### Abstract

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.

##### Keywords:

two-component Camassa-Holm system; Lagrangian system; global conservative solutions; conservative multipeakon solutions### 1 Introduction

Because of its capabilities of describing the dynamic behavior of water wave, the following Camassa-Holm (CH) equation

modeling the unidirectional propagation of shallow water waves in irrotational flow
over a flat bottom, with
*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 [1] 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

Considered herein is the two-component Camassa-Holm (CH2) shallow water system [14-16]

with
*ζ*:

It also has a bi-Hamiltonian structure corresponding to the Hamiltonian

and the Hamiltonian

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

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 [20], we develop a novel approach in this work to construct the multipeakon conservative solution for the CH2 system. Different from the work [20], 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

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

We first present the original 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

where
*P* as

then Eq. (1.2) can be rewritten as

Moreover, for regular solutions, we have that the total energy

is constant in time. Thus, Eq. (2.1) possesses the

where

We reformulate system (2.1) into a Lagrangian equivalent semilinear system as follows.

Let

and define the Lagrangian cumulative energy distribution *H* as

It is not hard to check that

Then it follows from (2.3) and (2.5) that

Throughout the following, we use the notation

After the change of variables
*P*, namely

where we have dropped the variable *t* for simplicity and taken that *y* is an increasing function for any fixed time *t* for granted (the validity will be proved later). Using
*P* in (2.7) as

From the definition of the characteristic, it follows that

Let us introduce another variable

where *P* and
*ξ* yields

which is semilinear w.r.t. the variables

System (2.10) can be regarded as an O.D.E. in the Banach space *E* given by

endowed with the norm

for any
*W* is a Banach space defined as

with the norm

### 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 [5])

*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*.

**Lemma 3.2***For any given*
*P**and*
*defined by* (2.8) *are locally Lipschitz continuous from**E**to*
*Moreover*, *we have*

*Proof* We write

where

where *R* is the operator from *E* to

Since the operator Λ (defined as in Section 2) is linear and continuous from
*R* is locally Lipschitz from *E* into
*E* into
*E* to
*E* to *W*, it then follows from Lemma 3.1 that
*E* to
*E* to
*P* defined by (2.8) is locally Lipschitz continuous from *E* to

**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*

*Proof* To establish the local existence of solutions, one proceeds as in Lemma 3.2, then
obtains that

with
*E*. We rewrite the solutions of system (2.10) as

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

**Definition 3.1** The set Γ is composed of all

with

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.

**Lemma 3.3***Given initial data*
*for some*
*we consider the local solution*
*of system* (2.10) *given by Theorem *3.1. *We have*

(i)
*for all*

(ii)
*for a*.*e*.
*and a*.*e*.

(iii)
*for all*

*Proof* (i) For given initial data

Note that
*U*, *P* and
*V*, *N* in

and on the other hand,

Thus,
*t*, we have
*ϖ* of
*t*, which is impossible. Hence,

(ii) Define the set

where

This implies that
*t* and therefore
*ξ*.

(iii) For any given

Let
*U*, *P* are bounded in

**Theorem 3.2***For any initial data*
*there exists a unique global solution*
*for system* (2.10). *Moreover*, *for all*
*if we equip* Γ *with the topology endowed with the**E*-*norm*, *then the map*
*defined as*

*is a continuous semigroup*.

*Proof* Let

Since
*ξ* for all *t* and
*ξ*, it follows from (3.5c) that

which implies

and therefore

We can obtain from the governing equation (2.10) that

Thus,

From the identity

which implies that

Therefore,
*P*. Let

After taking the

It follows from Gronwall’s lemma that

### 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.

We first introduce a set *G* as the set of relabeling functions defined by

where Id denotes the identity function. For any
*G* as

with a useful property: If
*f* is absolutely continuous,
*α* depending only on *c* and
*F* and

With the above useful property of
*F* is preserved by the governing equation (2.10).

Notice that the map
*G* on *F*, we then consider the quotient space
*F* w.r.t. the group action. The equivalence relation on *F* is defined as: for any
*X* and
*E*-norm, *i.e.*,

For any initial data

To derive the correspondence between the Lagrangian equivalent system and the original
system, we have to consider the space *D*, which characterizes the solutions in the original system:

where
*μ* is a positive finite Radon measure with

We now establish a bijection between
*D* to transport the continuous semigroup obtained in the Lagrangian equivalent system
(functions in
*D*).

We first introduce the mapping *M*, which corresponds to the transformation from the Lagrangian equivalent system into
the original system. In the other direction, we obtain the energy density *μ* in the original system, by pushing forward by *y* the energy density
*ν* by a measurable function *f* is defined by

for all Borel set *B*. Let

where

We are led to the mapping

**Definition 4.1** For any

where

**Remark 4.1** Note that
*y*, *U*, *V*, *N*, *H* in (4.2a)-(4.2c). Moreover, by the definition (4.2c), we have that

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

**Theorem 4.1***The maps**M**and**L**are invertible*, *that is*,

*Proof* Let
*X*. From the definition of

Using the fact that *y* is increasing and continuous, it follows that

and

Since

From the definition of

For any given
*y* is increasing and (4.4), it follows that
*x* such that
*y* is increasing, we have

Let
*g* be defined as before by (4.3). The same computation that leads to (4.5) now gives

Given
*i* tend to infinity. Since

By the definition of *g*, there exists an increasing sequence
*y* in (4.2a) that

We obtain that

Our next task is to transport the topology defined in
*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.

Let us define the distance
*D* as

which makes the bijection *L* between *D* and
*D* equipped with the metric

**Theorem 4.2***Given*
*if we denote*
*the corresponding trajectory*, *then*
*is a weak solution of the two*-*component Camassa*-*Holm equations* (2.1), *which constructs a continuous semigroup*. *Moreover*, *μ**is a weak solution of the following transport equation*:

*Furthermore*, *we have*

*and*

*Thus*, *the unique solution described here is a conservative weak solution of system* (2.1).

*Proof* To prove that

where
*ξ* for almost all *t*, we can use the change of variables

Since

On the other hand, using the change of variables
*y* is an increasing function, we have

It follows from the identity (3.5c) that

By comparing (4.15) and (4.16), we know that

Thus, the first identity in (4.13) follows directly from (4.14) and the second identity
in (4.13) follows in the same way. It is not hard to check that
*μ* in (4.1b), we can get that

which is constant in time from Lemma 3.3(iii). Thus, we have proved (4.11).

Since

for any Borel set *B*. Since *y* is one-to-one and

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

Solutions of the two-component Camassa-Holm system may experience wave breaking in
the sense that the solution develops singularities in finite time, while keeping the

where

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

We consider initial data

Without loss of generality, we assume that the
*u* defined on intervals

where the variables
*u* at the peaks. In the following, we will show that this property persists for conservative
solutions.

Let us define

which is a representative of

**Lemma 5.1***For given initial data*
*such that*
*the associated solution*
*of* (2.10) *belongs to*

*Proof* To prove this lemma, one proceeds as in Theorem 3.1 by using the contraction argument.
The Banach space *E* is replaced by

endowed with the norm

It suffices to show that *P* and

for a constant *C* depending only on
*P*. We compute the derivative of
*A* as follows:

Since
*P* and

System (5.6) is affine w.r.t.

where *C* is a constant depending only on

**Theorem 5.1***Let the initial data be given in* (5.3). *The solution given by Theorem *4.2 *satisfies*
*between the peaks*.

*Proof* Assuming that

Hence,

Let

For a given
*t*, we obtain, by using (2.10), (2.11) and (5.6), that

We differentiate (3.5c) w.r.t. *ξ* and get

After inserting the value of

It follows from (3.5c) and since

We claim that

for some polynomial *J*. Since
*X*,
*t*, from (3.5b), we have
*t* such that

Hence,

for some constant

which corresponds to the conservation of spatial angular momentum. For the multipeakons
at time

for all time *t* and all

For solutions with multipeakon initial data, we have the following result: If