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Mathematical Problems in Engineering

Volume 2014, Article ID 162472, 8 pages

http://dx.doi.org/10.1155/2014/162472
Research Article

Several Dynamical Properties for a Nonlinear Shallow Water Equation

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 18 April 2014; Accepted 8 May 2014; Published 22 May 2014

Academic Editor: Gongnan Xie

Copyright © 2014 Ls Yong and Haibo Yan. 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 third order dispersive shallow water equation including the Degasperis-Procesi model is investigated. The existence of weak solutions for the equation is proved in the space under certain assumptions. The Oleinik type estimate and   ( is a natural number) estimate for the solution are obtained.

1. Introduction

Constantin and Lannes [1] derived the shallow water wave equation where the constants , , , , , and satisfy certain conditions. As stated in [1], using suitable mathematical transformations, one can turn (1) into the form where , , , and are constants. Clearly, (2) contains both the Camassa-Holm and Degasperis-Procesi models.

The aim of this paper is to investigate the existence of weak solutions for the special case of (2). Namely, we study the shallow water equation where is a constant. Letting , and using (3), we derive the conservation law where . In fact, the conservation law (4) takes an important role in our further investigations of (3).

For , (3) reduces to the Degasperis-Procesi equation [2] which has been studied by many scholars (see [35]). Lundmark and Szmigielski [6] developed an inverse scattering approach for computing -peakon solutions to (5). The traveling wave solutions of (5) were investigated in Vakhnenko and Parkes [5]. Holm and Staley [7] studied stability of solitons and peakons numerically. Lin and Liu [8] proved the stability of peakons for the Degasperis-Procesi equation (5) under certain assumptions. The precise blowup scenario result, a blowup result, and the global existence of strong solutions and global weak solutions to (5) can be found in [9]. Matsuno [10] studied multisoliton solutions and their peakon limits. Analogous to the case of the Camassa-Holm equation, Henry [11] and Mustafa [12] showed that smooth solutions to (5) have infinite speed of propagation. For other methods to handle the problems relating to various dynamic properties of the Degasperis-Procesi equation and other shallow water equations, the reader is referred to [1315] and the references therein.

Coclite and Karlsen [16] established the existence, uniqueness, and stability of entropy weak solutions belonging to the class for (5). They obtained existence of at least one weak solution satisfying a restricted set of entropy inequalities in the space and extended these results to a class of generalized Degasperis-Procesi equations in [16].

Motivated by the desire to extend parts of the results presented in Coclite and Karlsen [16], we consider (3) with its Cauchy problem in the form which is equivalent to where is a constant and .

The objective of this paper is to study (3). We investigate the existence of weak solutions in the space under certain conditions. Several dynamical properties such as Oleinik type estimate and ( is a natural number) are obtained. As (3) includes the Degasperis-Procesi equation (5), parts of results presented in [16] are extended. Here we should mention that the generalized Degasperis-Procesi equation discussed in [16] does not include the model (3). We state that the ideas and approaches to prove our main results come from those in [16].

The rest of this paper is organized as follows. Section 2 establishes the , , and estimates for the viscous approximations of problem (6). The main result is given in Section 3.

2. Viscous Approximations and Estimates

Firstly, we give some notations.

Set . 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 .

For and nonnegative number , let denote the Frechet space of all continuous -valued functions on .

Defining and letting with and , we know that for any with .

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

To establish the existence of solutions to the Cauchy problem (6), we will analyze the limiting behavior of a sequence of smooth functions , where each function satisfies the viscous problem which is equivalent to the parabolic-elliptic system From the second identity of (11), we get

2.1. Estimates and Several Consequences

Several properties for the smooth function are given in the following Lemma.

Lemma 1. The following estimates hold for any with and : where is a constant independent of .

The proof of Lemma 1 is similar to that of Lemma 5 presented in [14]. Here we omit it.

Lemma 2. Provided that , for any fixed , there exists a unique global smooth solution to the Cauchy problem (11) belonging to with .

Proof. We omit the proof since it is similar to the one found in [16] or [17] by using .

Lemma 3. Assume that holds and is a solution of problem (10). Then, the following bounds hold for any : where is a positive constant independent of and .

Proof. Letting derives Multiplying the first equation of problem (11) by and integrating over yield For the left-hand side of this identity, using (16), we get

For the right-hand side of (18), we conclude where we have used (16) and integration by parts.

From (18), (19), and (20), we have

From (17), we obtain It follows from (16) that

Using (16), (21), and Lemma 1 derives that

The proof of Lemma 3 follows from (24).

We give some bounds on the nonlocal term , in which all are consequences of the bound in Lemma 3.

Lemma 4. Assume that holds. Then, where is a constant independent of and .

Proof. Using (11), we get From (29), we obtain (25). Using (14) and the Tonelli theorem, we have It follows from (31) that (26) and (27) hold. Using the second identity of problem (11), Lemma 3, and (26), we obtain (28).

Lemma 5. If , it holds that

Proof. Let functions and be such that . Multiplying the first equation in (11) with gives rise to

Choosing (modulo an approximation argument, see [16]) and then integrating the resulting equation over yield Using (26), we get from which we have (32).

2.2. and Estimates

In this subsection we establish several supplementary estimates for the viscous approximations, which also are consequences of the bound in Lemma 3. In particular, we prove that the sequence is bounded in , which yields strong compactness of this sequence. To this end, we need to assume that and .

Lemma 6. Assume that and hold. Then,

Proof. Setting , we know that satisfies the equation

If and satisfies , using the chain rule yields Choosing (modulo an approximation argument) and then integrating the resulting equation over give rise to Using (28), we have from which we obtain (36).

Lemma 7. Assume that holds. Then, where is independent of and .

Proof. Using and Lemma 6 derives (41). Using (27), (41), and the second equation of problem (11), we obtain inequality (42).

2.3. Estimate for Nature Number

Next we prove that the viscous approximations are bounded in for any nature number . From Lemmas 6 and 7, if , we have the inequality from which we derive

Lemma 8. Assume that holds. For any , it has

Proof. Choosing in (33), writing and integrating (46) over gives rise to

Integration by parts shows Letting , we have . Using Hölder’s inequality, (26), and (27), we obtain

Using (47)–(49) gives rise to from which we obtain (45) by Gronwall’s inequality.

2.4. Oleinik Type Estimate

Lemma 9 (Oleinik type estimate). Assume that . Then, for each with being fixed, where .

Proof. Setting , it follows from (11) and (42) that Considering the ordinary differential equation and using comparing theorem, we have which completes the proof.

3. Existence in

Using the estimates established in Section 2, we will show the existence of weak solutions to problem (6) under the assumption .

We state the concepts of weak solutions.

Definition 10 (weak solution). We call a function a weak solution of the Cauchy problem (6) provided that(i) and(ii) in ; that is, for all , there holds the identity where

Remark 11. It follows from part (i) of Definition 10 that for any and (see Lemma 5). Therefore, (55) makes sense.

We assume that where . Therefore, we have

Our main results are summarized in the following Theorem.

Theorem 12. Provided that and the solution satisfies (57), then there exists a weak solution to the Cauchy problem (6). The weak solution satisfies the following estimates for any : where is a positive constant independent of and .

The following Oleinik type estimate holds for a.e. : where

This theorem is an immediate consequence of Theorem 13 and results are presented in Section 2.

Theorem 13 (existence). Assume that and the solution satisfies (57). Then, there exists at least one weak solution to problem (6).

Proof. Using the estimates obtained in Section 2, we take a standard argument to see that there exists a sequence of strictly positive numbers tending to zero such that as

The previous estimates in Section 2 imply immediately that the limit function satisfies (59)-(60).

Let us now prove that as which follows from the following calculations:

From (62) to (64), we complete the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).

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