Research Article | Open Access
Periodic Conservative Solutions for a Modified Two-Component Camassa-Holm System with Peakons
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.
We consider in this paper a modified periodic two-component Camassa-Holm system with peakons , 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 , 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  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 .
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 . Moreover, an existence result for a class of lower regularity of solutions  and the global attractor with the periodic boundary condition  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  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.
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
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