Wave Breaking for the Modified Two-Component Camassa-Holm System

Abstract

Some new sufficient conditions to guarantee wave breaking for the modified two-component Camassa-Holm system are established.

1. Introduction

This paper concerns the following modified two-component Camassa-Holm system (MCH2, for simplicity): 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥=−𝑔𝜌𝜌_𝑥,𝜌𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥𝑢()=0,𝑡>0,𝑥∈ℝ,𝑥,𝑡=0=𝑢₀(𝑥)𝜌(),𝑥∈ℝ,𝑥,𝑡=0=𝜌₀(𝑥),𝑥∈ℝ, (1) where 𝜌(𝑥,𝑡)=(1−𝜕²_𝑥)(𝜌−𝜌₀)(𝑥,𝑡), 𝑢(𝑥,𝑡) expresses the velocity field, and 𝑔 is the downward constant acceleration of gravity in applications to shallow water waves. In this paper, we let 𝑔=1.

Let Λ=(1−𝜕²_𝑥)^(1/2); then the operator Λ⁻² can be denoted by its associated Green’s function 𝐺=(1/2)𝑒^−|𝑥| as Λ⁻²𝑓(𝑥)=(𝑥)=1𝐺∗𝑓2_ℝ𝑒^{−|𝑥−𝑦|}𝑓𝑦𝑑𝑦. (2)

Let 𝛾(𝑥,𝑡)=(𝜌−𝜌₀)(𝑥,𝑡) and (𝐺∗𝜌)(𝑥,𝑡)=𝛾(𝑥,𝑡). So system (1) is equivalent to the following one: 𝑢_𝑡+𝑢𝑢_𝑥+𝜕_𝑥𝐺∗𝑢²+12𝑢²_𝑥+12𝛾²−12𝛾²_𝑥𝛾=0,𝑡>0,𝑥∈ℝ,_𝑡+𝑢𝛾_𝑥+𝐺∗𝑢_𝑥𝛾_𝑥𝑥+𝑢_𝑥𝛾𝑢()=0,𝑡>0,𝑥∈ℝ,𝑥,𝑡=0=𝑢₀(𝑥)𝛾(),𝑥∈ℝ,𝑥,𝑡=0=𝛾₀(𝑥),𝑥∈ℝ. (3)

The MCH2 system admits peaked solutions in the velocity and average density and we refer it to reference [1]. The local posedness, precise blow-up scenarios, and the existence of strong solutions which blow up in finite time can be found in [2–5]. Note that the MCH2 system is a modified version of the 2-component Camassa-Holm (CH2, for simplicity) system to allow a dependence on the average density 𝜌 (or depth, in the shallow water interpretation) as well as the pointwise density 𝜌. Meanwhile, the MCH2 may not be integrable unlike the CH2 system. The characteristic is that it will amount to strengthening the norm for 𝜌 from 𝐿² to 𝐻¹ in the potential energy term [5]. Also, the MCH2 admits the following conserved quantity: 𝐸₁=_ℝ𝑢²+𝑢²_𝑥+𝛾²+𝛾²_𝑥𝑑𝑥. (4)

This paper mainly studies wave breaking phenomenon, and we aim at improving previous results which were proved in [3, 6]. Our method is partially motivated by [7]. The remaining of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, we establish a new blow-up criterion for the MCH2. Finally, we establish a similar criterion for the CH2 system in Section 4.

2. Preliminaries

In this section, we recall some results without the proofs for conciseness. The first one is concerning local well-posedness and blow-up scenario.

Lemma 1 (see [2]). Given 𝑋₀=(𝑢₀,𝛾₀)^𝑇∈𝐻^𝑠×𝐻^𝑠 to system (3), 𝑠≥3/2, there exists a maximal 𝑇=𝑇(‖𝑋₀‖_{𝐻𝑠×𝐻𝑠})>0, and a unique solution 𝑋=(𝑢,𝛾)^𝑇∈𝐻^𝑠×𝐻^𝑠 to system (3). Then the corresponding solutions blow up in finite time if and only if lim_𝑡→𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑥,𝑡)=−∞𝑜𝑟lim_𝑡→𝑇inf_𝑥∈ℝ𝛾_𝑥(𝑥,𝑡)=−∞. (5)

We also need to introduce the standard particle trajectory [8]. Let 𝑞(𝑥,𝑡) be the particle line evolved by the solution; that is, it satisfies 𝑞_𝑡𝑞()=𝑢𝑞,𝑡,0<𝑡<𝑇,𝑥∈ℝ,𝑥,0=𝑥,𝑥∈ℝ. (6)

Taking the derivative with respect to 𝑥, we get 𝑑𝑞_𝑡𝑑𝑥=𝑞_𝑥𝑡=𝑢_𝑥𝑞,𝑡𝑞_𝑥,𝑡∈(0,𝑇). (7) Hence 𝑞_𝑥(𝑥,𝑡)=exp^𝑡₀𝑢_𝑥𝑞,𝑠𝑑𝑠,𝑞_𝑥(𝑥,0)=1. (8) Thus, the map 𝑞(⋅,𝑡) is a diffeomorphism of the real line.

3. Blowup for the MCH2 System

In this section, we establish a new sufficient condition to guarantee blowup for system (3), which is an improvement of that in [3].

Theorem 2. Suppose 𝑋₀=(𝑢₀,𝛾₀)^𝑇∈𝐻^𝑠×𝐻^𝑠 to system (3), 𝑠>3/2 and 𝜌₀(𝑥₀)=0. And the initial data satisfies the following two conditions: (𝑖)𝜌₀𝑥₀≥0𝑜𝑛−∞,𝑥₀,𝜌₀𝑥₀≤0𝑜𝑛𝑥₀,∞,(𝑖𝑖)𝑢^′₀𝑥₀<−𝑢₀𝑥₀, (9)(10) for some point 𝑥₀∈ℝ. Then the solution 𝑋=(𝑢,𝛾)^𝑇 to our system (3) with initial value 𝑋₀ blows up in finite time.

Remark 3. In [17] conditions ∫^𝑥0_−∞𝑒^𝜉𝑦₀(𝜉)𝑑𝜉≥0 and ∫^∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉≤0 are needed to guarantee blowup, which implies condition (10). In addition, 𝑦₀(𝑥₀)=0 is required. So obviously Theorem 2 is an improvement of that in [3]. On the other hand, our condition is a local version and is easy to check. For nonlocal conditions, we refer to [5, 9].

Now we give a proof for Theorem 2.

Proof. Let us first consider the case 𝑋₀=(𝑢₀,𝛾₀)^𝑇∈𝐻²×𝐻². As in [10], we will look for (𝑑/𝑑𝑡)𝑢_𝑥(𝑞(𝑥,𝑡),𝑡). Applying 𝜕²_𝑥(𝐺∗𝑓)=𝐺∗𝑓−𝑓 to differentiate (3) with respect to 𝑥 yields 𝑢_𝑡𝑥+𝑢𝑢_𝑥𝑥1=−2𝑢²_𝑥+𝑢²+12𝛾²−12𝛾²_𝑥1−𝐺∗2𝑢²_𝑥+𝑢²+12𝛾²−12𝛾²_𝑥. (11)
Let 0<𝑇<𝑇^∗. Recalling that 𝑢∈𝐶¹([0,𝑇),𝐻²), we show that 𝑢 and 𝑢_𝑥 are continuous on [0,𝑇)×ℝ and 𝑥→𝑢(𝑡,𝑥) is Lipschitz, uniformly with respect to 𝑡 in any compact time interval in [0,𝑇).
We get 𝑑𝑢𝑑𝑡^𝑥𝑞𝑥₀=𝑢,𝑡,𝑡_𝑡𝑥+𝑢𝑢_𝑥𝑥𝑞𝑥₀=−1,𝑡,𝑡2𝑢²_𝑥+𝑢²+12𝛾²−12𝛾²_𝑥𝑡,𝑞𝑡,𝑥₀1−𝐺∗2𝑢²_𝑥+𝑢²+12𝛾²−12𝛾²_𝑥1≤−2𝑢²_𝑥+12𝑢², (12) where we used 𝐺∗(𝑢²+(1/2)𝑢²_𝑥)≥(1/2)𝑢², 𝛾²_𝑥(𝑥,𝑡)−𝛾²(𝑥,𝑡)≤𝛾²_𝑥(𝑞(𝑥₀,𝑡),𝑡)−𝛾²(𝑞(𝑥₀,𝑡),𝑡), and 𝜌(𝑞(𝑥₀,𝑡),𝑡)=0.
As 𝑑𝜌𝑞()𝑞𝑑𝑡𝑥,𝑡,𝑡_𝑥(𝑥,𝑡)=0, (13) we get 𝜌𝑞𝑥₀,𝑡,𝑡𝑞_𝑥𝑥₀,𝑡=𝜌₀𝑥₀=0; (14) it is easy to get 𝑞_𝑥(𝑥₀,𝑡)>0 in (8), so 𝜌(𝑞(𝑥₀,𝑡),𝑡)=0.
Consider 𝛾²_𝑥(𝑥,𝑡)−𝛾²(𝑥,𝑡)≤𝛾²_𝑥(𝑞(𝑥₀,𝑡),𝑡)−𝛾²(𝑞(𝑥₀,𝑡),𝑡); we can refer to [3].
The obvious factorization 𝑢²−𝑢²_𝑥=(𝑢−𝑢_𝑥)(𝑢+𝑢_𝑥); this leads us to study the functions of the form: 𝐼𝑥₀,𝑡=𝑒^{𝑞(𝑥0,𝑡)}𝑢−𝑢_𝑥𝑞𝑥₀,𝑥,𝑡,𝑡𝐼𝐼₀,𝑡=𝑒^{−𝑞(𝑥0,𝑡)}𝑢+𝑢_𝑥𝑞𝑥₀,𝑡,𝑡. (15)
Computing the derivatives with respect to 𝑡 using the definition of the flow map (6) gives 𝐼_𝑡𝑥₀,𝑡=𝑒^{𝑞(𝑥0,𝑡)}𝑢²−𝑢𝑢_𝑥+𝑢_𝑡+𝑢𝑢_𝑥−𝑢_𝑥𝑡+𝑢𝑢_𝑥𝑥𝑞𝑥₀,𝑡,𝑡=𝑒^{𝑞(𝑥0,𝑡)}−𝑢𝑢_𝑥+12𝑢²_𝑥−12𝛾²−𝛾²_𝑥+𝐺−𝜕_𝑥𝐺∗𝑢²+12𝑢²_𝑥+12𝛾²−𝛾²_𝑥≥𝑒^{𝑞(𝑥0,𝑡)}12𝑢²−𝑢𝑢_𝑥+12𝑢²_𝑥=12𝑒^{𝑞(𝑥0,𝑡)}𝑢−𝑢_𝑥²≥0. (16)
In fact, the next lemma will be used.

Lemma 4. Consider 𝐺±𝜕_𝑥𝐺∗𝑢²+12𝑢²_𝑥≥12𝑢². (17)

Proof. Consider 12𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑢²+𝑢²_𝑥(𝜉)𝑑𝜉≥𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑢𝑢_𝑥1𝑑𝜉=2𝑢²(𝑥)−12𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑢²(𝜉)𝑑𝜉. (18) So we get 12𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑢²+12𝑢²_𝑥(𝜉)1𝑑𝜉≥4𝑢². (19) The same computations also obtain that 12𝑒^𝑥𝑥_−∞𝑒^−𝜉𝑢²+12𝑢²_𝑥(𝜉)1𝑑𝜉≥4𝑢². (20) We have 𝐺−𝜕_𝑥𝐺=𝑒^−𝑥𝑥_−∞𝑒^𝜉𝑢²+12𝑢²_𝑥(𝜉)𝑑𝜉,𝐺+𝜕_𝑥𝐺=12𝑒^𝑥𝑥_−∞𝑒^−𝜉𝑢²+12𝑢²_𝑥(𝜉)𝑑𝜉; (21) taking the linear combination in the two last inequalities implies estimate (17).

Similarly, 𝐼𝐼_𝑡𝑥₀1,𝑡=−2𝑒^{−𝑞(𝑥0,𝑡)}𝑢+𝑢_𝑥²≤0. (22)

It is convenient to establish the following fundamental proposition.

Proposition 5. 𝑢 as in Theorem 2. Set 𝐼𝑥₀,𝑡=𝑒^{𝑞(𝑥0,𝑡)}𝑢−𝑢_𝑥𝑞𝑥₀,𝑥,𝑡,𝑡𝐼𝐼₀,𝑡=𝑒^{−𝑞(𝑥0,𝑡)}𝑢+𝑢_𝑥𝑞𝑥₀,𝑡,𝑡. (23) Then, for all 𝑥∈ℝ, the function 𝑡→𝐼(𝑥₀,𝑡) is monotonically increasing and 𝑡→𝐼𝐼(𝑡,𝑥₀) is monotonically decreasing.

It is easy to factorize 𝑢²−𝑢²_𝑥𝑞𝑥₀,𝑡,𝑡=𝐼𝑥₀,𝑡𝐼𝐼𝑥₀,𝑡; (24) from inequality (12) we get 𝑑𝑢𝑑𝑡^𝑥𝑞𝑥₀≤1,𝑡,𝑡2𝐼𝑥₀,𝑡𝐼𝐼𝑥₀,𝑡. (25)

Now let 𝑥₀ be such that 𝑢^′₀(𝑥₀)<−|𝑢₀(𝑥₀)|. Proposition 5 yields, for all 𝑡∈[0,𝑇), 𝐼𝑥₀,𝑡≥𝐼₀𝑥₀>0,𝐼𝐼𝑥₀,𝑡≤𝐼𝐼₀𝑥₀<0, (26) where we used 𝑢^′₀(𝑥₀)<−|𝑢₀(𝑥₀)|, then we get 𝐼₀(𝑥₀)>0 and 𝐼𝐼₀(𝑥₀)<0.

Assume, by contradiction, 𝑇=∞; set 𝐴(𝑡)=𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡); thus we get 𝐴^′(𝑡)≤12𝐼𝑥₀,𝑡𝐼𝐼𝑥₀≤1,𝑡2𝐼₀𝑥₀𝐼𝐼₀𝑥₀<0. (27)

Set 𝛽₀=(1/2)(𝑢₀^′2−𝑢²₀)(𝑥₀); then 𝐴(𝑡)≤𝐴(0)−𝛽₀𝑡; we can find 𝑡₀ such that (𝐴(0)−𝛽₀𝑡₀)²≥𝐸₁(𝐸₁=‖𝑢(𝑡)+𝛾(𝑡)‖²_𝐻1=‖𝑢₀+𝛾₀‖²_𝐻1). For 𝑡≥𝑡₀, then 𝐴(𝑡)≤𝐴(𝑡₀); we obtain 𝐴^′(𝑡)≤12𝐼𝑥₀,𝑡𝐼𝐼𝑥₀=1,𝑡2𝑢²−𝑢²_𝑥𝑞𝑥₀≤1,𝑡,𝑡212𝐸₁−𝐴(𝑡)²1≤−4𝐴(𝑡)². (28)

This implies that, for 𝑡≥𝑡₀, 𝐴(𝑡)≤4𝐴𝑡₀4−𝑡−𝑡₀𝐴𝑡₀. (29)

From above, 𝑢_𝑥(𝑞(𝑥₀,𝑡),𝑡) must blow up in finite time, and 𝑇^∗=𝑡₀+4/𝐴(𝑡₀)<∞, so the condition of the blowup scenario (5) is fulfilled.

4. Blowup for the CH2 System

In this section, we consider the following two-component Camassa-Holm system: 𝑢_𝑡+𝑢𝑢_𝑥+𝜕_𝑥𝐺∗𝑢²+12𝑢²_𝑥+𝛿2𝜌²𝜌=0,𝑡>0,𝑥∈ℝ,_𝑡+𝜌𝑢_𝑥=0,𝑡>0,𝑥∈ℝ. (30)

The CH2 system appears initially in [11]. Wave breaking mechanism was discussed in [3, 12–14]. The existence of global solutions was analyzed in [6, 15, 16]. This system also has the following conservation laws [17]: 𝐸₁=_ℝ𝑢²+𝑢²_𝑥+𝛿𝜌²𝐸𝑑𝑥,₂=_ℝ𝑢³+𝑢𝑢²_𝑥+𝛿𝑢𝜌²𝑑𝑥. (31)

In [6], a blow-up condition is established as 𝑦₀(𝑥₀)=0,∫^𝑥0_−∞𝑒^𝜉𝑦₀(𝜉)𝑑𝜉≥0 and ∫^∞_𝑥0𝑒^−𝜉𝑦₀(𝜉)𝑑𝜉≤0; here 𝑦₀(𝑥₀)=(1−𝜕²_𝑥)𝑢₀(𝑥₀). Similar to Theorem 2, we can do the following improvement.

Theorem 6. Suppose 𝑋₀=(𝑢₀,𝜌₀)^𝑇∈𝐻^𝑠×𝐻^𝑠−1 to system (30), 𝑠≥3/2, and 𝜌(𝑥₀)=0; furthermore 𝑢^′₀𝑥₀<−𝑢₀𝑥₀, (32) for some point 𝑥₀∈ℝ. Then the solution to our system (30) with initial value 𝑋₀ blows up in finite time.

The proof is similar to Theorem 2 and we omit it.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank anonymous reviewers for their valuable comments and careful reading.