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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 693010, 21 pages
http://dx.doi.org/10.1155/2012/693010
Research Article

The Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 3 August 2012; Accepted 21 September 2012

Academic Editor: Yong H. Wu

Copyright © 2012 Zhaowei Sheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the space × ; provided that its initial value belongs to the space . A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

1. Introduction

In [1], the author investigated the following weakly dissipative Camassa-Holm model where . When , (1.1) becomes the classical Camassa-Holm equation [2]. The authors in [1] obtained the local well-posedness of the solution for the model by using the Kato theorem. A necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are derived. The blow-up rate of the solution is discussed. It is also shown in [1] that the equation has global strong solutions, and these strong solutions decay to zero as time goes to infinite provided the potentials associated to their initial data are of one sign. However, the existence of global weak solutions in the space is not discussed in paper [1]. This will constitute the objective of this work.

More relevant for the present paper, here we state several works on the global weak solution for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions to the Camassa-Holm equation have been proved by Constantin and Escher [3], Constantin and Molinet [4], and Danchin [5, 6]. Xin and Zhang [7] proved that the global existence of the weak solution for the Camassa-Holm equation in the energy space without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under certain conditions on the solution [8]. Coclite et al. [9] investigated the global weak solutions for a generalized hyperelastic-rod wave equation or a generalized Camassa-Holm equation. The existence of a strongly continuous semigroup of global weak solutions for the generalized hyperelastic-rod equation with any initial value in the space was established in [9]. Under the sign condition imposed on the initial value, Yin et al. [10] proved the existence and uniqueness results of global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. For other dynamic properties about various generalized Camassa-Holm models and other partial differential equations, the reader is referred to [1116].

The aim of this work is to study the existence of global weak solutions for (1.1) in the space under the assumption . The limits of viscous approximations for the equation are used to establish the existence of the global weak solution. Here, we should mention that up to now, there have been no global existence results for weak solutions to the generalized Camassa-Holm equation (1.1).

The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we state the viscous problem and give a corresponding well-posedness result. An upper bound, higher integrability estimate, and basic compactness properties for the viscous approximations are also established in Section 3. Strong compactness of the derivative of the viscous approximations is obtained in Section 4, where the main result for the existence of (1.1) is proved.

2. Main Result

Consider the Cauchy problem for (1.1) which is equivalent to where operator . For a fixed , it has

In fact, problem (2.1) satisfies the following conservation law:

Now, we introduce the definition of a weak solution to the Cauchy problem (2.1) or (2.2).

Definition 2.1. A continuous function is said to be a global weak solution to the Cauchy problem (2.2) if(i); (ii); (iii) satisfies (2.2) in the sense of distributions and takes on the initial value pointwise.
The main result of this paper is stated as follows.

Theorem 2.2. Assume . Then, the Cauchy problem (2.1) or (2.2) has a global weak solution in the sense of Definition 2.1. Furthermore, this weak solution satisfies the following properties.(a)There exists a positive constant depending on and such that the following one-sided norm estimate on the first order spatial derivative holds (b)Let , , and . Then, there exists a positive constant depending only on , and such that the space higher integrability estimate holds

3. Viscous Approximations

Defining and setting the mollifier with and , we know that for any (see [10]). In fact, suitably choosing the mollifier, we have

The existence of a weak solution to the Cauchy problem (2.2) will be established by proving compactness of a sequence of smooth functions solving the following viscous problem:

The beginning point of our analysis is the following well-posedness result for problem (3.3).

Lemma 3.1. Provided that , then for any , there exists a unique solution to the Cauchy problem (3.3). Moreover, for any , it holds that or

Proof. For any and , we have . From Theorem 2.1 in [9] or Theorem 2.3 in [11], we conclude that problem (3.3) has a unique solution for an arbitrary .
We know that the first equation in system (3.3) is equivalent to the form from which we derive that which completes the proof.

From Lemma 3.1, we have Differentiating the first equation of problem (3.3) with respect to the variable and writing , we obtain

Lemma 3.2. Let , , and . Then, there exists a positive constant depending only on , , and , but independent of , such that the space higher integrability estimate holds where is the unique solution of problem (3.3).

Proof. The proof is a variant of the proof presented by Xin and Zhang [7] (also see Coclite et al. [9]). Let be a cut-off function such that and Consider the map , , and observe that We have
Differentiating the first equation of problem (3.3) with respect to the variable and writing and for simplicity, we obtain Multiplying (3.15) by , using the chain rule and integrating over , we have From (3.14), we get Using the Hölder inequality, (3.8) and (3.13), yields Integration by parts gives rise to From (3.13), (3.20), and the Hölder inequality, we have Using (3.13) and Lemma 3.1, we have From (3.13), it has Applying (3.8), the Hölder inequality, Lemma 3.1 and , we have From (3.24), we obtain The inequalities (3.15)–(3.23) and (3.25) derive the desired result (3.10).

Lemma 3.3. There exists a positive constant depending only on and such that where is the unique solution of system (3.3).

Proof. For simplicity, setting , we have The inequality (3.26) is proved in Lemma 3.2 (see (3.24)). Now, we prove (3.27). Using and (3.8) result in (3.27).
Applying the Tonelli theorem, (3.26) and (3.27), we get Since we have which, together with Lemma 3.1, we get (3.29) and (3.30). The proof of Lemma 3.3 is completed.

Lemma 3.4. Assume is the unique solution of problem (3.3). There exists a positive constant depending only on and such that the following one-sided norm estimate on the first order spatial derivative holds

Proof. From (3.9) and Lemma 3.3, we know that there exists a positive constant depending only on and such that . Therefore,
Let be the solution of Letting , due to the comparison principle for parabolic equations, we get
Using , we derive that where and . Setting , we obtain Letting , we have . Due to the comparison principle for ordinary differential equations, we get for all . Therefore, by this and (3.39), the estimate (3.36) is proved.

Lemma 3.5. There exists a sequence tending to zero and a function , such that where is the unique solution of (3.3).

Proof. For fixed , using Lemmas 3.1 and 3.3, and we obtain where depends on . Hence, is uniformly bounded in and (3.42) follows.
Observe that, for each ,
Moreover, is uniformly bounded in and . Then, (3.43) is valid.

Lemma 3.6. There exists a sequence tending to zero and a function such that for each

Proof. Using Lemma 3.3, we have the existence of pointwise convergence subsequence which is uniformly bounded in . This implies (3.47).

Throughout this paper we use overbars to denote weak limits (the space in which these weak limits are taken is with ).

Lemma 3.7. There exists a sequence tending to zero and two functions , such that for each and . Moreover,

Proof. (3.48) and (3.49) are direct consequence of Lemmas 3.1 and 3.2. Inequality (3.50) is valid because of the weak convergence in (3.49). Finally, (3.51) is a consequence of the definition of , Lemma 3.5 and (3.48).

In the following, for notational convenience, we replace the sequence , and by , and , separately.

Using (3.48), we conclude that for any convex function with bounded, Lipschitz continuous on and any we get

Multiplying (3.9) by yields

Lemma 3.8. For any convex with bounded, Lipschitz continuous on , it holds that in the sense of distributions on . Here, and denote the weak limits of and in , , respectively.

Proof. In (3.53), by the convexity of , (3.8), Lemmas 3.5, 3.6, and 3.7, sending gives rise to the desire result.

Remark 3.9. From (3.48) and (3.49), we know that almost everywhere in , where for . From Lemma 3.4 and (3.48), we have where is a constant depending only on and .

Lemma 3.10. In the sense of distributions on , it holds that

Proof. Using (3.9), Lemmas 3.5 and 3.6, (3.48), (3.49), and (3.51), the conclusion (3.57) holds by sending in (3.9).

The next lemma contains a generalized formulation of (3.57).

Lemma 3.11. For any with , it has in the sense of distributions on .

Proof. Let be a family of mollifiers defined on . Denote . The is the convolution with respect to the variable. Multiplying (3.57) by , it has Using the boundedness of and letting in the above two equations, we obtain (3.58).

4. Strong Convergence of and Existence for (1.1)

Following the ideas in [7] or [9], in this section, we improve the weak convergence of in (3.48) to strong convergence, and then we have an existence result for problem (3.3). Generally speaking, we will derive a “transport equation” for the evolution of the defect measure . Namely, we will prove that the measure is zero initially then it will continue to be zero at all later times .

Lemma 4.1. Assume . It holds that

Lemma 4.2. If , for each , it has where and , , .
The proof of Lemmas 4.1 and 4.2 is similar to those of Lemmas 6.1 and 6.2 in [9]. Here, we omit their proofs.

Lemma 4.3 (Coclite et al. [9]). Let . Then, for each , it holds that

Lemma 4.4. Assume . Then, for almost all

Proof. For an arbitrary , we let be sufficiently large (see Lemma 3.4). Using (3.54) minus (3.58), and the entropy (see Lemma 4.2) results in By the increasing property of and the convexity of , from (3.50), we have It follows from Lemma 4.3 that In view of Remark 3.9. Let . Applying (3.56) gives rise to In , it has From (4.6)–(4.10), we know that the following inequality holds in Integrating the resultant inequality over yields for almost all . Sending and using Lemma 4.2, we complete the proof.

Lemma 4.5. For any and , it holds that

Proof. Let . Subtracting (3.58) from (3.54) and using entropy , we deduce Since , we get By the convexity of , it has Using Remark 3.9 and Lemma 4.3 yields Inserting the inequalities from (4.15) to (4.18) into (4.14) gives Integrating the above inequality over , we obtain It follows from Lemma 4.3 that Using Remark 3.9 and (4.20), we have Applying the identity , we obtain (4.13).

Lemma 4.6. It holds that

Proof. Applying Lemmas 4.4 and 4.5 gives rise to From Lemma 3.6, we know that there exists a constant , depending only on , such that By Remark 3.9 and Lemma 4.3, it has Thus, by the convexity of the map , we get Using (4.25) derives Since is concave and choosing large enough, we have Then, from (4.24) and (4.29), it has By making of the Gronwall inequality and Lemmas 4.1 and 4.2, for each , we conclude that By the Fatou lemma, Remark 3.9, and (3.50), sending , we obtain which completes the proof.

Proof of the main result. Using (3.2), (3.4), and Lemma 3.5, we know that the Conditions (i) and (ii) in Definition 2.1 are satisfied. We have to verify (iii). Due to Lemma 4.6, we have From Lemma 3.5, (3.47), and (4.33), we know that is a distributional solution to problem (2.2). In addition, inequalities (2.5) and (2.6) are deduced from Lemmas 3.2 and 3.4. The proof of the main result is completed.

Acknowledgments

Thanks are given to the referees whose suggestions are very helpful to revise the paper. This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).

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