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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 532369, 26 pages
The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation
Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
Received 13 October 2011; Accepted 13 January 2012
Academic Editor: Sergey Piskarev
Copyright © 2012 Nan Li 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.
A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev space with is established via a limiting procedure. Provided that the initial value satisfies the sign condition and , it is shown that there exists a unique global solution for the equation in space .
Camassa and Holm  employed the Hamiltonian method to derive a completely integrable shallow water wave model which was alternatively established as a water wave equation in [2–4]. Equation (1.1) also models wave current interaction , while Dai  derived it as a model in elasticity (see ). In addition, it was pointed out in Lakshmanan  that the Camassa-Holm equation (1.1) could be relevant to the modeling of tsunami waves (see Constantin and Johnson ).
After the birth of the Camassa-Holm equation (1.1), many works have been carried out to probe its dynamic properties. For , (1.1) has travelling wave solutions of the form , called peakons, which describes an essential feature of the travelling waves of largest amplitude (see [10–14]). For , its solitary waves are stable solitons . It is shown in [16–18] that the inverse spectral or scattering approach is a powerful tool to handle the Camassa-Holm equation and analyze its dynamics. It is worthwhile to mention that (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group [19–21], and this geometric illustration leads to a proof that the least action principle holds. Xin and Zhang  proved the global existence of the weak solution in the energy space without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under some assumptions on the solution . Coclite et al.  extended the analysis presented in [22, 23] and obtained many useful dynamic properties to other partial differential equations (see [25–28] for an alternative approach). Li and Olver  established the local well posedness in the Sobolev space with for (1.1) and gave conditions on the initial data that lead to finite time blowup of certain solutions. It is shown in Constantin and Escher  that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [31–39] and the references therein.
Motivated by the work in Hakkaev and Kirchev  to investigate the generalization forms of the Camassa-Holm equation with high-order nonlinear terms, we study the following generalized Camassa-Holm equation: where is a natural number and . Obviously, (1.2) reduces to (1.1) if we set . As the Camassa-Holm equation (1.1) has been discussed by many mathematicians, we let the natural number in this paper.
The objective of this paper is to study (1.2). Its local well posedness of solutions in the Sobolev space with is developed by using the pseudoparabolic regularization method. Provided that and , the existence and uniqueness of the global solutions are established in space . It should be mentioned that the existence and uniqueness of global strong solutions for the nonlinear generalized Camassa-Holm models like (1.2) have never been investigated in the literatures.
2. Main Results
The space of all infinitely differentiable functions with compact support in is denoted by . 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 which is independent of parameter .
We consider the Cauchy problem of (1.2), which has the equivalent form
Now, we give our main results for problem (2.2).
Theorem 2.1. Suppose that the initial function belongs to the Sobolev space with . Then there is a , which depends on , such that there exists a unique solution of the problem (2.2) and
Theorem 2.2. Let and for all . Then problem (2.2) has a unique solution satisfying that
3. Local Well-Posedness
In order to prove Theorem 2.1, we consider the associated regularized problem where the parameter satisfies .
Lemma 3.1. Let and be real numbers such that . Then
Lemma 3.2. Let with . Then the Cauchy problem (3.1) has a unique solution where depends on . If , the solution exists for all time.
Proof. Assuming that , 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 lead to
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 (3.3). For any fixed , we get the following:
where may depend on . The algebraic property of with derives
Using the first inequality of Lemma 3.1, we have
where may depend on . From (3.5)–(3.7), we obtain that
where and is independent of . Choosing sufficiently small such that , we know that is a contraction. Applying the above inequality yields 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 .
For , using the first equation of system (3.1) derives from which we have the conservation law The proof of the global existence result is a routine argument by using (3.11) (see Xin and Zhang ).
Lemma 3.3 (Kato and Ponce ). If , then is an algebra. Moreover where is a constant depending only on .
Lemma 3.4 (Kato and Ponce ). Let . If and , then
Lemma 3.5. Let , and the function is a solution of problem (3.1) and the initial data . Then the following inequality holds:
For , there is a constant independent of such that
For , there is a constant independent of such that
Proof. The inequality and (3.11) derives (3.14).
Using and the Parseval equality gives rise to
For , applying to both sides of the first equation of system (3.1) and integrating with respect to by parts, we have the identity We will estimate the terms on the right-hand side of (3.18) separately. For the second term, by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have Similarly, for the first term in (3.18), we have Using the above estimate to the third term yields that For the fourth term, using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain that in which we have used .
For the last term in (3.18), using results in For , it follows from (3.22) that For , applying Lemma 3.3 derives It follows from (3.19)–(3.25) that there exists a constant depending only on such that Integrating both sides of the above inequality with respect to results in (3.15).
To estimate the norm of , we apply the operator to both sides of the first equation of system (3.1) to obtain the equation Applying to both sides of (3.27) for gives rise to For the right hand of (3.28), we have Since using Lemma 3.3, and , we have Using the Cauchy-Schwartz inequality and Lemmas 3.1 and 3.3 yields that in which we have used (3.25).
Applying (3.29)–(3.32) into (3.28) yields the inequality for a constant . This completes the proof of Lemma 3.5.
Setting with and , we know that for any . From Lemma 3.2, it derives that the Cauchy problem has a unique solution .
Furthermore, we have the following.
Lemma 3.7. For , it holds that where is a constant independent of .
Lemma 3.9. If with such that . Let be defined as in system (3.34). Then there exist two positive constants and , which are independent of , such that the solution of problem (3.34) satisfies for any .
Proof. Using notation and differentiating both sides of the first equation of problem (3.34) or (3.27) with respect to give rise to
Letting be an integer and multiplying the above equation by and then integrating the resulting equation with respect to yield the equality
Applying the Hölder's inequality yields that
Since as for any , integrating both sides of (3.44) with respect to and taking the limit as result in the estimate
Using the algebraic property of with and (3.40) yields that
where we have used (3.16) and (3.40). Using (3.48), we have
where is a constant independent of . Moreover, for any fixed , there exists a constant such that . Using (3.16) and (3.40) yields that
Making use of the Gronwall's inequality to (3.15) with , and (3.40) gives rise to
From (3.36), (3.37), (3.50), and (3.51), one has
For , it follows from (3.46), (3.49), and (3.52) that
It follows from the contraction mapping principle that there is a such that the equation has a unique solution . Using the Theorem presented at page 51 in Li and Olver  or Theorem II in section I.1 presented in  yields that there are constants and , which are independent of , such that for arbitrary , which leads to the conclusion of Lemma 3.9.
Lemma 3.10 (Li and Olver ). If and are functions in , then
Lemma 3.11. For with , , , and a natural number , it holds that
The proof of this Lemma can be found in .
Lemma 3.12. For problem (3.34), and , there exist two positive constants and , which are independent of , such that the following inequalities hold for any sufficiently small and
Proof. If , , we obtain that
From Lemma 3.9, we know that there exist two constants and (both independent of ) such that
Applying the inequality (3.15) with and the bounded property of solution (see (3.40) and (3.60)), we have where in which we have used (3.36) and (3.37).
From (3.61) and (3.62) and using the Gronwall's inequality, we get the following: from which we know that (3.57) holds.
In a similar manner, for and , applying (3.40) and (3.60) to (3.15), we have which results in (3.58) by using Gronwall's inequality.
From (3.16), for , we have which leads to (3.59) by (3.58).
Lemma 3.13. If and , then for any functions defined on , it holds that
The proof of this lemma can be found in .
Our next step is to demonstrate that is a Cauchy sequence. Let and be solutions of problem (3.34), corresponding to the parameters and , respectively, with , and let . Then satisfies the problem
Lemma 3.14. For , , there exists such that the solution of (3.34) is a Cauchy sequence in .
Proof. For with , multiplying both sides of (3.69) by and then integrating with respect to give rise to
It follows from the Schwarz inequality that
Using the first inequality in Lemma 3.10, we have where . For the last three terms in (3.72), using Lemmas 3.1 and 3.13, , , the algebra property of with and (3.40), we have Using (3.67), we derives that the inequality holds for some constant , where . Using the algebra property of with , and Lemma 3.12, we have for . Then it follows from (3.57)–(3.59) and (3.73)–(3.76) that there is a constant depending on such that the estimate holds for any , where if and if . Integrating (3.77) with respect to , one obtains the estimate Applying the Gronswall inequality, (3.37) and (3.39) yields that for any .
Multiplying both sides of (3.69) by and integrating the resultant equation with respect to , one obtains that From Lemma 3.13, we have From Lemma 3.11, it holds that Using the Cauchy-Schwartz inequality and the algebra property of with , for , we have in which we have used Lemma 3.1 and the bounded property of and (see Lemma 3.12). It follows from (3.80)–(3.84) and (3.57)–(3.59) and (3.79) that there exists a constant depending on such that where . Integrating (3.85) with respect to leads to the estimate