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Global Conservative and Multipeakon Conservative Solutions for the Modified Camassa-Holm System with Coupling Effects
This paper investigates the continuation of solutions to the modified coupled two-component Camassa-Holm system after wave breaking. The underlying problem is rather challenging due to the mutual coupling effect between two components in the system. By introducing a novel transformation that makes use of a skillfully defined characteristic and a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which the global conservative solution is obtained, which further allows for the establishment of the multipeakon conservative solution of the system. The results obtained herein are deemed useful for understanding the inevitable phenomenon near wave breaking.
We consider here the following modified coupled two-component Camassa-Holm system with peakons : System (1) is a modified version of the new coupled two-component Camassa-Holm system in the following equation; namely, which, as an extension of the Camassa-Holm (CH) equation, has been established by Fu and Qu to allow for peakon solitons in the form of a superposition of multipeakons. By parameterizing for system (2), it then takes the form of (1), which can be rewritten as a Hamiltonian system with the Hamiltonian , where , , and .Particularly, when (or ), the degenerated (1) has the same peakon solitons as the CH equation. We are interested in such system because it exhibits the following conserved quantities, as can be easily verified:
Note that, when , system (1) is reduced to the scalar Camassa-Holm equation as follows: The CH equation, which models the unidirectional propagation of shallow water waves over a flat bottom, has a bi-Hamiltonian structure  and is completely integrable [4–6]. The CH equation has attracted considerable attention because it has peaked solitons [4, 7] and experiences wave breaking [4, 8]. The presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [8, 9]. After wave breaking, the solutions of the CH equation can be continued uniquely as either global conservative [10–13] or global dissipative solutions .
As one of the integrable multicomponent generalizations of the CH equation, system (1) has been shown to be locally well posed with global strong solutions which blow up in finite time [1, 2]. Moreover, the existence issue for a class of local weak solutions for the modified coupled CH2 system was also addressed in . It has been known that the continuation of solutions for the system beyond wave breaking has been a challenging problem. In our recent work , the problem of continuation beyond wave breaking for the modified coupled CH2 system was studied by applying an approach that reformulates the system (1) into a semilinear system of O.D.E. taking values in a Banach space. Such treatment makes it possible to investigate the continuity of the solutions beyond collision time, leading to the uniquely global solutions of this system. Also the global dissipative and multipeakon dissipative solutions of this system have been established in [16, 17], while, as far as the authors’ concern, there is no effort made in the literature on the study of the global conservative as well as multipeakon conservative solutions of such system, another important feature associated with the system. Motivated by our recent work [15–17], in this paper we develop a new approach to establish a global and stable solution for the modified coupled CH2 system, which is conservative and further allows for the construction of the multipeakon conservative solution. The approach utilized in this paper makes use of a novel system transformation, which is different from  and is based on a skillfully defined characteristic and a set of newly introduced variables, where the associated energy is introduced as an additional variable so as to obtain a well-posed initial-value problem, facilitating the study on the behavior of wave breaking. It should be stressed that both global stable solution and multipeakon solution are important aspects related to the solutions near wave breaking, while there is no effort made in the literature on the study of multipeakon property of system (1), which is another motivation of this work. Our inspiration of investing the underlying issue mainly also stems from the early work [10, 11] in the study of the global conservative solution of the CH equation and  where the multipeakon solution is obtained for the CH equation. In this work a coupled system is dealt with where the mutual effect between two components makes the analysis more complicated than a single one as considered in [10, 11, 13]. By utilizing the novel transformation method, the inherent difficulty is circumvented and then the global conservative solutions of (1) are obtained, which then allows for the establishment of the multipeakon conservative solution of system (1).
The remainder of this paper is organized as follows. Section 2 presents the basic equations. In Section 3, by introducing a set of Lagrangian variables, we transform the original system into an equivalent semilinear system and derive the global solutions of the equivalent system. We obtain a global continuous semigroup of weak conservative solutions for the original system in Section 4 and the multipeakon conservative solution in Section 5.
2. The Original System
We first introduce an operator , which can be expressed by its associated Green’s function such as , for all , where denotes the spatial convolution. Thus we can rewrite (1) as a form of a quasilinear evolution equation: Let us define , , , and as Then (1) can be rewritten as For regular solutions, we get that the total energy is constant in time. Thus (8) possesses the -norm conservation law defined as where denotes the solution of system (8). Note that , and so Young’s inequality ensures that .
3. Global Solutions of the Lagrangian Equivalent System
We reformulate system (8) as follows. For a given initial data , we define the corresponding characteristic as the solution of and we define the Lagrangian cumulative energy distribution as It is not hard to check that Then it follows from (11) and (13) that
Throughout the following, we use the notation
In the following, we drop the variable for simplification. Here, we take as an increasing function for any fixed time for granted (later on we will prove this). Then after the change of variables and , we obtain the following expressions for and (); namely, Since , then , , , and can be rewritten as From the definition of the characteristic, it is not hard to check that We introduce another variable with . It will turn out that . With these new variables, we now derive an equivalent system of (8) as follows: where and are given in (18), while , , , and are given in (19). We regard system (21) as a system of ordinary differential equations in the Banach space endowed with the norm for any . Here is a Banach space with the norm given by . Note that .
Differentiating (21) with respect to the variable yields which are semilinear with respect to the variables , , , and .
To obtain the uniqueness of solutions, one proceeds as follows. By proving that all functions on the right-hand side of (21) are locally Lipschitz continuous, the local existence of solutions will follow from the standard theory of ordinary differential equations in Banach spaces. In a second step, we will then prove that this local solution can be extended globally in time. Note that global solutions of (21) 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 1. The set is composed of all such that (i) (ii) (iii) where and .
Lemma 2. Let and let , or let be two locally Lipschitz maps. Then, the product is also a locally Lipschitz map from to or from to .
Theorem 3. Given initial data , there exists a time depending only on such that the system (21) admits a unique solution in .
Proof. To obtain the local existence of solutions, it suffices to show that , given by
with , is a Lipchitz function on any bounded set of which is a Banach space.
Our main task is to prove the Lipschitz continuity of and () given by (18) and (19) from to . We first prove that given in (19) is locally Lipschitz continuous from to and the others follow in the same way. Let us write where denotes the indicator function of a given set . Let We rewrite as where is the operator from to given as Since the operator (given in Section 2) is linear and continuous from to and is continuously embedded in , we have . It is not hard to check that is locally Lipschitz continuous from to and therefore from to . Thus is locally Lipschitz continuous from to . Since the mapping is locally Lipschitz continuous from to , by Lemma 2, we deduce that is locally Lipschitz continuous from to . Similarly, is also locally Lipschitz continuous and therefore is locally Lipschitz continuous. One proceeds in the same way and proves that , , , and defined by (18) and , , and defined by (19) are locally Lipschitz continuous from to . We rewrite the solutions of (21) as Thus the theorem follows from the standard contraction argument of ordinary differential equations.
It remains to prove the existence of global solutions of (21). Theorem 3 gives us the existence of local solutions of (21) for initial data in . In the following, we will only consider initial data that belongs to given by . To obtain that the solution of (21) belongs to , we have to specify the initial condition for (24). Let We have . For , is taken as , and is given as , 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. Given initial data , one considers the local solution of (21) with initial data for some . One then gets that for all . Moreover, for a.e. , , for a.e. , and exists and is independent of time for all .
Proof. We first show that for all . For any given initial data , we get that the local solution of (21) belongs to , which satisfies (25) for all . We now show that (27) holds for any and therefore a,e.. Consider a fixed and drop it in the notation if there is no ambiguity. On the one hand, it follows from (24) that
and, on the other hand,
Hence, . Notice that ; then for all and (27) has been proved. We now prove the inequalities in (26). Set . Assume that . Since is continuous with respect to , we have . It follows from (27) that . Furthermore, (24) implies that and . If , then which implies that for all by the uniqueness of the solution of system (24). 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 that for all , which contradicts the fact that . Thus we have proved for all . We now prove that for all . This follows from (27) when . If , then from (27). As we have seen, would imply that for some in a punctured neighborhood of , which is impossible. Hence, for all . Now we get that for all . If for some , it then follows that which implies that for all , which contradicts the fact that for all . Hence, . This completes the proof that for all .
We now prove that for almost all . 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 . Since , it thus follows from (37) that for almost every . This implies that for almost all and therefore is strictly increasing and invertible with respect to .
For any given , since and , we know that exist. We have the following: Let . Since , , , are bounded in and as for all , (38) implies that for all . Since , it follows that for all .
Theorem 5. For any initial data , there exists a unique global solution for the system (21). Moreover, for all , we have , which constructs a continuous semigroup.
Proof. To ensure that the local solution of system (21) can be extended to a global solution, it suffices to show that Since is an increasing function with respect to for all and , we have . We now consider a fixed and drop it for simplification. Since when and , for a.e. , it follows from (27) that which implies that and therefore Similarly, We can obtain from the governing equation (21) that and then . We can also get from the governing equation (21) that From the identity , we can deduce that which implies that Therefore, . It is not hard to know that . Similarly, one can obtain that the bounds hold for , , , , , , and . Let Using the integrated version of (21) and (24), after taking the -norms on both sides, we obtain It follows from Gronwall’s inequality that . Hence, we infer that the map defined as generates a continuous semigroup from the standard theory of ordinary differential equations.
4. Global Conservative Solutions of the Original System
To obtain the global conservative solution of the original system, we have to establish the correspondence between the Lagrangian equivalent system and the original system.
Let us first introduce the subsets and of given by where is defined as And, for any , the subsets of are given by with a useful characterization. If (), then a.e. Conversely, if is absolutely continuous, and there exists such that a.e., and then for some depending only on and . With this useful characterization of , it is not hard to prove that the space is preserved by the governing equation (21). Notice that the map defines a group action of on ; we consider the quotient space of with respect to the group action. The equivalence relation on is defined as follows: for any , if there exists such that , we claim that and are equivalent.
We denote the projection by . For any , we introduce the mapping 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, and any topology defined on is naturally transported into by this isomorphism. 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 . Let us denote by the continuous semigroup which to any initial data associates the solution of (21). The system (8) is invariant with respect to relabeling. That is, for any , , for an and . Thus the map given by is well-defined, which generates a continuous semigroup.
To obtain a semigroup of solution for (8), we have to consider the space , which characterizes the solutions in the original system: where and is a positive finite Radon measure with as its absolute continuous part.
We now establish a bijection between and to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in ) into the original system (functions in ).
We first introduce the mapping , which transforms the original system into the Lagrangian equivalent system defined as follows.
Definition 6. For any , let with . We define as the equivalence class of .
Remark 7. From the definition of , , , , , in (55)–(57), we can check that , which also satisfies (25). Moreover, we have from (57), which implies that . Furthermore, if is absolutely continuous, then and
for all .
We are led to the mapping , which corresponds to the transformation from 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 the Lagrangian equivalent system, where the push-forward of a measure by a measurable function is defined as for all the Borel set . Give any element , and let be defined as where and . We get that , which does not depend on the representative of that we choose. We denote by the map to any and given by (60)-(61), which conversely transforms the Lagrangian equivalent system into the original system.
We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.
Theorem 8. The maps and are well-defined and . That is,
Proof. Let in be given. We consider as a representative of and given by (60)-(61) for this particular . From the definition of , we have . Let be the representative of in given by Definition 6. We have to prove that and therefore . Let Using the fact that is increasing and continuous, it follows that