Wave breaking and infinite propagation speed for a modified two-component Camassa-Holm system with
© Lv et al.; licensee Springer. 2014
Received: 21 January 2014
Accepted: 14 March 2014
Published: 28 March 2014
In this paper, we investigate the modified two-component Camassa-Holm equation with on the real line. Firstly, we establish sufficient conditions on the initial data to guarantee that the corresponding solution blows up in finite time for the modified two-component Camassa-Holm (MCH2) system. Then an infinite propagation speed for MCH2 is proved in the following sense: the corresponding solution with compactly supported initial data does not have compact x-support in its lifespan.
MSC:37L05, 35Q58, 26A12.
KeywordsMCH2 blow-up infinite propagation speed
where , , u denotes the velocity field, and ρ is related to the free surface density with the boundary assumptions; expresses an averaged or filtered density, κ is a nonnegative dissipative parameter, g is the downward constant acceleration of gravity in applications to shallow water waves. For convenience we assume in this paper. Moreover, u and γ satisfy the boundary conditions: and as .
Obviously, under the constraints of and , system (1.1) reduces to the Camassa-Holm equation, which was derived physically by Camassa and Holm in  (found earlier by Fokas and Fuchssteniner  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. There have been extensive studies on Camassa-Holm equation. Now, we mention some results that are related to our results. Firstly, wave breaking for a large class of initial data has been established in [3–6]. Recently, Zhou and his collaborators  give a direct proof for McKean’s theorem . In addition, the large time behavior for the support of momentum density of the Camassa-Holm equation was studied in . An interesting phenomenon of the propagation speed for the Camassa-Holm equation with was presented by Zhou and his collaborators in their work  in the sense that a strong solution of the Cauchy problem with compact initial profile cannot be compactly supported at any later time unless it is the zero solution. Meanwhile, for the same problem about the equation , we refer to  for details.
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, possessed the peakon, multi-kink solutions and the bi-Hamiltonian structure [13, 14] and is always integrable. The wave breaking mechanism was discussed in [15–17] and the existence of global solutions was analyzed in [12, 16, 18]. A geometric investigation can be found in [15, 19].
We cannot obtain the conservation of norm for the CH2 system.
In what follows, we always assume and .
This paper is organized as follows. In Section 2, we will present some results, which will be used in this paper. In Section 3, we will establish several sufficient conditions to guarantee that the corresponding strong solution brows up. In Section 4, we will investigate the infinite propagation speed of MCH2 with .
In this section, for completeness, we recall some elementary results. We list them and skip their proofs for conciseness. Local well-posedness for the MCH2 system (1.3) can be obtained by Kato’s semigroup theory .
Lemma 2.2 
Lemma 2.3 
with constants . If the initial data , then the solution to (2.2) goes to −∞ before t tends to .
which is always positive before the blow-up time. Therefore, the function is an increasing diffeomorphism of a line.
In this section, we establish sufficient conditions on the initial data to guarantee blow-up for system (1.3). We start this section with the following useful lemma.
for all , where is the initial value of . □
The next result describes the precise blow-up scenarios for sufficiently regular solutions to system (1.3).
This contradicts the assumption. Conversely, the Sobelev embedding result (with ) implies that if Theorem 3.2 holds, the solution blows up in finite time, which completes the proof of Theorem 3.2. □
We state our first criterion via the associated initial potential as follows.
for some point . Then the solution to our system (1.3) with initial value blows up in finite time.
Remark 3.1 This theorem is similar to the result proved by Zhou in .
In order to arrive at our result, we need the following three claims.
Claim 1. for all t in its lifespan.
Therefore, thanks to (2.4) we obtain , for all t in its lifespan.
Claim 2. For any fixed t, for all .
Claim 3. is decreasing, for all .
where we used the Claim 2.
for all , which implies can be extended to infinity.
where we used .
Let , then (3.22) is an equation of type (3.1) with . The proof is completed by applying Lemma 2.3. □
we find that the inequalities (3.19), (3.20) still hold. As is well known, McKean  states that only the sign of the initial potential , not the size of it, affects the wave breaking phenomenon. Similar to his theorem, we apply a similar initial potential to the two-component case, and it reveals that the sign of the initial density also plays an important role.
Then we give the second criterion in this paper.
Then the corresponding solution to system (1.3) blows up in finite time.
This completes the proof. □
Finally, we give the third criterion.
This completes the proof. □
4 Infinite propagation speed
In this section, we consider the infinite propagation speed for system (1.1). It can be shown as follows.
where and denote continuous nonvanishing functions with and for . Furthermore, is a strictly decreasing function, while is an increasing function.
Therefore, is an increasing function in the lifespan. From (4.1), it follows that for .
Similarly, it is easy to see that is decreasing with . Therefore, for .
Taking , we obtain what we want. Then the theorem is proved. □
This work is partially supported by NSFC (Grant No. 11101376), NSFC (Grant No. 11226176) and ZJNSF (Grant No. LQ13A010008).
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