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

Weiyi Zhu

Wujun Lv

CiteSeerX

Research Article
Wave Breaking for the Modified Two-Component
Camassa-Holm System
Wujun Lv and Weiyi Zhu
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Correspondence should be addressed to Weiyi Zhu; zwy@zjnu.cn
Received 8 April 2014; Accepted 21 April 2014; Published 5 May 2014
Academic Editor: Yonghui Xia
Copyright © 2014 W. Lv and W. Zhu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Somenew sufficient conditions to guaranteewave breaking for themodified two-componentCamassa-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
𝑥
𝑢
𝑥𝑥
− 𝑢𝑢
𝑥𝑥𝑥
= −𝑔𝜌𝜌
𝑥
,
𝑡 > 0, 𝑥 ∈ R,
𝜌
𝑡
+ (𝜌𝑢)
𝑥
= 0, 𝑡 > 0, 𝑥 ∈ R,
𝑢 (𝑥, 𝑡 = 0) = 𝑢
0
(𝑥) , 𝑥 ∈ R,
𝜌 (𝑥, 𝑡 = 0) = 𝜌
0
(𝑥) , 𝑥 ∈ R,
(1)
where 𝜌(𝑥, 𝑡) = (1 − 𝜕2
𝑥
)(𝜌 − 𝜌
0
)(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.
LetΛ = (1−𝜕2
𝑥
)
(1/2); then the operatorΛ−2 can be denoted
by its associated Green’s function 𝐺 = (1/2)𝑒−|𝑥| as
(Λ
−2
𝑓) (𝑥) = (𝐺 ∗ 𝑓) (𝑥) =
1
2
∫
R
𝑒
−|𝑥−𝑦|
𝑓 (𝑦) 𝑑𝑦. (2)
Let 𝛾(𝑥, 𝑡) = (𝜌 − 𝜌
0
)(𝑥, 𝑡) and (𝐺 ∗ 𝜌)(𝑥, 𝑡) = 𝛾(𝑥, 𝑡). So
system (1) is equivalent to the following one:
𝑢
𝑡
+ 𝑢𝑢
𝑥
+ 𝜕
𝑥
𝐺 ∗ (𝑢
2
+
1
2
𝑢
2
𝑥
+
1
2
𝛾
2
−
1
2
𝛾
2
𝑥
) = 0,
𝑡 > 0, 𝑥 ∈ R,
𝛾
𝑡
+ 𝑢𝛾
𝑥
+ 𝐺 ∗ ((𝑢
𝑥
𝛾
𝑥
)
𝑥
+ 𝑢
𝑥
𝛾) = 0, 𝑡 > 0, 𝑥 ∈ R,
𝑢 (𝑥, 𝑡 = 0) = 𝑢
0
(𝑥) , 𝑥 ∈ R,
𝛾 (𝑥, 𝑡 = 0) = 𝛾
0
(𝑥) , 𝑥 ∈ R.
(3)
TheMCH2 systemadmits 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 to𝐻1
in the potential energy term [5]. Also, the MCH2 admits the
following conserved quantity:
𝐸
1
= ∫
R
(𝑢
2
+ 𝑢
2
𝑥
+ 𝛾
2
+ 𝛾
2
𝑥
) 𝑑𝑥. (4)
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 520218, 4 pages
http://dx.doi.org/10.1155/2014/520218
2 Abstract and Applied Analysis
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]). Given𝑋
0
= (𝑢
0
, 𝛾
0
)
𝑇
∈ 𝐻
𝑠
×𝐻
𝑠 to system
(3), 𝑠 ≥ 3/2, there exists a maximal 𝑇 = 𝑇(‖𝑋
0
‖
𝐻
𝑠
×𝐻
𝑠) > 0,
and a unique solution 𝑋 = (𝑢, 𝛾)𝑇 ∈ 𝐻𝑠 × 𝐻𝑠 to system (3).
Then the corresponding solutions blow up in finite time if and
only if
lim
𝑡→𝑇
inf
𝑥∈R
{𝑢
𝑥
(𝑥, 𝑡)} = −∞ 𝑜𝑟 lim
𝑡→𝑇
inf
𝑥∈R
{𝛾
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
𝑡
= 𝑢 (𝑞, 𝑡) , 0 < 𝑡 < 𝑇, 𝑥 ∈ R,
𝑞 (𝑥, 0) = 𝑥, 𝑥 ∈ R.
(6)
Taking the derivative with respect to 𝑥, we get
𝑑𝑞
𝑡
𝑑𝑥
= 𝑞
𝑥𝑡
= 𝑢
𝑥
(𝑞, 𝑡) 𝑞
𝑥
, 𝑡 ∈ (0, 𝑇) . (7)
Hence
𝑞
𝑥
(𝑥, 𝑡) = exp{∫
𝑡
0
𝑢
𝑥
(𝑞, 𝑠) 𝑑𝑠} , 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 𝑋
0
= (𝑢
0
, 𝛾
0
)
𝑇
∈ 𝐻
𝑠
× 𝐻
𝑠 to system
(3), 𝑠 > 3/2 and 𝜌
0
(𝑥
0
) = 0. And the initial data satisfies the
following two conditions:
(𝑖) 𝜌
0
(𝑥
0
) ≥ 0 𝑜𝑛 (−∞, 𝑥
0
) ,
𝜌
0
(𝑥
0
) ≤ 0 𝑜𝑛 (𝑥
0
,∞) ,
(9)
(𝑖𝑖) 𝑢
󸀠
0
(𝑥
0
) < −
󵄨
󵄨
󵄨
󵄨
𝑢
0
(𝑥
0
)
󵄨
󵄨
󵄨
󵄨
, (10)
for some point 𝑥
0
∈ R. Then the solution 𝑋 = (𝑢, 𝛾)𝑇 to our
system (3) with initial value 𝑋
0
blows up in finite time.
Remark 3. In [17] conditions ∫𝑥0
−∞
𝑒
𝜉
𝑦
0
(𝜉)𝑑𝜉 ≥ 0 and
∫
∞
𝑥
0
𝑒
−𝜉
𝑦
0
(𝜉)𝑑𝜉 ≤ 0 are needed to guarantee blowup, which
implies condition (10). In addition, 𝑦
0
(𝑥
0
) = 0 is required.
So obviouslyTheorem 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 𝑋
0
= (𝑢
0
, 𝛾
0
)
𝑇
∈ 𝐻
2
×
𝐻
2. 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
𝑥𝑥
= −
1
2
𝑢
2
𝑥
+ 𝑢
2
+
1
2
𝛾
2
−
1
2
𝛾
2
𝑥
− 𝐺 ∗ (
1
2
𝑢
2
𝑥
+ 𝑢
2
+
1
2
𝛾
2
−
1
2
𝛾
2
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, uniformlywith respect to 𝑡 in any compact
time interval in [0, 𝑇).
We get
𝑑
𝑑𝑡
𝑢
𝑥
(𝑞 (𝑥
0
, 𝑡) , 𝑡)
= (𝑢
𝑡𝑥
+ 𝑢𝑢
𝑥𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡)
= (−
1
2
𝑢
2
𝑥
+ 𝑢
2
+
1
2
𝛾
2
−
1
2
𝛾
2
𝑥
) (𝑡, 𝑞 (𝑡, 𝑥
0
))
− 𝐺 ∗ (
1
2
𝑢
2
𝑥
+ 𝑢
2
+
1
2
𝛾
2
−
1
2
𝛾
2
𝑥
)
≤ −
1
2
𝑢
2
𝑥
+
1
2
𝑢
2
,
(12)
whereweused𝐺∗(𝑢2+(1/2)𝑢2
𝑥
) ≥ (1/2)𝑢
2, 𝛾2
𝑥
(𝑥, 𝑡)−𝛾
2
(𝑥, 𝑡) ≤
𝛾
2
𝑥
(𝑞(𝑥
0
, 𝑡), 𝑡) − 𝛾
2
(𝑞(𝑥
0
, 𝑡), 𝑡), and 𝜌(𝑞(𝑥
0
, 𝑡), 𝑡) = 0.
As
𝑑
𝑑𝑡
𝜌 (𝑞 (𝑥, 𝑡) , 𝑡) 𝑞
𝑥
(𝑥, 𝑡) = 0, (13)
we get
𝜌 (𝑞 (𝑥
0
, 𝑡) , 𝑡) 𝑞
𝑥
(𝑥
0
, 𝑡) = 𝜌
0
(𝑥
0
) = 0; (14)
it is easy to get 𝑞
𝑥
(𝑥
0
, 𝑡) > 0 in (8), so 𝜌(𝑞(𝑥
0
, 𝑡), 𝑡) = 0.
Consider 𝛾2
𝑥
(𝑥, 𝑡)−𝛾
2
(𝑥, 𝑡) ≤ 𝛾
2
𝑥
(𝑞(𝑥
0
, 𝑡), 𝑡)−𝛾
2
(𝑞(𝑥
0
, 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:
𝐼 (𝑥
0
, 𝑡) = 𝑒
𝑞(𝑥
0
,𝑡)
(𝑢 − 𝑢
𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡) ,
𝐼𝐼 (𝑥
0
, 𝑡) = 𝑒
−𝑞(𝑥
0
,𝑡)
(𝑢 + 𝑢
𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡) .
(15)
Abstract and Applied Analysis 3
Computing the derivatives with respect to 𝑡 using the
definition of the flow map (6) gives
𝐼
𝑡
(𝑥
0
, 𝑡) = 𝑒
𝑞(𝑥
0
,𝑡)
[𝑢
2
− 𝑢𝑢
𝑥
+ (𝑢
𝑡
+ 𝑢𝑢
𝑥
)
− (𝑢
𝑥𝑡
+ 𝑢𝑢
𝑥𝑥
) ] (𝑞 (𝑥
0
, 𝑡) , 𝑡)
= 𝑒
𝑞(𝑥
0
,𝑡)
[− 𝑢𝑢
𝑥
+
1
2
𝑢
2
𝑥
−
1
2
(𝛾
2
− 𝛾
2
𝑥
) + (𝐺 − 𝜕
𝑥
𝐺)
∗(𝑢
2
+
1
2
𝑢
2
𝑥
+
1
2
(𝛾
2
− 𝛾
2
𝑥
))]
≥ 𝑒
𝑞(𝑥
0
,𝑡)
(
1
2
𝑢
2
− 𝑢𝑢
𝑥
+
1
2
𝑢
2
𝑥
)
=
1
2
𝑒
𝑞(𝑥
0
,𝑡)
(𝑢 − 𝑢
𝑥
)
2
≥ 0.
(16)
In fact, the next lemma will be used.
Lemma 4. Consider
(𝐺 ± 𝜕
𝑥
𝐺) ∗ (𝑢
2
+
1
2
𝑢
2
𝑥
) ≥
1
2
𝑢
2
. (17)
Proof. Consider
1
2
𝑒
−𝑥
∫
𝑥
−∞
𝑒
𝜉
(𝑢
2
+ 𝑢
2
𝑥
) (𝜉) 𝑑𝜉
≥ 𝑒
−𝑥
∫
𝑥
−∞
𝑒
𝜉
𝑢𝑢
𝑥
𝑑𝜉 =
1
2
𝑢
2
(𝑥) −
1
2
𝑒
−𝑥
∫
𝑥
−∞
𝑒
𝜉
𝑢
2
(𝜉) 𝑑𝜉.
(18)
So we get
1
2
𝑒
−𝑥
∫
𝑥
−∞
𝑒
𝜉
(𝑢
2
+
1
2
𝑢
2
𝑥
) (𝜉) 𝑑𝜉 ≥
1
4
𝑢
2
. (19)
The same computations also obtain that
1
2
𝑒
𝑥
∫
𝑥
−∞
𝑒
−𝜉
(𝑢
2
+
1
2
𝑢
2
𝑥
) (𝜉) 𝑑𝜉 ≥
1
4
𝑢
2
. (20)
We have
(𝐺 − 𝜕
𝑥
𝐺) = 𝑒
−𝑥
∫
𝑥
−∞
𝑒
𝜉
(𝑢
2
+
1
2
𝑢
2
𝑥
) (𝜉) 𝑑𝜉,
(𝐺 + 𝜕
𝑥
𝐺) =
1
2
𝑒
𝑥
∫
𝑥
−∞
𝑒
−𝜉
(𝑢
2
+
1
2
𝑢
2
𝑥
) (𝜉) 𝑑𝜉;
(21)
taking the linear combination in the two last inequalities
implies estimate (17).
Similarly,
𝐼𝐼
𝑡
(𝑥
0
, 𝑡) = −
1
2
𝑒
−𝑞(𝑥
0
,𝑡)
(𝑢 + 𝑢
𝑥
)
2
≤ 0. (22)
It is convenient to establish the following fundamental
proposition.
Proposition 5. 𝑢 as in Theorem 2. Set
𝐼 (𝑥
0
, 𝑡) = 𝑒
𝑞(𝑥
0
,𝑡)
(𝑢 − 𝑢
𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡) ,
𝐼𝐼 (𝑥
0
, 𝑡) = 𝑒
−𝑞(𝑥
0
,𝑡)
(𝑢 + 𝑢
𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡) .
(23)
Then, for all 𝑥 ∈ R, the function 𝑡 → 𝐼(𝑥
0
, 𝑡) is monotonically
increasing and 𝑡 → 𝐼𝐼(𝑡, 𝑥
0
) is monotonically decreasing.
It is easy to factorize
(𝑢
2
− 𝑢
2
𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡) = 𝐼 (𝑥
0
, 𝑡) 𝐼𝐼 (𝑥
0
, 𝑡) ; (24)
from inequality (12) we get
𝑑
𝑑𝑡
𝑢
𝑥
(𝑞 (𝑥
0
, 𝑡) , 𝑡) ≤
1
2
𝐼 (𝑥
0
, 𝑡) 𝐼𝐼 (𝑥
0
, 𝑡) . (25)
Now let 𝑥
0
be such that 𝑢󸀠
0
(𝑥
0
) < −|𝑢
0
(𝑥
0
)|. Proposition 5
yields, for all 𝑡 ∈ [0, 𝑇),
𝐼 (𝑥
0
, 𝑡) ≥ 𝐼
0
(𝑥
0
) > 0, 𝐼𝐼 (𝑥
0
, 𝑡) ≤ 𝐼𝐼
0
(𝑥
0
) < 0, (26)
where we used 𝑢󸀠
0
(𝑥
0
) < −|𝑢
0
(𝑥
0
)|, then we get 𝐼
0
(𝑥
0
) > 0 and
𝐼𝐼
0
(𝑥
0
) < 0.
Assume, by contradiction, 𝑇 = ∞; set 𝐴(𝑡) =
𝑢
𝑥
(𝑞(𝑥
0
, 𝑡), 𝑡); thus we get
𝐴
󸀠
(𝑡) ≤
1
2
𝐼 (𝑥
0
, 𝑡) 𝐼𝐼 (𝑥
0
, 𝑡) ≤
1
2
𝐼
0
(𝑥
0
) 𝐼𝐼
0
(𝑥
0
) < 0. (27)
Set𝛽
0
= (1/2)(𝑢
󸀠2
0
−𝑢
2
0
)(𝑥
0
); then𝐴(𝑡) ≤ 𝐴(0)−𝛽
0
𝑡; we can
find 𝑡
0
such that (𝐴(0) − 𝛽
0
𝑡
0
)
2
≥ 𝐸
1
(𝐸
1
= ‖𝑢(𝑡) + 𝛾(𝑡)‖
2
𝐻
1 =
‖𝑢
0
+ 𝛾
0
‖
2
𝐻
1). For 𝑡 ≥ 𝑡0, then 𝐴(𝑡) ≤ 𝐴(𝑡0); we obtain
𝐴
󸀠
(𝑡) ≤
1
2
𝐼 (𝑥
0
, 𝑡) 𝐼𝐼 (𝑥
0
, 𝑡) =
1
2
(𝑢
2
− 𝑢
2
𝑥
) (𝑞 (𝑥
0
, 𝑡) , 𝑡)
≤
1
2
(
1
2
𝐸
1
− 𝐴(𝑡)
2
)
≤ −
1
4
𝐴(𝑡)
2
.
(28)
This implies that, for 𝑡 ≥ 𝑡
0
,
𝐴 (𝑡) ≤
4𝐴 (𝑡
0
)
4 − (𝑡 − 𝑡
0
) 𝐴 (𝑡
0
)
. (29)
From above, 𝑢
𝑥
(𝑞(𝑥
0
, 𝑡), 𝑡) must blow up in finite time,
and 𝑇∗ = 𝑡
0
+ 4/𝐴(𝑡
0
) < ∞, 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
+ 𝜕
𝑥
(𝐺 ∗ (𝑢
2
+
1
2
𝑢
2
𝑥
+
𝛿
2
𝜌
2
)) = 0,
𝑡 > 0, 𝑥 ∈ R,
𝜌
𝑡
+ (𝜌𝑢)
𝑥
= 0, 𝑡 > 0, 𝑥 ∈ R.
(30)
4 Abstract and Applied Analysis
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]:
𝐸
1
= ∫
R
(𝑢
2
+ 𝑢
2
𝑥
+ 𝛿𝜌
2
) 𝑑𝑥,
𝐸
2
= ∫
R
(𝑢
3
+ 𝑢𝑢
2
𝑥
+ 𝛿𝑢𝜌
2
) 𝑑𝑥.
(31)
In [6], a blow-up condition is established as 𝑦
0
(𝑥
0
) = 0,
∫
𝑥
0
−∞
𝑒
𝜉
𝑦
0
(𝜉)𝑑𝜉 ≥ 0 and ∫∞
𝑥
0
𝑒
−𝜉
𝑦
0
(𝜉)𝑑𝜉 ≤ 0; here 𝑦
0
(𝑥
0
) =
(1−𝜕
2
𝑥
)𝑢
0
(𝑥
0
). Similar toTheorem 2, we can do the following
improvement.
Theorem 6. Suppose 𝑋
0
= (𝑢
0
, 𝜌
0
)
𝑇
∈ 𝐻
𝑠
× 𝐻
𝑠−1 to system
(30), 𝑠 ≥ 3/2, and 𝜌(𝑥
0
) = 0; furthermore
𝑢
󸀠
0
(𝑥
0
) < −
󵄨
󵄨
󵄨
󵄨
𝑢
0
(𝑥
0
)
󵄨
󵄨
󵄨
󵄨
, (32)
for some point 𝑥
0
∈ R. Then the solution to our system (30)
with initial value 𝑋
0
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.
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Wave Breaking for the Modified Two-Component Camassa-Holm System

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

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