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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 872187, 15 pages
http://dx.doi.org/10.1155/2012/872187
Research Article

The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation

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

Received 9 January 2012; Accepted 28 March 2012

Academic Editor: Yonghong Wu

Copyright © 2012 Shaoyong Lai and Aiyin Wang. 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

A nonlinear partial differential equation containing the famous Camassa-Holm and Degasperis-Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well-posedness of solutions for the equation in the Sobolev space with . Although the -norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space with is proved under the assumptions and .

1. Introduction

Constantin and Lannes [1] derived the shallow water equation where the constants , and satisfy certain conditions. Under several restrictions on the coefficients of model (1.1), the large time well-posedness was established on a time scale provided that the initial value belongs to with , and the wave-breaking phenomena were also discussed in [1]. As stated in [1], using suitable mathematical transformations, one can turn (1.1) into the form where , and are constants. Obviously, (1.2) is a generalization of both the Camassa-Holm equation [2] and the Degasperis-Procesi model [3] Equations (1.3) and (1.4) are bi-Hamiltonian and arise in the modeling of shallow water waves. These two equations pertain to waves of medium amplitude (cf. the discussions in [1, 4]) and accommodate wave-breaking phenomena. Moreover, the Camassa-Holm and Degasperis-Procesi models admit peaked (periodic as well as solitary) traveling waves capturing the main feature of the exact traveling wave solutions of the greatest height of the governing equations for water waves (cf. [5, 6]). For other dynamic properties about (1.3) and (1.4), the reader is referred to [720].

Recently, Lai and Wu [21] investigate (1.2) in the case where , , , and . The well-posedness of global solutions is established in [21] in Sobolev space with under certain assumptions on the initial value. The local strong and weak solutions for (1.2) are discussed in [22] in the case where , , and are arbitrary constants.

Motivated by the desire to extend the work in [22], we investigate the following generalized model of  (1.2): where , , and are arbitrary constants, and is a positive integer.

The aim of this paper is to investigate (1.5). Since , and are arbitrary constants, we do not have the result that the norm of the solution of (1.5) remains constant. We will apply the Kato theorem [23] to prove the existence and uniqueness of local solutions for (1.5) in the space provided that the initial value belongs to . Moreover, it is shown that there exists a weak solution of (1.5) in lower-order Sobolev space with .

The structure of this paper is as follows. The main results are given in Section 2. The existence and uniqueness of the local strong solution for the Cauchy problem (1.5) are proved in Section 3. The existence of weak solutions is established in Section 4.

2. Main Results

Firstly, we give some notations.

The space of all infinitely differentiable functions with compact support in is denoted by . We let be the space of all measurable functions such that . We define with the standard norm . For any real number , we let denote the Sobolev space with the norm defined by where . Here, we note that the norms , and depend on variable .

For and nonnegative number , let denote the space of functions with the properties that for each , and the mapping is continuous and bounded.

For simplicity, throughout this paper, we let denote any positive constant which is independent of parameter and set .

In order to study the existence of solutions for (1.5), we consider its Cauchy problem in the form where , and are arbitrary constants. Now, we give the theorem to describe the local well-posedness of solutions for problem (2.2).

Theorem 2.1. Let with , then the Cauchy problem (2.2) has a unique solution where depends on .

For a real number with , suppose that the function is in , and let be the convolution of the function and such that the Fourier transform of satisfies , , and for any . Thus one has . It follows from Theorem 2.1 that for each satisfying , the Cauchy problem has a unique solution , in which may depend on . However, one will show that under certain assumptions, there exist two constants and , both independent of , such that the solution of problem (2.3) satisfies for any , and there exists a weak solution for problem (2.2). These results are summarized in the following two theorems.

Theorem 2.2. If with such that , let be defined as in system (2.3), then there exist two constants and , which are independent of , such that the solution of problem (2.3) satisfies for any .

Theorem 2.3. Suppose that with and , then there exists a such that problem (2.2) has a weak solution in the sense of distribution and .

3. Proof of Theorem 2.1

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 only on :(I) for with and (i.e., is quasi-m-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 [23])
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
In fact, problem (2.2) can be written as which is equivalent to
We set with constant , , , , , and . We know that is an isomorphism of onto . In order to prove Theorem 2.1, we only need to check that and satisfy assumptions (I)–(III).

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

Proofs of the above Lemmas 3.13.3 can be found in [24] or [25].

Lemma 3.4 (see [23]). Let and be real numbers such that , then

Lemma 3.5. Let with and , then is bounded in and satisfies

Proof. Using the algebra property of the space with and , we have from which we obtain (3.12).
Applying Lemma 3.4, , , we get which completes the proof of (3.12).

Proof of Theorem 2.1. Using the Kato theorem, Lemmas 3.1, 3.2, 3.3, and 3.5, we know that system (3.11) or problem (2.2) has a unique solution

4. Proofs of Theorems 2.2 and 2.3

Before establishing the proofs of Theorems 2.2 and 2.3, we give several lemmas.

Lemma 4.1 (Kato and Ponce [26]). If , then is an algebra. Moreover, where is a constant depending only on .

Lemma 4.2 (Kato and Ponce [26]). Let . If and , then where .
Using the first equation of problem (2.2) gives rise to from which one has

Lemma 4.3. Let , and the function is a solution of the problem (2.2) and the initial data , then it holds that where .
For , there is a constant depending only on such that If , there is a constant depending only on such that

Proof. Using and (4.4) derives (4.5).
We write (1.5) in the equivalent form
Applying and the Parseval’s equality gives rise to
For , applying on both sides of (4.8), noting the above equality, and integrating the new equation with respect to by parts, we obtain the equation
We will estimate each of the terms on the right-hand side of (4.10). For the first and the fourth terms, using integration by parts, the Cauchy-Schwartz inequality, and Lemmas 4.1-4.2, we have where only depends on . Using the above estimate to the second term yields For the third term, using Lemma 4.1 gives rise to
For the last term, using Lemma 4.1 repeatedly, we get
It follows from (4.10)–(4.14) that which results in (4.6). Applying the operator on both sides of (4.8) yields the equation Multiplying both sides of (4.16) by for and integrating the resultant equation by parts give rise to On the right-hand side of (4.17), we have in which we have used Lemma 4.1. As by using , we have
Using the Cauchy-Schwartz inequality and Lemma 4.1 yields Applying (4.18)–(4.23) to (4.17) yields the inequality for a constant .

Lemma 4.4. For , and , the following estimates hold for any with where is a constant independent of .

The proof of this lemma can be found in [21].

Applying Lemmas 4.3 and 4.4, we can now state the following lemma, which plays an important role in proving existence of weak solutions.

Lemma 4.5. For and , there exists a constant independent of , such that the solution of problem (2.3) satisfies where .

Proof. The proof can be directly obtained from Lemma 4.4 and inequality (4.5).

Proof of Theorem 2.2. Using notation and differentiating (4.16) with respect to give rise to Using we get
Letting be an integer and multiplying (4.27) by and then integrating the resulting equation with respect to yield the equality Applying the Hölder’s inequality, we get or where Since as for any , integrating (4.32) with respect to and taking the limit as result in the estimate Using the algebraic property of with and Lemma 4.5 leads to where is independent of , and means that there exists a sufficiently small such that . From Lemma 4.5, we have
Applying (4.25), (4.34), (4.35), and (4.37) and writing out the subscript of , we obtain
It follows from the contraction mapping principle that there is a such that the equation has a unique solution . From (4.39), we know that the variable only depends on and . Using the theorem presented on page  51 in [16] or Theorem  2 in Section 1.1 in [27] derives that there are constants and independent of such that for arbitrary , which leads to the conclusion of Theorem  2.

Remark 4.6. Under the assumptions of Theorem 2.2, there exist two constants and , both independent of , such that the solution of problem (2.3) satisfies for any . This states that in Lemma 4.5, there exists a independent of such that (4.26) holds.
Using Theorem 2.2, Lemma 4.5, (4.6), (4.7), notation , and Gronwall’s inequality results in the inequalities where , and . 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 .

Proof of Theorem 2.3. From Theorem 2.2, we know that is bounded in the space . Thus, the sequences, , and are weakly convergent to , , , and in for any , separately. Hence, 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 . As weakly converges to in , it results that almost everywhere. Thus, we obtain .

Acknowledgment

This work is supported by the Applied and Basic Project of Sichuan Province (2012JY0020).

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