and given by ((63a) and (63b)), which transforms the Lagrangian equivalent system into the original system.

We are led to the mapping , which corresponds to the transformation from the original system into the Lagrangian equivalent system defined as follows.

Definition 6. For any , let

where . We define as the equivalence class of . Here, the function associated with any positive periodic Radon measure is given by

which is lower-semicontinuous, increasing and moreover, for all in .

Remark 7. From the definition of , ,, in ((64), (65), (66), and (67)), we can check that and satisfies (41a), (41b), and (41c), respectively. Moreover, by definition (67), we have . Thus .

We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.

Theorem 8. The mappings and are invertible; that is,

Proof. Let be given. We denote by a representative of and given by ((63a) and (63b)) for this particular . It follows from the definition of that . Let be the representative of in given by (64), (65), (66), and (67). It suffices to show that and therefore . Set

Using the fact that is increasing and continuous, it is not hard to know that

and for any in . It follows from (63b) that

Since and , we have

From (72) and the definition of , we get

It then follows from the definition of that

For any given , using the fact that is increasing and (71), we have . Assume , and there then exists such that and (75) implies that . Therefore, , which contradicts the fact that . Hence, we have

Since , , and , we have

It is not hard to know that

It then follows that .

Given in , we denote by the representative of in given by (64), (65), (66), and (67). Then, let . Let be defined as before by (70). The same computation that leads to (74) now gives

Given , we consider an increasing sequence converging to which is guaranteed by (65), such that . Let tend to infinity and since is lower semicontinuous, we have . Take and then we get

By the definition of , there exists an increasing sequence converging to such that . It follows from the definition of in (65) that . Passing to the limit, we obtain which, together with (80), yields

We obtain that by comparing (81) and (79). It is clear from the definitions that . Hence, and .

Now we have established a bijection between the two equivalent systems, and thus the topology defined in can be transported into . Let us define the distance on as

which makes the bijection between and into an isometry. Since equipped with is a complete metric space, it is not hard to check that equipped with the metric is also a complete metric space. For each , we define the mapping as

Our main results are given as follows.

Theorem 9. Let be given. If one denotes by the corresponding trajectory, is a weak solution of the modified coupled two-component Camassa-Holm system (13), which constructs a continuous semigroup. Moreover, is a weak solution of the following transport equation for the energy density

Furthermore, one has

and for almost all , there holds

Thus, the unique solution described here is a conservative weak solution of the system (13).

Proof. To prove is a weak solution of the original system (13), it suffices to show that,   for all with compact support,

where , , , and are given by (13). Let the solution of (29) be a representative of . Since is Lipschitz and invertible w. r. t. for almost all , we can use the change of variables and get

Using the identities and , it then follows from (29) that

On the other hand, using the change of variables and and since is an increasing function, we obtain that

It then follows from (41c) that

By comparing (89) and (91), we can see that

Hence, the first identity in (87) holds. The second identity in (87) follows in the same way. One can easily check that is solution of (84). It follows from the definitiong in (70) that

Since is periodic, (63b) implies that

which is constant in time from (29). Thus, we have proved (85). Since almost everywhere for almost every , then it follows from (41c) that

for any Borel set . Using the fact that is one-to-one and , almost everywhere, (95) implies that

Hence, (86) holds. This completes the proof.

5. Conclusion

We consider in this paper the periodic conservative property of the solutions for the modified coupled two-component Camassa-Holm system with peakons. By introducing a skillfully defined characteristic, together with a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which periodic conservative solution is obtained. It is the transformation together with the associated properties that overcomes the difficulties inherent in the multicomponent system and allows for the continuity of the solution beyond collision time, leading to the desired results.

Acknowledgments

The authors would like to thank the referees for constructive suggestions and comments. The paper is supported by the Major State Basic Research Development Program 973 (no. 2012CB215202) and the National Natural Science Foundation of China (no.  61134001).

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