Global conservative and multipeakon conservative solutions for the two-component Camassa-Holm system
© Wang and Song; licensee Springer. 2013
Received: 6 March 2013
Accepted: 26 June 2013
Published: 10 July 2013
The continuation of solutions for the two-component Camassa-Holm system after wave breaking is studied in this paper. The global conservative solution is derived first, from which a semigroup and a multipeakon conservative solution are established. In developing the solution, a system transformation based on a skillfully defined characteristic and a set of newly introduced variables is used. It is the transformation, together with the associated properties, that allows for the establishment of the results for continuity of the solution beyond collision time.
modeling the unidirectional propagation of shallow water waves in irrotational flow over a flat bottom, with representing the fluid velocity at time t in the horizontal direction, has attracted considerable attention [1–10]. The CH equation is a quadratic order water wave equation in an asymptotic expansion for unidirectional shallow water waves described by the incompressible Euler equations, which was found earlier by Fuchssteiner and Fokas  as a bi-Hamiltonian generalization of the KdV equation. It is completely integrable [2, 3] and possesses an infinite number of conservation laws. A remarkable property of the CH equation is the existence of the non-smooth solitary wave solutions called peakons [2, 11]. The peakon , , is smooth except at its crest and the tallest among all waves of fixed energy. Another remarkable fact for the CH equation is that it can model wave breaking [2, 12], which means that the solution remains bounded while its slope becomes unbounded in finite time [12, 13], setting it apart from the classical soliton equations such as KdV. After wave breaking, the solutions of the CH equation can be continued uniquely as either global conservative [4–6] or global dissipative solutions .
The Cauchy problem for the two-component Camassa-Holm system has been studied extensively [18–24]. It was shown that the CH2 system is locally well posed with initial data , . The system also has global strong solutions which blow-up in finite time [19, 21, 22] and a global weak solution . However, the problem about continuation of the solutions beyond wave breaking, although interesting and important, has not been explicitly addressed yet. In our recent work , we studied the continuation beyond wave breaking by applying an approach that reformulated system (1.2) as a semilinear system of O.D.E. taking values in a Banach space. Such treatment makes it possible to investigate the continuity of the solution beyond collision time, leading to a global conservative solution where the energy is conserved for almost all times.
It should be stressed that both global conservation and multipeakon conservation are two important aspects worthy of investigation. To our best knowledge, however, little effort has been made in studying the multipeakon conservation associated with the CH2 system in the literature. As a compliment and extension to the previous work , we develop a novel approach in this work to construct the multipeakon conservative solution for the CH2 system. Different from the work , we reformulate the problem by utilizing a skillfully defined characteristic and a new set of variables, of which the associated energy serves as an additional variable to be introduced such that a well-posed initial-value problem can be obtained, making it convenient to study the dynamic behavior of wave breaking. Because of the introduction of the new variables, we are able to establish the multipeakon conservative solution from the global conservative solution for the CH2 system.
Some related earlier works [4, 5] studied the global existence of solutions to the CH equation. However, the system considered in this work is a heavily coupled one, in which the mutual effect between the two components makes the analysis quite complicated and involved as compared with the system with a single component as studied in [4, 5]. The key and novel effort made in this work to circumvent the difficulty is the utilization of the skillfully defined characteristic and the new set of variables, as well as careful estimates for each iterative approximate component of the solutions, which allows us to establish the global conservative solutions of system (1.2). It is shown that the multipeakon structure is preserved by the semigroup of a global conservative solution and the multipeakon solution is obtained by carefully computing the convolution equations and (), where, in contrast to the existing works, the inherent mutual effect between the two components is well reflected.
The remainder of this paper is organized as follows. Section 2 presents the transformation from the original system to a Lagrangian semilinear system. The global solutions of the equivalent semilinear system are obtained in Section 3, which are transformed into the global conservative solutions of the original system in Section 4. Finally, we establish the multipeakon conservative solutions for the original system in Section 5.
2 The original system and the equivalent Lagrangian system
where . Since , Young’s inequality ensures .
We reformulate system (2.1) into a Lagrangian equivalent semilinear system as follows.
which is semilinear w.r.t. the variables , , , and .
with the norm . Note that .
3 Global solutions of the equivalent system
In this section, we prove that the equivalent system admits a unique global solution. We first obtain the Lipschitz bounds we need on P and .
Lemma 3.1 (See )
Let and , or be two locally Lipschitz maps. Then the product is also a locally Lipschitz map from E to , or from E to W.
Since the operator Λ (defined as in Section 2) is linear and continuous from to , and is continuously embedded in , we have . It is not hard to know that R is locally Lipschitz from E into and therefore from E into . Thus, is locally Lipschitz from E to . Since the mapping is locally Lipschitz from E to W, it then follows from Lemma 3.1 that is locally Lipschitz from E to . Similarly, is also locally Lipschitz and therefore is locally Lipschitz from E to . We can obtain that P defined by (2.8) is locally Lipschitz continuous from E to in the same way. By using the chain rule, the formulas in (3.1) are obtained by direct computation, see [, p.129]. □
Theorem 3.1 Let any be given. System (2.10) admits a unique local solution defined on some time interval , where T depends only on .
Then the theorem follows from the standard contraction argument on Banach spaces. □
Theorem 3.1 gives us the existence of local solutions to (2.10) for initial data in E. It remains to prove that the local solutions can be extended to global solutions. Note that the global solutions of (2.10) may not exist for all initial data in E. However, they exist when the initial data belongs to the set Γ, which is defined as follows.
with , where .
The global existence of the solution for initial data in Γ relies essentially on the fact that the set Γ is preserved by the flow as the next lemma shows.
for all ,
for a.e. and a.e. ,
for all .
- (ii)Define the set . It follows from Fubini’s theorem that(3.6)
- (iii)For any given , since and , we know that exist. We have(3.7)
Let . Since U, P are bounded in and as , it then follows from (3.7) that for all . Since , it follows that for all . □
is a continuous semigroup.
Thus, . The governing equation (2.10) also implies that and .
It follows from Gronwall’s lemma that , which implies that is a continuous semigroup by the standard O.D.E. theory. □
4 Global solutions for the original system
We transform the global solution of the equivalent system (2.10) into the global conservative solution of the original system (2.1) in this section. It suffices to establish the correspondence between the Lagrangian equivalent system and the original system.
With the above useful property of , it is not hard to prove that the space F is preserved by the governing equation (2.10).
Notice that the map given by defines a group action of G on F, we then consider the quotient space of F w.r.t. the group action. The equivalence relation on F is defined as: for any , if there exists such that , we claim that X and are equivalent. We denote the projection by . For any , we introduce the map given by . It is not hard to prove that when , and for any and . Hence, we can define the map as for any representative of . For any , we have . Hence, . Note that any topology defined on is naturally transported into by this isomorphism, that is, if we equip with the metric induced by the E-norm, i.e., for all , which is complete, then for any , the topology on is defined by a complete metric given by .
For any initial data , we denote the continuous semigroup with the solution of system (2.10) by . As we indicated earlier, Eq. (2.1) is invariant w.r.t. relabeling. That is, , for any and . Thus, the map defined by is valid, which generates a continuous semigroup.
where and μ is a positive finite Radon measure with as its absolute continuous part.
We now establish a bijection between and D to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in ) into the original system (functions in D).
where , . We have that , which does not depend on the representative of we choose. We denote by the mapping to any and given by (4.1a) and (4.1b), which transforms the Lagrangian equivalent system into the original system.
We are led to the mapping , which conversely transforms the original system into the Lagrangian equivalent system defined as follows.
where . We define as the equivalence class of .
Remark 4.1 Note that , which satisfies (3.5a)-(3.5c) from the definition of y, U, V, N, H in (4.2a)-(4.2c). Moreover, by the definition (4.2c), we have that . Thus, .
We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.
For any given , using the fact that y is increasing and (4.4), it follows that . If , there then exists x such that and (4.6) implies that . Conversely, since y is increasing, we have , which implies that . This is a contradiction. Hence, we have that . Since , it follows directly from the definitions that , , and . Hence, .
We obtain that by comparing (4.9) and (4.7). It is clear from the definitions that . Hence, and . □
Our next task is to transport the topology defined in into D, which is guaranteed by the fact that we have established a bijection between the two equivalent systems and then obtained a continuous semigroup of solutions for the original system.
Thus, the unique solution described here is a conservative weak solution of system (2.1).
which is constant in time from Lemma 3.3(iii). Thus, we have proved (4.11).
Thus, (4.12) holds and the proof is completed. □
5 Multipeakon solutions of the original system
We derive a new system of ordinary differential equations for the multipeakon solutions which is well posed even when collisions occur in this section, and the variables are used to characterize multipeakons in a way that avoids the problems related to blowing up.
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. Clearly, if the remain distinct, system (5.2) allows for a global smooth solution. In the case where has the same sign for all , the remain distinct, and (5.2) admits a unique global solution. In this case, the peakons are traveling in the same direction. However, when two peakons have opposite signs, collisions may occur, and if so, system (5.2) blows up.
where the variables denote the position of the peaks, and the variables denote the values of u at the peaks. In the following, we will show that this property persists for conservative solutions.
which is a representative of in the Lagrangian equivalent system, that is, . Let . We claim that the functions , , and belong to and even belong to , as the next lemma shows.
Lemma 5.1 For given initial data such that , the associated solution of (2.10) belongs to .
where C is a constant depending only on , which is bounded. Thus does not blow up from Gronwall’s lemma, and therefore the solution is globally defined in . □
Theorem 5.1 Let the initial data be given in (5.3). The solution given by Theorem 4.2 satisfies between the peaks.
for all time t and all . Thus, . □
For solutions with multipeakon initial data, we have the following result: If vanishes at some point in the interval , then vanishes everywhere in . Furthermore, for the given initial multipeakon solution , let be the solution of system (2.10) with initial data given by (5.4a)-(5.4e), then between adjacent peaks, if , the solution is twice differentiable with respect to the space variable, and for , we have that .
We now start the derivation of a system of ordinary differential equations for multipeakons.
Now we can summarize the result as follows.
Theorem 5.2 Let , and for with a multipeakon initial data given by (5.3). Then with initial data there exists a global solution of (5.13), (5.33), (5.34). For each time t, is defined as the solution of the Dirichlet problem with boundary conditions , on each interval . Thus, is a conservative solution of the two-component Camassa-Holm system, which is the multipeakon conservative solution.
The paper is supported by the Major State Basic Research Development Program 973 (No. 2012CB215202), the National Natural Science Foundation of China (No. 61134001) and the Fundamental Research Funds for the Central Universities (No. CDJXS12170003). The authors would like to thank the referees for constructive suggestions and comments.
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