Two blow-up criteria of solutions to a modified two-component Camassa-Holm system
© Ma et al.; licensee Springer. 2014
Received: 4 November 2013
Accepted: 20 January 2014
Published: 4 February 2014
In this paper, we establish two sufficient conditions on the initial data to guarantee a blow-up phenomenon for the modified two-component Camassa-Holm (MCH2) system.
MSC:37L05, 35Q58, 26A12.
where , u denotes the velocity field, g is the downward constant acceleration of gravity as applied to shallow water waves, is the average density, and is taken to be a constant. For convenience we let in this paper. The MCH2 system does admit peaked solutions in the velocity and average density; we refer to Ref.  for details. There the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply a singularity in the pointwise density ρ at the point of vertical slope. Some other recent works can be found in [2, 3]. We find that the MCH2 system is expressed in terms of an averaged or filtered density in analogy to the relation between momentum and velocity by setting . Note that the MCH2 system is a version of the CH2 system modified to allow for a dependence on the average density (or depth, in the shallow water interpretation) as well as the pointwise density ρ.
The CH2 system appeared initially in , and recently Constantin and Ivanov in  gave a demonstration about its derivation in view of the fluid shallow water theory from the hydrodynamic point of view. This generalization, similar to the Camassa-Holm equation, possesses the peakon, multi-kink solutions and the bi-Hamiltonian structure [7, 8] and is integrable. Well-posedness and the wave breaking mechanism were discussed in [9–11] and the existence of global solutions was analyzed in [6, 10, 12]. The geometric investigation can be found in [13, 14].
Obviously, under the constraint of , this system reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in  (found earlier by Fokas and Fuchssteiner  as a bi-Hamiltonian generalization of the KdV equation) by directly approximating the Hamiltonian for Euler’s equation in the shallow water region with representing the free surface above a flat bottom. Some satisfactory results have been obtained recently, for instance, see Refs. [17–21]. Moreover, wave breaking criteria for a large class of initial data have been established in [18, 20–22]. In , McKean established the necessary and sufficient condition of wave breaking, while Zhou and his collaborators gave a new and direct proof in  for McKean’s theorem. In , Xin and Zhang showed global existence of weak solutions but uniqueness was obtained only under a priori assumption that is known to hold only for initial data such that is a sign-definite random measure. The solitary waves of the Camassa-Holm equation are peaked solutions and are orbitally stable ; see also  for a very related rod equation. Recently, an asymptotic analysis was given in . If , the CH2 system which includes both velocity and density variables in the dynamics is actually an extension of the CH equation. Although possessing peaked solutions in the velocity, the CH2 system does not admit singular solutions in the density profile. Its mathematical properties have been studied further in many works [6–10, 27, 28].
In Section 2, we recall some preliminary results on well-posedness and blow-up scenario. In Section 3, two detailed blow-up criteria are presented.
In this section, for completeness, we recall some elementary results and skip their proofs. Local well-posedness for the MCH2 system can be obtained by Kato’s semi-group theory . In , the authors gave a detailed description on the well-posedness theorem.
Theorem 2.1 
The next result describes the precise blow-up scenario for sufficiently regular solutions to system (1.2).
Theorem 2.2 
which is always positive before the blow-up time. Therefore, the function is an increasing diffeomorphism of the line.
Now we give our two blow-up theorems.
where . Then the solution to system (1.2) with the initial value blows up in finite time.
If , the theory becomes the blow-up theorem in  for the Camassa-Holm. As has nothing to do with the initial data, so we add the initial energy to condition (ii).
In order to reach our result, we need the following claim.
Claim 1 is decreasing, for all , where T is the maximal existence time of the solution.
where we used (3.3) and (3.4).
Then the initial assumption makes obvious. So our claim is proved.
Before completing the proof, we need the following technical lemma.
Lemma 3.1 
Let , then given the condition (i) and due to the claim and the expression of , we get and . Using the above lemma, (3.10) is an equation of type (3.11) with . We can conclude that under the conditions (i) and (ii), the solution to system (1.2) blows up in finite time. □
Theorem 3.2 Suppose , , there exists δ satisfying when , , when , and when , . Some portion of the positive part of lies to the left of some portion of its negative part with the changing sign point at , then the solution to system (1.2) with the initial value blows up in finite time.
Then concerning the sign of and , we have four cases.
Case 1: , .
Case 2: , .
Case 3: , .
Case 4: , .
The cases for or are easy to handle.
First, we can find that Case 3 is equivalent to Case 2.
By the same reasoning, we have .
In order to get the monotonous property of and , we need the following claim.
Since and from the above equation, we get . Therefore the claim holds.
Claim 3 For any fixed t, , for any and all .
This completes the proof of the claim.
which implies is a strictly increasing function, while is a strictly decreasing one for a nontrivial solution.
Now we prove Case 1.
Then we need the following lemma to finish our proof for Case 1.
Lemma 3.2 
with constant . If we have the initial datum , then the solution goes to −∞ before t tends to .
Through this lemma, we can see that goes to −∞ within finite time and Case 1 has been proved.
Now we prove Case 2.
We will prove that after some time Case 2 will change to Case 1. So it is sufficient to show that there exists a time , such that as .
That is to say .
Case 3.1 For any x, y satisfying , there exists a constant , such that .
Case 3.2. There exist some points, say , such that is unbounded.
Claim 4 For any two adjacent points belong to , say e and f with , it satisfies as .
So the claim is true.
for large enough and in the time interval it is increasing.
Putting (3.21) and (3.22) into (3.20), we know that becomes negative. This is a contradiction.
Therefore, we finish the proof for Case 2.
Finally we finish the proof of our theorem with proving Case 4.
We want to prove that Case 4 can be reduced to the first or the third case, so it is sufficient to prove that there exists a time , such that as .
Then a contradiction is obtained from (3.23): taken at is summable with respect to t, but is not.
So there exists a time T, such that when , . This completes the proof for Case 4.
for all . Then the theorem still holds.
Remark 3.3 This blow-up theorem has nothing to do with the initial energy . It is the sign of the initial density and the sign of that play an important role in wave breaking, it is not the size of them that affects it. It is very similar to the necessary and sufficient blow-up condition for the Camassa-Holm equation given by McKean in . □
This work is partially supported by ZJNSF (Grant No. LQ13A010008) and NSFC (Grant No. 11226176). Thanks are also given to the anonymous referees for careful reading and suggestions.
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