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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 164876, 17 pages
On the Study of Local Solutions for a Generalized Camassa-Holm Equation
School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
Received 23 May 2012; Revised 30 June 2012; Accepted 18 July 2012
Academic Editor: Yong Hong Wu
Copyright © 2012 Meng Wu. 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.
The pseudoparabolic regularization technique is employed to study the local well-posedness of strong solutions for a nonlinear dispersive model, which includes the famous Camassa-Holm equation. The local well-posedness is established in the Sobolev space with via a limiting procedure.
In recent years, extensive research has been carried out worldwide to study highly nonlinear equations including the Camassa-Holm (CH) equation and its various generalizations [1####^~^~^~^~^~^####x2013;6]. It is shown in [7####^~^~^~^~^~^####x2013;9] that the inverse spectral or scattering approach is a powerful technique to handle the Camassa-Holm equation and analyze its dynamics. It is pointed out in [10####^~^~^~^~^~^####x2013;12] that the CH equation gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group, and this geometric illustration leads to a proof that the Least Action Principle holds. Li and Olver  established the local well-posedness to the CH model in the Sobolev space with and gave conditions on the initial data that lead to finite time blow-up of certain solutions. Constantin and Escher  proved that the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. Hakkaev and Kirchev  investigated a generalized form of the Camassa-Holm equation with high order nonlinear terms and obtained the orbit stability of the traveling wave solutions under certain assumptions. Lai and Wu  discussed a generalized Camassa-Holm model and acquired its local existence and uniqueness. Recently, Li et al.  investigated the generalized Camassa-Holm equation where is a natural number and . The authors in  assume that the initial value satisfies the sign condition and establish the global existence of solutions for (1.1).
In this paper, we will study the following generalization of (1.1): where is a natural number, , and is a constant.
The objective of this paper is to study the local well-posedness of (1.2). Its local well-posedness of strong solutions in the Sobolev space with is investigated by using the pseudoparabolic regularization method. Comparing with the work by Li et al. , (1.2) considered in this paper possesses a conservation law different to that in  (see Lemma 3.2 in Section 3). Also (1.2) contains a dissipative term , which causes difficulty to establish its local and global existence in the Sobolev space. It should be mentioned that the existence and uniqueness of local strong solutions for the generalized nonlinear Camassa-Holm models like (1.2) have never been investigated in the literatures.
2. Main Result
Firstly, we introduce several notations.
is the space of all measurable functions such that . We define with the standard norm . For any real number , denotes the Sobolev space with the norm defined by where .
For and nonnegative number , denotes the Frechet space of all continuous -valued functions on . We set . For simplicity, throughout this paper, we let denote any positive constant that is independent of parameter .
We consider the Cauchy problem of (1.2)
Now, we give our main results for problem of (2.2).
Theorem 2.1. Suppose that the initial function belongs to the Sobolev space with and is a constant. Then, there is a , which depends on , such that problem (2.2) has a unique solution satisfying
3. Local Well-Posedness
In order to prove Theorem 2.1, we consider the associated regularized problem where the parameter satisfies .
Lemma 3.1. For and and letting be an integer such that , are uniformly continuous bounded functions that converge to 0 at .
Lemma 3.2. If is a solution to problem (3.1), it holds that
Proof. Using Lemma 3.1, we have . The integration by parts results in
Direct calculation and integration by parts give rise to in which we have used (3.3). From (3.4), we obtain the conservation law (3.2).
Lemma 3.3. Let . The function is a solution of problem (3.1) and the initial value . Then, the following inequality holds: For , there is a constant independent of such that For , there is a constant independent of such that
The proof of this lemma is similar to that of Lemma####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;3.5 in . Here we omit it.
Lemma 3.4. Let and be real numbers such that . Then,
Lemma 3.5. Let with . Then, the Cauchy problem (3.1) has a unique solution , where depends on . If , the solution exists for all time.
Proof. Letting , we know that is a bounded linear operator. Applying the operator on both sides of the first equation of system (3.1) and then integrating the resultant equation with respect to over the interval , we get Suppose that both and are in the closed ball of radius about the zero function in and is the operator in the right-hand side of (3.9). For any fixed , we obtain where may depend on . The algebraic property of with derives Using the first inequality of Lemma 3.4 gives rise to where may depend on . From (3.11)-(3.12), we obtain where and is independent of . Choosing sufficiently small such that , we know that is a contraction. Similarly, it follows from (3.10) that Choosing sufficiently small such that , we deduce that maps to itself. It follows from the contraction-mapping principle that the mapping has a unique fixed point in . It completes the proof.
Setting with and , we know that for any . From Lemma 3.5, it derives that the Cauchy problem has a unique solution .
Furthermore, we have the following.
Lemma 3.6. For , it holds that where is a constant independent of .
Lemma 3.8. Suppose with such that . Let be defined as in system (3.17). Then, there exist two positive constants and , which are independent of , such that the solution of problem (3.17) satisfies for any .
Lemma 3.9 (see Li and Olver ). If and are functions in , then
Lemma 3.10 (see Lai and Wu ). For with , , and a natural number , it holds that
Lemma 3.11 (see Lai and Wu ). If and , then for any functions defined on , it holds that
Lemma 3.12. For problem (3.17), , and , there exist two positive constants and , which are independent of , such that the following inequalities hold for any sufficiently small and :
Our next step is to demonstrate that is a Cauchy sequence. Let and be solutions of problem (3.17), corresponding to the parameters and , respectively, with , and let . Then, satisfies the problem
Lemma 3.13. For , , there exists such that the solution of (3.17) is a Cauchy sequence in .
Proof. For with , multiplying both sides of (3.29) by and then integrating with respect to give rise to
It follows from the Schwarz inequality that
Using the first inequality in Lemma 3.9, we have
where . For the last three terms in (3.32), using Lemmas 3.4 and 3.12, , , the algebra property of with , and (3.23), we have
Using (3.26), we derive that the inequality
holds for some constant , where . Using the algebra property of with , and Lemma 3.11, we have for . Then, it follows from (3.28) and (3.33)####^~^~^~^~^~^####x2013;(3.37) that there is a constant depending on such that the estimate
holds for any , where if and if . Integrating (3.38) with respect to , one obtains the estimate
Applying the Gronwall inequality and using (3.20) and (3.22) yield
for any .
Multiplying both sides of (3.29) by and integrating the resultant equation with respect to , one obtains From Lemma 3.12, we have From Lemma 3.10, it holds that Using the Cauchy-Schwartz inequality and the algebra property of with , for , we have in which we have used Lemma 3.4 and the bounded property of and (see Remark 3.7). It follows from (3.41)####^~^~^~^~^~^####x2013;(3.46) and the inequalities (3.28) and (3.40) that there exists a constant depending on such that where . Integrating (3.47) with respect to leads to the estimate It follows from the Gronwall inequality and (3.48) that where is independent of and .
Then, (3.22) and the above inequality show that Next, we consider the convergence of the sequence . Multiplying both sides of (3.29) by and integrating the resultant equation with respect to , we obtain
It follows from inequalities (3.28) and the Schwartz inequality that there is a constant depending on and such that Hence which results in
It follows from (3.40) and (3.50) that as , in the norm. This implies that is a Cauchy sequence in the spaces and , respectively. The proof is completed.
4. Proof of the Main Result
We consider the problem Letting be the limit of the sequence and taking the limit in problem (4.1) as , from Lemma 3.13, we know that is a solution of the problem and hence is a solution of problem (4.2) in the sense of distribution. In particular, if , is also a classical solution. Let and be two solutions of (4.2) corresponding to the same initial value such that , . Then, satisfies the Cauchy problem For any , applying the operator to both sides of (4.3) and integrating the resultant equation with respect to , we obtain the equality By the similar estimates presented in Lemma 3.13, we have Using the Gronwall inequality leads to the conclusion that for . This completes the proof.
Thanks are given to the referees whose suggestions were very helpful in improving the paper. This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).
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