Abstract
This paper is concerned with blowup phenomena and global existence for the periodic twocomponent DullinGottwaldHolm system. We first obtain several blowup results and the blowup rate of strong solutions to the system. We then present a global existence result for strong solutions to the system.
MSC: 35G25, 35L05.
Keywords:
periodic twocomponent DullinGottwaldHolm system; blowup; blowup rate; global existence1 Introduction
In this paper, we consider the following periodic twocomponent DullinGottwaldHolm (DGH) system:
System (1.1) has been recently derived by Zhu et al. in [1] by following Ivanov’s approach [2]. It was shown in [1] that the DGH system is completely integrable and can be written as a compatibility condition of two linear systems
and
where ξ is a spectral parameter. Moreover, this system has the following two Hamiltonians:
and
For and , (1.1) becomes the DGH equation [3]
where A and α are two positive constants, modeling unidirectional propagation of surface waves on a shallow layer of water which is at rest at infinity, stands for fluid velocity. It is completely integrable with a biHamiltonian and a Lax pair. Moreover, its traveling wave solutions include both the KdV solitons and the CH peakons as limiting cases [3]. The Cauchy problem of the DGH equation has been extensively studied, cf.[413].
For , , system (1.1) becomes the twocomponent CamassaHolm system [2]
where is in connection with the free surface elevation from scalar density (or equilibrium), and the parameter A characterizes a linear underlying shear flow. System (1.2) describes water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case indicates the presence of an underlying current. A large amount of literature was devoted to the Cauchy problem (1.2); see [1422].
The Cauchy problem (1.1) has been discussed in [1]. Therein Zhu and Xu established the local wellposedness to system (1.1), derived the precise blowup scenario and investigated the wave breaking for it. The aim of this paper is to further study the blowup phenomena for strong solutions to (1.1) and to present a global existence result.
Our paper is organized as follows. In Section 2, we briefly state some needed results including the local well posedness of system (1.1), the precise blowup scenario and some useful lemmas to study blowup phenomena and global existence. In Section 3, we give several new blowup results and the precise blowup rate. In Section 4, we present a new global existence result of strong solutions to (1.1).
Notation Given a Banach space Z, we denote its norm by . Since all space of functions is over , for simplicity, we drop in our notations if there is no ambiguity.
2 Preliminaries
In this section, we will briefly give some needed results in order to pursue our goal.
With , we can rewrite system (1.1) as follows:
Note that if , is the kernel of , then for all , . Here we denote by ∗ the convolution. Using this identity, we can rewrite system (2.1) as follows:
The local wellposedness of the Cauchy problem (2.1) can be obtained by applying Kato’s theorem. As a result, we have the following wellposedness result.
Lemma 2.1[1]
Given the initial data, , there exists a maximaland a unique solution
of (2.1). Moreover, the solutiondepends continuously on the initial dataand the maximal time of existenceis independent ofs.
Consider now the following initial value problem:
whereudenotes the first component of the solutionto (2.1).
Lemma 2.2[1]
Letbe the solution of (2.1) with the initial data, . Then Eq. (2.3) has a unique solution. Moreover, the mapis an increasing diffeomorphism of ℝ with
Lemma 2.3[1]
Letbe the solution of (2.1) with the initial data, , andbe the maximal time of existence. Then we have
Moreover, if there exists ansuch that, thenfor all.
Next, we give two useful conservation laws of strong solutions to (2.1).
Lemma 2.4[1]
Letbe the solution of (2.1) with the initial data, , and letbe the maximal time of existence. Then, for all, we have
Lemma 2.5Letbe the solution of (2.1) with the initial data, , and letbe the maximal time of existence. Then, for all, we have
Proof By the first equation in (2.1), we have
This completes the proof of the lemma. □
Then we state the following precise blowup mechanism of (2.1).
Lemma 2.6[1]
Letbe the solution of (2.1) with the initial data, , and letbe the maximal time of existence. Then the solution blows up in finite time if and only if
Lemma 2.7[23]
Letand. Then, for every, there exists at least one pointwith
and the functionmis almost everywhere differentiable onwith
Lemma 2.8[24]
with the best possible constantclying within the range. Moreover, the best constantcis.
Lemma 2.9[25]
Ifis such that, then, for every, we have
Moreover,
Lemma 2.10[26]
Assume that a differentiable functionsatisfies
whereC, Kare positive constants. If the initial datum, then the solution to (2.4) goes to −∞ beforettends to.
3 Blowup phenomena
In this section, we discuss the blowup phenomena of system (2.1). Firstly, we prove that there exist strong solutions to (2.1) which do not exist globally in time.
Theorem 3.1Let, , andTbe the maximal time of the solutionto (2.1) with the initial data. If there is somesuch thatand
then the corresponding solution to (2.1) blows up in finite time.
Proof Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for some . Here we assume to prove the above theorem.
Define now
By Lemma 2.7, we let be a point where this infimum is attained. It follows that
Differentiating the first equation in (2.2) with respect to x and using the identity , we have
Since the map given by (2.3) is an increasing diffeomorphism of ℝ, there exists a such that . In particular, . Note that , we can choose . It follows that . By Lemma 2.3 and the condition , we have
Valuating (3.1) at and using Lemma 2.7, we obtain
here we used the relations and . Note that . By Lemma 2.4 and Lemma 2.8, we get
and
It follows that
where . Since , Lemma 2.10 implies
Applying Lemma 2.6, the solution blows up in finite time. □
Theorem 3.2Let, , andTbe the maximal time of the solutionto (2.1) with the initial data. Assume that. If there is somesuch thatand for any,
then the corresponding solution to (2.1) blows up in finite time.
Proof By Lemma 2.5, we have . Using Lemma 2.4 and Lemma 2.9, we obtain
and
Following a similar proof in Theorem 3.1, we have
where . Following the same argument as in Theorem 3.1, we deduce that the solution blows up in finite time. □
Letting and in Theorem 3.2, we have the following result.
Corollary 3.1Let, , andTbe the maximal time of the solutionto (2.1) with the initial data. Assume that. If there is somesuch thatand
then the corresponding solution to (2.1) blows up in finite time.
Remark 3.1 Note that system (2.1) is variational under the transformation and even . Thus, we cannot get a blowup result according to the parity of the initial data as we usually do.
Next, we will give more insight into the blowup mechanism for the wavebreaking solution to system (2.1), that is, the blowup rate for strong solutions to (2.1).
Theorem 3.3Letbe the solution to system (2.1) with the initial data, , satisfying the assumption of Theorem 3.1, andTbe the maximal time of the solution. Then we have
Proof As mentioned earlier, here we only need to show that the above theorem holds for .
Define now
By the proof of Theorem 3.1, there exists a positive constant such that
Let . Since by Theorem 3.1, there is some with and . Since m is locally Lipschitz, it is then inferred from (3.5) that
A combination of (3.5) and (3.6) enables us to infer
Since m is locally Lipschitz on and (3.6) holds, it is easy to check that is locally Lipschitz on . Differentiating the relation , , we get
with absolutely continuous on . For . Integrating (3.7) on to obtain
that is,
By the arbitrariness of the statement of Theorem 3.3 follows. □
4 Global existence
In this section, we will present a global existence result.
Theorem 4.1Let, , andTbe the maximal time of the solutionto (2.1) with the initial data. Iffor all, then the corresponding solutionexists globally in time.
Proof Define
By Lemma 2.7, we let be a point where this infimum is attained. It follows that
Since the map given by (2.3) is an increasing diffeomorphism of ℝ, there exists an such that .
Set and . Valuating (3.1) at and using Lemma 2.7, we obtain
where . By Lemma 2.4, Lemma 2.8 and , we have
By Lemmas 2.22.3, we know that has the same sign with for every . Moreover, there is a constant such that because of for all . Next, we consider the following Lyapunov positive function:
Differentiating (4.2) with respect to t and using (4.1), we obtain
By Gronwall’s inequality, we have
Thus,
This completes the proof by using Lemma 2.6. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This research is partially supported by the Doctoral Research Foundation of Zhengzhou University of Light Industry.
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