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The entropy weak solution to a generalized Degasperis-Procesi equation
Journal of Inequalities and Applications volume 2013, Article number: 409 (2013)
A nonlinear generalization of the Degasperis-Procesi equation is investigated. The well-posedness of entropy weak solutions for the Cauchy problem of the equation is established in the space .
The objective of this work is to study the well-posedness in the space for the generalized Degasperis-Procesi equation
where is a constant and . Letting be an initial condition for Eq. (1), we derive the inequality
where and are positive constants. In our further investigation, we only assume that
The formal integrability of Eq. (4) was found in . It was shown in  that Eq. (4) possesses a bi-Hamiltonian structure with an infinite sequence of conserved quantities and has exact peakon solutions. Dullin et al.  proved that the Degasperis-Procesi equation can be obtained from the shallow water elevation equation by an appropriate Kodama transformation. The traveling wave solutions of Eq. (4) were found in Lundmark and Szmigielski  and Vakhnenko and Parkes . Lin and Liu  established the -stability of peakons for Degasperis-Procesi Eq. (4) under certain assumptions imposing on the initial value. The local well-posedness of Eq. (4) with initial data , and the precise blow-up scenario were analyzed in . Lenells  classified all weak traveling wave solutions. Matsuno  studied multisoliton solutions and their peakon limits. The properties of infinite speed of propagation of Eq. (4) were established in Henry  and Mustafa . 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 [12–20] and the references therein.
Recently, Coclite and Karlsen  established the existence, uniqueness and stability of entropy weak solutions belonging to the class for Eq. (4). They obtained the existence of at least one weak solution satisfying a restricted set of entropy inequalities in the space . In Coclite and Karlsen , the well-posedness of entropy weak solution is investigated in the space .
which is equivalent to
where is a constant and .
The objective of this paper is to study problem (5). We establish the existence, uniqueness and stability of entropy weak solutions belonging to the space under condition (3). One of our contributions in this work is that we derive inequality (2), which leads us to establishing our main results. Here we state that we will adopt the well-known and celebrated Kruzkov technique (see ), which was originally introduced to analyze hyperbolic conservation laws.
The rest of this paper is organized as follows. Section 2 establishes several estimates for the viscous approximations of problem (5). The existence, uniqueness and stability of entropy weak solutions for problem (6) are presented in Section 3.
2 Viscous approximations and estimates
and letting with and , we know that for any with . We let () be the space of all measurable functions h such that . We define with the standard norm .
For simplicity, throughout this article, we let c denote any positive constants which are independent of parameter ε.
Several properties for the smooth functions are stated in the following lemma.
Lemma 2.1 The following estimates hold for any ε with and :
where c is a constant independent of ε.
The proof of the above lemma can be found in .
To establish the existence of solutions to Cauchy problem (5), 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 (8), we get
Lemma 2.2 Provided that , for any fixed , there exists a unique global smooth solution to Cauchy problem (7) belonging to with .
Here we state that the following lemma takes an important role in our further study of Eq. (1).
Lemma 2.3 Assume that holds and is a solution of problem (7). Then the following bounds hold for any :
where , and c are positive constants independent of ε and t.
We give some bounds on the nonlocal term , which all are consequences of the bound in Lemma 2.3.
Lemma 2.4 Assume that holds. Then
where c is a constant independent of ε and t.
The proofs of Lemmas 2.3 and 2.4 are similar to those of Lemmas 2.2, 2.3 and 2.4 in Coclite and Karlsen . Here we omit them.
Lemma 2.5 If , it holds that
using Lemma 2.4, we have
Setting , we get
Using and the comparison principle for the parabolic equations, we obtain the desired result (15). □
Lemma 2.6 (Oleinik-type estimate)
Assume that (3) holds and . Then
where the constant depends on T.
We omit the proof of this lemma since it is similar to the proof of Lemma 6 in .
Definition 2.1 (Weak solution)
We call a function a weak solution of Cauchy problem (8) provided
in , that is, , the following identity holds:(20)
Definition 2.2 (Entropy weak solution)
We call a function an entropy weak solution of Cauchy problem (8) if
u is a weak solution in the sense of Definition 2.1,
for any , and
for any convex entropy with corresponding entropy flux defined by , the following holds:(22)
that is, ,
satisfy (23). Using the Kruzkov entropy fluxes, we see that the weak formulation (20) is a consequence of the entropy formulation (23).
3 Main result
Now we give the following stability result of entropy weak solutions for Eq. (1).
Theorem 3.1 (-stability)
Assume that u and v are two entropy weak solutions of Eq. (1) with initial data and satisfying (3). For an arbitrary , it holds that
where c depends on , , , , and T.
Letting in Theorem 3.1 and assuming , we know for any .
Lemma 3.1 Let be a family of functions defined on such that
and the family
is compact in for any convex , where . Then there exist a sequence , , and a map , , such that
Lemma 3.2 Let Ω be a bounded open subset of , . Suppose that the sequence of distributions is bounded in and
where lies in a compact subset of and lies in a bounded subset of . Then lies in a compact subset of .
Lemma 3.3 Suppose that . Then there exist a subsequence of and a limit function
Proof Let be any convex entropy function which is compactly supported, and let be the corresponding entropy flux defined by . We write
are distributions. We claim that
Applying Lemmas 2.3, 2.4 and 2.5, we have
Hence, (30) follows. Therefore, from Lemmas 3.1 and 3.2, we know that there exist a subsequence and a limit function u satisfying (26) such that as
Using Lemma 2.5, from (34) and (35), we get (27). □
Lemma 3.4 Suppose that holds. Then
where the sequence and the function u are constructed in Lemma 3.3.
The proof is similar to that of Lemma 9 in . Here we omit it.
Theorem 3.2 (Existence)
Assume that (3) holds. Then there exists at least one entropy weak solution to problem (7).
Proof Let . It follows from (8) that
From Lemmas 2.1 and 3.3, we derive that the function u presented in Lemma 3.3 is a weak solution of problem (8) in the sense of Definition 2.1. We have to verify that u satisfies the entropy inequalities in Definition 2.2. Let be a convex entropy with flux q defined by . The convexity of η and (8) yield
Therefore, the entropy inequalities follow from Lemmas 3.3 and 3.4. □
From Theorems 3.1 and 3.2, we have the following theorem.
Theorem 3.3 Assume that (3) holds. Then Cauchy problem (7) has a unique entropy weak solution in the sense of Definition 2.2.
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The authors are 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 both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
The authors declare that they have no competing interests.
The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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Lai, S., Li, N. & Fan, S. The entropy weak solution to a generalized Degasperis-Procesi equation. J Inequal Appl 2013, 409 (2013). https://doi.org/10.1186/1029-242X-2013-409
- entropy weak solutions
- bounded solution