Abstract

The global dissipative and multipeakon dissipative behavior of the two-component Camassa-Holm shallow water system after wave breaking was studied in this paper. The underlying approach is based on a skillfully defined characteristic and a set of newly introduced variables which transform the original system into a Lagrangian semilinear system. It is the transformation, together with the associated properties, that allows for the continuity of the solution beyond collision time to be established, leading to a uniquely global dissipative solution, which constructs a semigroup, and the multipeakon dissipative solution.

1. Introduction

In view of the wide applications in fluid dynamics, nonlinear optics, biochemistry, microbiology, physics, and many other fields, the study of the dynamic behavior of shallow water wave represents an important subject of research [1–5]. The Camassa-Holm (CH) equation [1] has been widely used to model the unidirectional propagation of shallow water waves over a flat bottom. The nonstandard properties of the CH equation set it apart from the classical soliton equations such as KdV, the first two remarkable of which are that it has peaked solitons [1, 6] and is able to model wave breaking [1, 2]. The presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [2, 5]. After wave breaking, the solutions of the CH equation become uniquely as either global conservative [7–9] or global dissipative solutions [10–12].

In this paper, we focus on the two-component Camassa–Holm shallow water system [13–15]: with , , ( or in the “short wave” limit, ). Here represents the horizontal velocity of the fluid and is in connection with the free-surface elevation from equilibrium (or scalar density). This system appears originally in [16] and then derived by Constantin and Ivanov [13] in the context of shallow water theory. System (1) is an extension of the Camassa-Holm (CH) equation by combing its integrability property with compressibility or free-surface elevation dynamics in its shallow water interpretation [6, 17]; analogously it is formally integrable [13–15] in the sense that it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter :

The Cauchy problem for the two-component Camassa-Holm (CH2) system has been studied extensively [18–22]. It has been shown that the CH2 system is locally well-posed with initial data , [18]. The system also has global strong solutions which blow up in finite time [19, 21, 22] and a global weak solution [23]. However, the problem about continuation of the solutions beyond wave breaking, although interesting and important, has not been explicitly addressed yet to the best of our knowledge. In our recent work [20], we studied the continuation beyond wave breaking by applying an approach based on a novel transformation that transforms system (1) into an equivalent semilinear system of original differential equation. Such treatment makes it possible to investigate the continuity of the solution beyond collision time, leading to the multipeakon conservative solution and a global conservative solution where the energy is conserved for almost all times.

It should be noted that both conservation and dissipation are important features associated with the system. The global conservation property of the CH2 system has been obtained in [20], and the dissipative behavior of the modified coupled two-component Camassa-Holm system, different from the CH2 system, has also been established in [24, 25]; however, there is no effort made in the literature on the study of the global dissipative and the multipeakon dissipative solutions of such system studied in this paper to the best of the authors’ knowledge. In this work, we explore a new approach to establish a global and stable dissipative solution of the CH2 system, which allows for the construction of the global dissipative multipeakon solution of this system. Our study is also motivated by the early work [10, 11] in the study of the global dissipative solution of the CH equation. The main difference is that here we deal with a coupled system and consider explicitly the mutual effect between two components, which literally makes the analysis more challenging as compared to the single one considered in [10, 11]. The key to circumvent the difficulty is to use the skillfully defined characteristic and a new set of variables, which allows for the establishment of the global dissipative solutions of system (1). Furthermore, it is useful to understand whether or not system (1) has the multipeakon solution as with the CH equation [12], an important aspect related to the solutions near wave breaking. And this work develops such multipeakon solution and confirms that the semigroup of the global dissipative solution preserves the multipeakon structure.

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 dissipative solutions of the original system in Section 4. We establish the multipeakon dissipative solutions for the original system in Section 5. To demonstrate the feature of the solution, two numerical examples are considered in Section 6. Finally the paper is closed in Section 7.

2. The Original System and the Equivalent Lagrangian System

For simplicity, we consider here the associated evolution for positive times (of course, one would get similar results for negative times just by changing the initial condition into ). Let us introduce an operator , which can be expressed by its associated Green’s function such as for all . If we define as where , then (1) can be rewritten as Moreover, for regular solutions, we have the total energy is constant in time. Thus (4) possesses the -norm conservation law defined as where . Since with , Young’s inequality ensures .

We reformulate system (4) into a Lagrangian equivalent semilinear system as follows. Let denote the solution of system (4). For given initial data , we define the corresponding characteristic as the solution of where the variable is identified with a “particle.” The change in Lagrangian energy distribution along the particle path is given by It is not hard to check that Then it follows from (7) and (9) that Throughout the following, we use the notation In the following, we drop the variable for simplicity. After the change of variables and , we obtain the following expressions for and ; namely, where we have taken that is an increasing function for any fixed time for granted (the validity will be proved later) and have used the fact that . From the definition of the characteristic, it follows that Let us introduce another variable , such as (it will turn out that ). With these new variables, we now derive an equivalent system of (4): where and are given by (12). Differentiating the first two equations in (14) with respect to yields which is semilinear with respect to the variables , , and . Dissipative solutions differ from conservative solutions when particles collide, that is, where for in an interval of positive length. If we solve (14) and (15), we obtain the conservative solution. However, to obtain the dissipative solution, we impose that when particles collide, they lose their energy; that is, if for some , then we set . One can show that implies so that the system (15) implies that for . Thus, we can define to be the first time when vanishes; namely, However, from this definition in (16), it is not clear whether is measurable or not. We now replace this definition by the following one. Let where is a dense countable subset of and the sets are measurable for all , with for . Let us set with being the indicator function of the set . It is not hard to know that is by construction measurable and increasing with respect to , which is also bounded by . Thus, we define and is a measurable function. One can check that this definition coincided with (16). Hence, the expressions for and in (12) become and the modified system to be solved here reads where is the indicator function of the set , which can be regarded as an O.D.E. in the Banach space .

3. Global Solutions of the Equivalent System

To obtain the global existence of solutions, we start from a contraction argument that offers the local existence of solutions, which is proved in the Banach space . Note that global solutions of (20) 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

with , and is given by for , where is the following subset of : We write system (21) in a compact form that where . For , we set . Note that, in this definition, we do not reset the energy density to zero after collision but keep the value it reached just before the collision. Let us define (the subset of ) as follows:

Theorem 2. Given initial data , there exists a time such that the system (20) and (21) admit a unique solution in .

Proof. To obtain the local existence of solutions, it suffices to show that given by (25) is a contraction, and, therefore, there exists a unique fixed point which is solution to (20) and (21).
Our main task is to prove the estimates for and defined by (19). Let us write as with where Young’s inequalities imply that where we have denoted by a generic constant. The estimate for can be obtained in exactly the same way. We thus obtain that given in (25) is a contraction, and the local existence of solutions follows from the standard contraction argument of ordinary differential equations.

When proving existence and stability, a priori control is essential. Let us introduce the set and therefore there exists an such that for , and in addition and the set and therefore, by taking large enough depending only on and , is a decreasing function and are increasing functions of time for , and in addition We have because , and which implies .

Lemma 3. Given for some constant , one denotes the solution of (20), (21) by with initial data . Let . Then the following hold.(i)Consider for all and almost all . Thus implies .(ii)Consider for all and a constant .

Proof. (i) We know that (35)–(37) hold for almost every at because . We consider a fixed that we suppress in the notation. On the one hand, it follows from (21) that and on the other hand Hence, , which together with yields for all , and (36) has been proved. It follows directly from the definition of and that for . For , we have as . The continuity (with respect to ) of implies and, therefore, for all because the variable does not change for .
(ii) We consider a fixed and drop it in the notation. Let us denote the Euclidean norm of by . Since , we have We have for a constant depending on the norm chosen for the matrix . We infer from (25) that for a constant depending only on and . We denote generically by such constant. Thus, . Gronwall’s lemma implies that , and therefore Since and , it follows from (36) that , and therefore , which yields