Global Conservative Solutions of a Generalized Two-Component Camassa-Holm System

Abstract

The Cauchy problem for a generalized two-component Camassa-Holm system is investigated. Following the idea of fixed points and using new sets of independent and dependent variables, the existence of the global conservative solutions for the system is established.

1. Introduction

We consider a generalized two-component Camassa-Holm system which is a model for wave motion on shallow water; describes the horizontal velocity of the fluid, , , is horizontal deviation of the free surface elevation from equilibrium (or depth, in the shallow-water interpretation), and , where indicates locally averaged density. The classical Camassa-Holm equation in [1] has the form

Combining its integrability property with free surface elevation dynamics in [2], the authors have extended the Camassa-Holm equation to two-component Camassa-Holm system (CH2):

Moreover, the local well-posedness and blow-up solutions for CH2 have been established in [3, 4]. It was shown that the system has strong solutions which blow up in finite time [5]. Using the localization analysis in the transport equation, Gui and Liu [6] obtained the global solutions of (3) in the Sobolev space with . Hu [7] considered the weakly dissipative CH2 which includes a nonlinearly dissipative term , where is a differential operator or a quasidifferential operator. It has been shown that CH2 system possesses peakon solutions in and cornerlike solutions in , but singular solutions do not exist for its density variable [8]. By setting , Holm et al. [8] derived the modified CH2 (MCH2): where the is averaged or filtered density. Then, this slight modification of the system CH2 leads to the system MCH2. In [9], the global conservative and dissipative solutions of (4) have been testified.

The Cauchy problem for (2) has been studied extensively. It has been shown that this equation has global strong solutions [10–12]. On the other hand, it has global weak solutions in [13–15]. Weak solutions for a weakly dissipative Camassa-Holm equation have been obtained in [16]. Moreover, the global and dissipative solutions have been established in [17–20]. It is worth remarking that Bressan and Constantin [17] derived the existence of global conservative solutions to (2). The essential point in [17] is to introduce some independent and dependent variables.

For (, is constant and is positive integer), the generalized Camassa-Holm equation has been discussed in [21, 22], and the existence of travelling waves solutions and peaked solitary wave solutions has been found. Mustafa [23] and Yin [24] obtained results of local well-posedness. The global conservative solutions and dissipative solutions of (5) have been established in [25]. However, the conservative solutions for the problem (1) have not been discussed yet. The aim of this paper is to establish the existence of global conservations for the problem (1) and prove uniqueness and continuous dependence on the initial data.

This paper is organized as follows. The preliminary is given in Section 2. In Section 3, we come up with an equivalent semilinear system for the problem (1) and the global solution of this system is constructed. In Section 4, we prove that the semilinear system yields global conservation solutions to the problem (1).

2. Preliminaries

2.1. The Basic Equations

Firstly, we give some notations.

We define Banach space with the norm and let , ; then for all . We set ( is a constant) and ; then , .

We rewrite the equivalent form of (1) as follows: with initial data and in which are defined by where , , and . If is in , due to , , and , we deduce that Since , Young’s inequality ensures that .

Definition 1. For a solution of the Cauchy problem (6) on , it means a Hölder continuous function is defined on with the following properties. At each fixed it has . Moreover, the maps and are Lipschitz continuous from to , satisfying the initial condition and where (9) is understood as equalities between functions in .

For smooth solutions, we have the conservation law Indeed, differentiating the first equation of (6) with respect to and using the identity , we have where . Using problem (6) and the above equality, we have where .

In the same way, for the second equation in (6), we have

Combining (12) with (13), we deduce that Thus, (10) holds.

2.2. A New Set of Independent and Dependent Variables

Let be the initial data. For an energy variable , let the nondecreasing map be defined by

Assuming that the solution to (6) remains Lipschitz continuous for , we derive an equivalent system of (6) by using the independent variables .

Let be the characteristic starting at such that

Moreover, we write

The following further variables will be used: , , and , defined as

Obviously, is defined up to multiples of . Notice that (15) implies

For future use, we write the identities

Using identity (21) yields

Furthermore, we have

In the above formulae, we use the change of variables , the validity of which will be checked in Section 4, and write the convolution as an integral over the variable . Using identities (20)–(22), we obtain expressions for and in terms of the new variable , namely, From (6) and (16), the evolution equations for and in the new variables take the form where and are given in (24)–(29).

Next, to derive an evolution equation for the variable , we observe from (16) that holds. We have the equality

Therefore, it has Applying (6), (9), and (16)–(18) yields

3. Global Solutions of the Semilinear System

Let initial data be given. From Section 2, we rewrite the corresponding Cauchy problem (6) for the variables in the form with initial data

We regard (35) as an ordinary differential equation in the Banach space: with the norm

Then the solution of the Cauchy problem means a fixed point of the integral transformation: where

Proving the local Lipschitz continuity of the right-hand side of (35), the local existence of solution follows from the standard theorem for ordinary differential equations in Banach spaces. Then, we show the conservation of energy property expressed by (10). Moreover, we prove that this local solution can be extended globally in time.

Theorem 2. If , then the Cauchy problem (35)-(36) has a unique solution for all in the sense of Definition 1.

Proof.
Step  1 (local solution). In order to establish the local existence of solution, it suffices to show that the operator determined by the right-hand side of (35) mapping to is Lipschitz continuous on every bounded domain in the form of for any constants .
Applying Sobolev’s inequality , , we know that the maps are all Lipschitz continuous from . Now, we only need to prove the Lipschitz continuity of the maps defined in (24)–(29) from to . This will also imply the Lipschitz continuity of these maps from to . To get this goal, we first observe that, as long as , it holds that
Therefore, for any , it holds that
Introducing the exponentially decaying function we show that
Next, we begin to show , . We only give the estimates for , since the estimates for others are similar. It follows from definition (24) that
Using standard properties of convolutions, we obtain where we have used and (8). Differentiating (24), we get thus
Therefore, we have and .
To establish the Lipschitz continuity of (44), it suffices to show that their partial derivatives are uniformly bounded as ranges inside the domain . We observe that these derivatives are bounded operators from the appropriate space into . For sake of illustration, we will work out the detailed estimate for . All other derivatives can be estimated by the same way.
For a given point , the partial derivative is the linear operator defined by
Thus,
By use of and , the above operator norm satisfies
From (51), is the linear operator defined by whose norm is bounded by
Combining (57) and (59) yields the boundedness of as a linear operator from to . The bounds on the other partial derivatives in (54) are obtained by a similar way.
The local existence of a solution to the Cauchy problem (35)-(36) on some small time interval with now follows from the standard theory of ordinary differential equations in Banach space. Thus, we have showed that the right-hand side of (35) is Lipschitz continuity on a neighbourhood of the initial data in the space .
Step  2 (extension to a global solution). To ensure that the local solution of (35) constructed above can be extended to a global solution defined for all , it suffices to show that the quantity remains uniformly bounded on any bounded time interval. The a priori bounds on (60) will actually follow from the conservation of the total energy (10). In the following, we rederive this energy conservation property in terms of the new variables and .
As long as the local solution of (35) is defined, we claim that
Using (35) and (25) yields
From (35), we get
Thus, we have
Moreover, at the initial time , using (20) and (36), we get
Therefore, we obtain that (61) remains valid for all times as long as the solution is defined. The (62) holds true by the same way.
To prove (63), we proceed as follows. From (35), we deduce that
On the other hand, from (24)–(29), we have
Applying (61)-(62) and (68)-(69), we have which implies (63).
We can now rewrite the total energy (10) in terms of the new variables. According to (70), we have that the energy remains constant in time, along any solution of (35)-(36).
From (61) and (71), we have
Applying (8) and (72), we get . Moreover, it holds that
Notice that
Similar calculations yield
Noting (35) and (76), we deduce that, as long as the solution is defined,
Thus, we have
Since , using Gronwall’s inequality, it holds that
From (35), it derives that
Moreover, (35) implies
From (61)–(63) and (72)–(73), we get uniform bounds of , , , on bounded time interval, respectively. The estimates on and will follow from bounds on the -norms of and , . To get this goal, letting be the right-hand side of (79), we have . Indeed, from (8) and (35), we have
It follows from (51) that where
We deduce
The estimate for is similar. Note that , ; similar calculations show that the -norms of , , , are uniformly bounded. This proves the boundedness of and for in bounded interval.
Finally, the equations in (35) imply that
Combining the bounds , with (77), we have that is uniformly bounded for in bounded interval. Thus,
By the previous bounds, it is clear that remains bounded on bounded time interval. This shows that the solution of (35) can be extended globally in time.

For future use, we record here a property of the solution. Namely, consider the set of times

Then, we claim

Its validity will be proved in the next section.

4. Solutions to the Original Equation

We now show that the global solution of the system (35) yields a global conservative solution to system (6) in the original variables .

Let us start with a global solution to (35). Define

For each fixed , the function provides a solution to the Cauchy problem:

We claim that a solution of (6) can be obtained by setting

Theorem 3. Let provide a global solution to (35)-(36). Then the pair of functions defined by (90)–(92) provide the global solution to the problem (6)-(7).
The solution constructed in this way has the following properties. The energy is almost always conserved; namely,
Furthermore, consider the sequence of initial data , such that
Then the corresponding solutions converge to uniformly for in any bounded sets.

Proof. Using the uniform bound in (72) and combining with (90), we have the estimate
Applying the definition of in (15), we obtain
Therefore, the image of the continuous map is the half-plane . Now we claim for all and a.e. . Indeed, from (35) and (61) we have
On the other hand, (90) implies
Since the function is measurable, identity (97) holds true for almost every at . By the above calculation it remains true for all .
Next, we prove the set that defined in (88) has measure zero. Indeed, if , then . Using (61), we get By (64), we have which implies that satisfying is isolated. Thus, satisfying is also isolated. Since , it infers
From (97), we get is nondeceasing. Moreover, if and , then
Hence, throughout the interval of integration. Therefore, from (61)-(62), we have which proves that the map in (92) is well defined for all and .
If holds, using the fact , we have . This implies . For every fixed , we have
Since the measure of is zero, the equality holds true for almost all .
By Sobolev’s inequality, this implies the uniform Hölder continuity with the exponent of and as functions of . By (35) and the bounds , , and , we get that is Hölder continuous with the exponent . Indeed where we choose such that the characteristic passes through the point . Notice that . This implies that is uniform Hölder continuous with the exponent . The same computation shows that is uniform Hölder continuous with the exponent .
We now prove the Lipschitz continuity of with values in . Consider any interval . For a given point , we choose such that the characteristic passes through the point . By (35) and (72), it follows that
Integrating over the whole real line, in view of the boundedness of , (8), (79), and (105), we obtain where constant depends only on . This clearly implies the Lipschitz continuity of the map in terms of the variable. Similar calculation shows the Lipschitz continuity of as a map .
Since is a reflexive space, the left-hand side of (6) is well defined for almost all . Note that we have proved that the right-hand side of (6) also lies in for almost all . To establish the equality between these two sides, we observe that
On the other hand, recalling (88), for every the map is one-to-one. Then, the change of variable formulae (24)–(29) yields
Hence, identity (6) is valid for almost all . This implies that is a global solution of the generalized two-component Camassa-Holm system in the sense of Definition 1.
From (71) and (105), we obtain identity (93) for almost all .
Finally, let be a sequence of initial data converging to in . In this case, from (35) and the boundedness of , , , we can infer . Recalling (15) and (36) at time , we have
Now from (35) and the bounds of , , , , , we obtain
Thus, Gronwall’s inequality implies that , , uniformly for in bounded sets. Returning to the original coordinates, it yields that uniformly on bounded sets since all functions , , , are uniformly Hölder continuous.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors acknowledge Teacher Ls Yong’s help in modifying the English language of this paper.