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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 285040, 15 pages
The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation
Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China
Received 17 January 2011; Revised 3 August 2011; Accepted 8 August 2011
Academic Editor: Yuri V. Rogovchenko
Copyright © 2011 Shaoyong Lai. 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.
Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in space with . In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space with is developed.
Camassa and Holm  used the Hamiltonian method to derive a completely integrable wave equation by retaining two terms that are usually neglected in the small amplitude, shallow water limit. Its alternative derivation as a model for water waves can be found in Constantin and Lannes  and Johnson . Equation (1.1) also models wave current interaction , while Dai  derived it as a model in elasticity (see Constantin and Strauss ). Moreover, 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 ).
In fact, a huge amount of work has been carried out to investigate the dynamic properties of (1.1). For , (1.1) has traveling wave solutions of the form , called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations (see [9–11]). For , its solitary waves are stable solitons [6, 11]. It was shown in [12–14] that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation. Equation (1.1) is a completely integrable infinite-dimensional Hamiltonian system (in the sense that for a large class of initial data, the flow is equivalent to a linear flow at constant speed ). It should be emphasized that (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group (see [16, 17]), and this geometric illustration leads to a proof that the Least Action Principle holds. It is worthwhile to mention that Xin and Zhang  proved that 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 conditions on the solution . Coclite et al.  extended the analysis presented in [18, 19] and obtained many useful dynamic properties to other equations (also see [21–24]). 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 was 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. After wave breaking, the solution can be continued uniquely either as a global conservative weak solution  or a global dissipative solution . For peakons, these possibilities are explicitly illustrated in the paper . For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water models, the reader is referred to [10, 28–32] and the references therein.
In this paper, motivated by the work in [25, 33], we study the following generalized Camassa-Holm equation where and are natural numbers, and , , and are arbitrary constants. Obviously, (1.2) reduces to (1.1) if we set , , and . Actually, Wu and Yin  consider a nonlinearly dissipative Camassa-Holm equation which includes a nonlinearly dissipative term , where is a differential operator or a quasidifferential operator. Therefore, we can regard the term as a nonlinearly dissipative term for the dissipative Camassa-Holm equation (1.2).
Due to the term in (1.2), the conservation laws in previous works [10, 25] for (1.1) lose their powers to obtain some bounded estimates of the solution for (1.2). A new conservation law different from those presented in [10, 25] will be established to prove the local existence and uniqueness of the solution to (2.3) subject to initial value with . We should address that all the generalized versions of the Camassa-Holm equation in previous works (see [17, 25, 34]) do not involve the nonlinear term . Lai and Wu  only studied a generalized Camassa-Holm equation in the case where and is an odd number. Namely, (1.2) with and arbitrary positive integer was not investigated in .
The main tasks of this paper are two-fold. Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for (1.2) with any and arbitrary positive integer in space with . Secondly, it is shown that the existence of weak solutions in lower order Sobolev space with . The ideas of proving the second result come from those presented in Li and Olver .
2. Main Results
Firstly, we give some notation.
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 .
In order to study the existence of solutions for (1.2), we consider its Cauchy problem in the form which is equivalent to
Now, we state our main results.
Theorem 2.2. Suppose that with and . Then there exists a such that (1.2) subject to initial value has a weak solution in the sense of distribution and .
3. Local Well-Posedness
We consider the abstract quasilinear evolution equation Let and be Hilbert spaces such that is continuously and densely embedded in , and let be a topological isomorphism. Let be the space of all bounded linear operators from to . If , we denote this space by . We state the following conditions in which , and are constants depending on .(i) for with and (i.e., is quasi--accretive), uniformly on bounded sets in .(ii), where is bounded, uniformly on bounded sets in . Moreover, (iii) extends to a map from into is bounded on bounded sets in , and satisfies
Kato Theorem (see )
Assume that (i), (ii), and (iii) hold. If , there is a maximal depending only on , and a unique solution to problem (3.1) such that Moreover, the map is a continuous map from to the space
Lemma 3.1. The operator with , belongs to .
Lemma 3.2. Let with and . Then for all . Moreover,
Lemma 3.3. For , and , it holds that for and
Lemma 3.4 (see ). Let and be real numbers such that . Then
Lemma 3.5. Let with , then is bounded on bounded sets in and satisfies
4. Existence of Weak Solutions
For , using the first equation of system (2.2) derives from which we have the conservation law
Lemma 4.1 (Kato and Ponce ). If , then is an algebra. Moreover, where is a constant depending only on .
Lemma 4.2 (Kato and Ponce ). Let . If and , then
Lemma 4.3. Let and the function is a solution of problem (2.2) and the initial data . Then the following inequality holds
For , there is a constant , which only depends on , , , , and , such that
For , there is a constant , which only depends on , and , such that
Proof. Using and (4.2) derives (4.5).
Using and the Parseval equality gives rise to
For , applying to both sides of the first equation of system (2.3) and integrating with respect to by parts, we have the identity We will estimate the terms on the right-hand side of (4.9) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have Using the above estimate to the second term yields For the third term, using the Cauchy-Schwartz inequality and Lemma 4.1, we obtain
For the last term in (4.9), using Lemma 4.1 repeatedly results in It follows from (4.9) to (4.13) that there exists a constant depending only on and the coefficients of (1.2) such that Integrating both sides of the above inequality with respect to results in inequality (4.6).
To estimate the norm of , we apply the operator to both sides of the first equation of system (2.3) to obtain the equation Applying to both sides of (4.15) for gives rise to For the right-hand side of (4.16), we have Since using Lemma 4.1, and , we have Using the Cauchy-Schwartz inequality and Lemma 4.1 yields Applying (4.17)–(4.20) into (4.16) yields the inequality This completes the proof of Lemma 4.3.
Defining and setting with and , we know that for any and .
It follows from Theorem 2.1 that for each the Cauchy problem has a unique solution .
Lemma 4.4. Under the assumptions of problem (4.23), the following estimates hold for any with and where is a constant independent of .
The proof of this Lemma can be found in Lai and Wu .
Lemma 4.5. If with such that . Let be defined as in system (4.23). Then there exist two positive constants and , which are independent of , such that the solution of problem (4.23) satisfies for any .
Proof. Using notation and differentiating both sides of the first equation of problem (4.23) or (4.15) 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
Since as for any , integrating both sides of the inequality (4.28) with respect to and taking the limit as result in the estimate
Using the algebra property of with yields ( means that there exists a sufficiently small such that )
in which Lemma 4.3 is used. Therefore, we get
From (4.30) and (4.32), one has
From Lemma 4.4, 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  or Theorem 2 in Section 1.1 presented in  yields that there are constants and independent of such that for arbitrary , which leads to the conclusion of Lemma 4.5.
Using Lemmas 4.3 and 4.5, notation and Gronwall's inequality results in the inequalities where , and depends on . It follows from Aubin's compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function strongly in the space for and converges to strongly in the space for . Thus, we can prove the existence of a weak solution to (2.2).
Proof of Theorem 2.2. From Lemma 4.5, we know that is bounded in the space . Thus, the sequences and are weakly convergent to and in for any , respectively. Therefore, satisfies the equation with and . Since is a separable Banach space and is a bounded sequence in the dual space of , there exists a subsequence of , still denoted by , weakly star convergent to a function in . It derives from the weakly convergent to in that almost everywhere. Thus, we obtain .
The author is very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper. This work is supported by the Key Project of Chinese Ministry of Education (109140).
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