Several Dynamical Properties for a Nonlinear Shallow Water Equation

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.


Mathematical Problems in Engineering
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 [13][14][15] and the references therein.
Coclite and Karlsen [16] established the existence, uniqueness, and  1 () stability of entropy weak solutions belonging to the class  1 () ∩ () for (5).They obtained existence of at least one weak solution satisfying a restricted set of entropy inequalities in the space  2 () ∩  4 () 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  > 0 is a constant and The objective of this paper is to study (3).We investigate the existence of weak solutions in the space  1 () ∩ () under certain conditions.Several dynamical properties such as Oleinik type estimate and  2 () ( 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  2 , , and  ∞ estimates for the viscous approximations of problem (6).The main result is given in Section 3.

Viscous Approximations and Estimates
Firstly, we give some notations.
For simplicity, throughout this paper, we let  0 denote any positive constant, which is independent of parameter  and time .
To establish the existence of solutions to the Cauchy problem (6) which is equivalent to the parabolic-elliptic system From the second identity of (11), we get 2.1. 2 Estimates and Several Consequences.Several properties for the smooth function  0, are given in the following Lemma.
Lemma 1.The following estimates hold for any  with 0 <  < 1/4 and  ≥ 0: where  0 is a constant independent of .
The proof of Lemma 1 is similar to that of Lemma 5 presented in [14].Here we omit it.
Proof.We omit the proof since it is similar to the one found in [16] or [17] by using  0, ∈  ∞ ().Lemma 3. Assume that  0 ∈  2 () holds and   is a solution of problem (10).Then, the following bounds hold for any  ≥ 0: where  0 is a positive constant independent of  and .

Proof.
Letting derives Multiplying the first equation of problem (11) by 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) The proof of Lemma 3 follows from (24).
We give some bounds on the nonlocal term   , in which all are consequences of the  2 bound in Lemma 3.
We state the concepts of weak solutions.
We assume that where Our main results are summarized in the following Theorem.
Theorem 12. Provided that  0 ∈  1 () ∩  2 () ∩ () 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  ∈ (0, ): This theorem is an immediate consequence of Theorem 13 and results are presented in Section 2.
Proof.Using the estimates obtained in Section 2, we take a standard argument to see that there exists a sequence of strictly positive numbers {  } ∞ =1 tending to zero such that as  → ∞    →  a.e. in  + × ,    →  in  1 loc ( + × ) . (62) The previous estimates in Section 2 imply immediately that the limit function  satisfies (59)-(60).