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The stability of local strong solutions for a shallow water equation
Journal of Inequalities and Applications volume 2014, Article number: 410 (2014)
We establish the stability of local strong solutions for a shallow water equation which includes the Degasperis-Procesi equation provided that its initial value lies in the Sobolev space with . The key element in our analysis is that the norm of the solutions keeps finite for all finite time t.
From the propagation of shallow water waves over a flat bed, Constantin and Lannes  derived the equation
where a, b, k and m are constants. We know that the Camassa-Holm and Degasperis-Procesi models are special cases of Eq. (2). Lai and Wu  established the well-posedness of local strong solutions and obtained the existence of local weak solutions for Eq. (2).
The aim of this paper is to investigate a special case of Eq. (2). Namely, we study the shallow water equation
where and are constants. Letting , and using Eq. (3), we derive the conservation law
where and is the Fourier transform of with respect to variable x. In fact, the conservation law (4) plays an important role in our further investigations of Eq. (3).
Various dynamic properties for Eq. (5) have been acquired by many scholars. Escher et al.  and Yin  studied the global weak solutions and blow-up structures for Eq. (5), while the blow-up structure for a generalized periodic Degasperis-Procesi equation was obtained in . Lin and Liu  established the stability of peakons for Eq. (5) under certain assumptions on the initial value. For other dynamic properties of the Degasperis-Procesi (5) and other shallow water models, the reader is referred to [8–19] and the references therein.
The objective of this work is to establish the stability of local strong solutions for the generalized Degasperis-Procesi equation (3) under the condition that we let the initial value belong to the space with . Here we address that the stability of local strong solutions for Eq. (3) has never been established in the literature. Our main approaches come from those presented in .
This paper is organized as follows. Section 2 gives several lemmas. The main result and its proof are presented in Section 3.
2 Several lemmas
The Cauchy problem of Eq. (3) is written in the form
which is equivalent to
where for any or .
Let and , we have
Lemma 2.1 For problem (6) with , it holds that
In addition, there exist two positive constants and depending only on m such that
Proof Letting and and using Eq. (3), we have and
from which we complete the proof. □
Lemma 2.2 ()
If with , there exist maximal and a unique local strong solution to problem (6) such that
Firstly, we study the differential equation
Lemma 2.3 Let , and let be the maximal existence time of the solution to problem (10). Then problem (10) has a unique solution . Moreover, the map is an increasing diffeomorphism of R with for .
Proof From Lemma 2.2, we have and . Thus we conclude that both functions and are bounded, Lipschitz in space and in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (10) has a unique solution .
Differentiating (10) with respect to x yields
which leads to
For every , using the Sobolev embedding theorem yields
It is inferred that there exists a constant such that for . This completes the proof. □
Lemma 2.4 Assume with . Let T be the maximal existence time of the solution g to Eq. (3). Then we have
where constant depends on m, k.
Proof Let , we have for all and . Using a simple density argument presented in , it suffices to consider to prove this lemma. If T is the maximal existence time of the solution g to Eq. (3) with the initial value such that . From (7), we obtain
from (16), we have
from which we get
where c is a positive constant independent of t. Using (18) results in
Using the Sobolev embedding theorem to ensure the uniform boundedness of for with , from Lemma 2.3, for every , we get a constant such that
We deduce from the above equation that the function is strictly increasing on R with as long as . It follows from (20) that
Lemma 2.5 Assume . Then
where is a constant independent of t.
Proof Using (7), we get
It follows from (23)-(24) and Lemma 2.1 that (22) holds. □
Lemma 2.6 Assume that and are two local strong solutions of equation (3) with initial data , , respectively. Then, for any , it holds that
where depends on t, f, , , and .
Proof We have
in which we have used the Tonelli theorem and Lemma 2.4. The proof is completed. □
We define to be a function which is infinitely differentiable on such that , for and . For any number , we let . Then we know that is a function in and
Assume that the function is locally integrable in . We define an approximation function of u as
We call a Lebesgue point of the function if
At any Lebesgue points of the function , we have . Since the set of points which are not Lebesgue points of has measure zero, we get as almost everywhere.
We introduce notation connected with the concept of a characteristic cone. For any , we define . Let ℧ designate the cone . We let designate the cross section of the cone ℧ by the plane , .
Let , where , and for an arbitrary . The space of all infinitely differentiable functions with compact support in is denoted by .
Lemma 2.7 ()
Let the function be bounded and measurable in cylinder . If for and any number , then the function
Lemma 2.8 ()
Let be bounded. Then the function
satisfies the Lipschitz condition in u and v, respectively.
Lemma 2.9 Let g be the strong solution of problem (7), and . Then
where k is an arbitrary constant.
Proof Let be an arbitrary twice smooth function on the line . We multiply the first equation of problem (7) by the function , where . Integrating over and transferring the derivatives with respect to t and x to the test function f, for any constant k, we obtain
in which we have used .
Integration by parts yields
Let be an approximation of the function and set . Using the properties of , (29), (30) and sending , we have
which completes the proof. □
In fact, the proof of (28) can also be found in .
For and with , using Lemma 2.2, we know that there exists such that two local strong solutions and of Eq. (3) satisfy
3 Main result
Now, we give the main result of this work.
Theorem 3.1 Assume that and are two local strong solutions of Eq. (3) with initial data , . For in (32), it holds that
where c depends on , , , and T.
Proof For arbitrary and , we assume that outside the cylinder
where and . The function is defined in (26). Note that
Using the Kruzkov device of doubling the variables  and Lemma 2.9, we have
Similarly, we have
from which we obtain
We will show that
In fact, the first two terms in the integrand of (39) can be represented in the form
From Lemma 2.4 and the assumptions on solutions , , we have and . From Lemma 2.8, we know that satisfies the Lipschitz condition in and , respectively. By the choice of η, we have outside the region
Considering the estimate and the expression of function , we have
where the constant c does not depend on h. Using Lemma 2.7, we obtain as . The integral does not depend on h. In fact, substituting , , , and noting that
Using Lemma 2.7, we have as . Using (44), we have
From (42), (46), (48), (49) and (50), we prove that inequality (40) holds.
and choose two numbers ρ and , . In (40), we choose
We note that the function outside the cone ℧ and outside the set ⊎. For , we have the relations
Applying (53)-(55) and (40), we have the inequality
Using Lemma 2.6 and letting and , we obtain
By the properties of the function for , we have
where c is independent of h. Letting
from which we obtain
Similarly, we have
It follows from (61) and (62) that
Send , , and note that
Thus, from (57), (58), (63)-(64), we have
from which we complete the proof by using the Gronwall inequality. □
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This work is supported by the National Natural Science Foundation of China (11471263).
The authors declare that they have no competing interests.
The article is a joint work of four authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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Cite this article
Lai, S., Yan, H., Chen, H. et al. The stability of local strong solutions for a shallow water equation. J Inequal Appl 2014, 410 (2014). https://doi.org/10.1186/1029-242X-2014-410
- strong solutions
- shallow water equation