1. Introduction
This paper concerns the following modified two-component Camassa-Holm system (MCH2, for simplicity):
(1)
u
t
-
u
x
x
t
+
3
u
u
x
-
2
u
x
u
x
x
-
u
u
x
x
x
=
-
g
ρ
ρ
-
x
,
t
>
0
,
x
∈
ℝ
,
ρ
t
+
(
ρ
u
)
x
=
0
,
t
>
0
,
x
∈
ℝ
,
u
(
x
,
t
=
0
)
=
u
0
(
x
)
,
x
∈
ℝ
,
ρ
(
x
,
t
=
0
)
=
ρ
0
(
x
)
,
x
∈
ℝ
,
where
ρ
(
x
,
t
)
=
(
1
-
∂
x
2
)
(
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
.
Let
Λ
=
(
1
-
∂
x
2
)
(
1
/
2
)
; then the operator
Λ
-
2
can be denoted by its associated Green’s function
G
=
(
1
/
2
)
e
-
|
x
|
as
(2)
(
Λ
-
2
f
)
(
x
)
=
(
G
*
f
)
(
x
)
=
1
2
∫
ℝ
e
-
|
x
-
y
|
f
(
y
)
d
y
.
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:
(3)
u
t
+
u
u
x
+
∂
x
G
*
(
u
2
+
1
2
u
x
2
+
1
2
γ
2
-
1
2
γ
x
2
)
=
0
,
t
>
0
,
x
∈
ℝ
,
γ
t
+
u
γ
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
.
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
L
2
to
H
1
in the potential energy term [5]. Also, the MCH2 admits the following conserved quantity:
(4)
E
1
=
∫
ℝ
(
u
2
+
u
x
2
+
γ
2
+
γ
x
2
)
d
x
.
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
X
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
∈
H
s
×
H
s
to system (3). Then the corresponding solutions blow up in finite time if and only if
(5)
lim
t
→
T
inf
x
∈
ℝ
{
u
x
(
x
,
t
)
}
=
-
∞
o
r
lim
t
→
T
inf
x
∈
ℝ
{
γ
x
(
x
,
t
)
}
=
-
∞
.
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
(6)
q
t
=
u
(
q
,
t
)
,
0
<
t
<
T
,
x
∈
ℝ
,
q
(
x
,
0
)
=
x
,
x
∈
ℝ
.
Taking the derivative with respect to
x
, we get
(7)
d
q
t
d
x
=
q
x
t
=
u
x
(
q
,
t
)
q
x
,
t
∈
(
0
,
T
)
.
Hence
(8)
q
x
(
x
,
t
)
=
exp
{
∫
0
t
u
x
(
q
,
s
)
d
s
}
,
q
x
(
x
,
0
)
=
1
.
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:
(9)
(
i
)
ρ
0
(
x
0
)
≥
0
o
n
(
-
∞
,
x
0
)
,
ρ
0
(
x
0
)
≤
0
o
n
(
x
0
,
∞
)
,
(10)
(
i
i
)
u
0
′
(
x
0
)
<
-
|
u
0
(
x
0
)
|
,
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
∫
-
∞
x
0
e
ξ
y
0
(
ξ
)
d
ξ
≥
0
and
∫
x
0
∞
e
-
ξ
y
0
(
ξ
)
d
ξ
≤
0
are needed to guarantee blowup, which implies condition (10). In addition,
y
0
(
x
0
)
=
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
X
0
=
(
u
0
,
γ
0
)
T
∈
H
2
×
H
2
. As in [10], we will look for
(
d
/
d
t
)
u
x
(
q
(
x
,
t
)
,
t
)
. Applying
∂
x
2
(
G
*
f
)
=
G
*
f
-
f
to differentiate (3) with respect to
x
yields
(11)
u
t
x
+
u
u
x
x
=
-
1
2
u
x
2
+
u
2
+
1
2
γ
2
-
1
2
γ
x
2
-
G
*
(
1
2
u
x
2
+
u
2
+
1
2
γ
2
-
1
2
γ
x
2
)
.
Let
0
<
T
<
T
*
. Recalling that
u
∈
C
1
(
[
0
,
T
)
,
H
2
)
, 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
(12)
d
d
t
u
x
(
q
(
x
0
,
t
)
,
t
)
=
(
u
t
x
+
u
u
x
x
)
(
q
(
x
0
,
t
)
,
t
)
=
(
-
1
2
u
x
2
+
u
2
+
1
2
γ
2
-
1
2
γ
x
2
)
(
t
,
q
(
t
,
x
0
)
)
-
G
*
(
1
2
u
x
2
+
u
2
+
1
2
γ
2
-
1
2
γ
x
2
)
≤
-
1
2
u
x
2
+
1
2
u
2
,
where we used
G
*
(
u
2
+
(
1
/
2
)
u
x
2
)
≥
(
1
/
2
)
u
2
,
γ
x
2
(
x
,
t
)
-
γ
2
(
x
,
t
)
≤
γ
x
2
(
q
(
x
0
,
t
)
,
t
)
-
γ
2
(
q
(
x
0
,
t
)
,
t
)
, and
ρ
(
q
(
x
0
,
t
)
,
t
)
=
0
.
As
(13)
d
d
t
ρ
(
q
(
x
,
t
)
,
t
)
q
x
(
x
,
t
)
=
0
,
we get
(14)
ρ
(
q
(
x
0
,
t
)
,
t
)
q
x
(
x
0
,
t
)
=
ρ
0
(
x
0
)
=
0
;
it is easy to get
q
x
(
x
0
,
t
)
>
0
in (8), so
ρ
(
q
(
x
0
,
t
)
,
t
)
=
0
.
Consider
γ
x
2
(
x
,
t
)
-
γ
2
(
x
,
t
)
≤
γ
x
2
(
q
(
x
0
,
t
)
,
t
)
-
γ
2
(
q
(
x
0
,
t
)
,
t
)
; we can refer to [3].
The obvious factorization
u
2
-
u
x
2
=
(
u
-
u
x
)
(
u
+
u
x
)
; this leads us to study the functions of the form:
(15)
I
(
x
0
,
t
)
=
e
q
(
x
0
,
t
)
(
u
-
u
x
)
(
q
(
x
0
,
t
)
,
t
)
,
I
I
(
x
0
,
t
)
=
e
-
q
(
x
0
,
t
)
(
u
+
u
x
)
(
q
(
x
0
,
t
)
,
t
)
.
Computing the derivatives with respect to
t
using the definition of the flow map (6) gives
(16)
I
t
(
x
0
,
t
)
=
e
q
(
x
0
,
t
)
[
u
2
-
u
u
x
+
(
u
t
+
u
u
x
)
-
(
u
x
t
+
u
u
x
x
)
]
(
q
(
x
0
,
t
)
,
t
)
=
e
q
(
x
0
,
t
)
[
(
u
2
+
1
2
u
x
2
+
1
2
(
γ
2
-
γ
x
2
)
)
-
u
u
x
+
1
2
u
x
2
-
1
2
(
γ
2
-
γ
x
2
)
+
(
G
-
∂
x
G
)
*
(
u
2
+
1
2
u
x
2
+
1
2
(
γ
2
-
γ
x
2
)
)
]
≥
e
q
(
x
0
,
t
)
(
1
2
u
2
-
u
u
x
+
1
2
u
x
2
)
=
1
2
e
q
(
x
0
,
t
)
(
u
-
u
x
)
2
≥
0
.
In fact, the next lemma will be used.
Lemma 4.
Consider
(17)
(
G
±
∂
x
G
)
*
(
u
2
+
1
2
u
x
2
)
≥
1
2
u
2
.
Proof.
Consider
(18)
1
2
e
-
x
∫
-
∞
x
e
ξ
(
u
2
+
u
x
2
)
(
ξ
)
d
ξ
≥
e
-
x
∫
-
∞
x
e
ξ
u
u
x
d
ξ
=
1
2
u
2
(
x
)
-
1
2
e
-
x
∫
-
∞
x
e
ξ
u
2
(
ξ
)
d
ξ
.
So we get
(19)
1
2
e
-
x
∫
-
∞
x
e
ξ
(
u
2
+
1
2
u
x
2
)
(
ξ
)
d
ξ
≥
1
4
u
2
.
The same computations also obtain that
(20)
1
2
e
x
∫
-
∞
x
e
-
ξ
(
u
2
+
1
2
u
x
2
)
(
ξ
)
d
ξ
≥
1
4
u
2
.
We have
(21)
(
G
-
∂
x
G
)
=
e
-
x
∫
-
∞
x
e
ξ
(
u
2
+
1
2
u
x
2
)
(
ξ
)
d
ξ
,
(
G
+
∂
x
G
)
=
1
2
e
x
∫
-
∞
x
e
-
ξ
(
u
2
+
1
2
u
x
2
)
(
ξ
)
d
ξ
;
taking the linear combination in the two last inequalities implies estimate (17).
Similarly,
(22)
I
I
t
(
x
0
,
t
)
=
-
1
2
e
-
q
(
x
0
,
t
)
(
u
+
u
x
)
2
≤
0
.
It is convenient to establish the following fundamental proposition.
Proposition 5.
u
as in Theorem 2. Set
(23)
I
(
x
0
,
t
)
=
e
q
(
x
0
,
t
)
(
u
-
u
x
)
(
q
(
x
0
,
t
)
,
t
)
,
I
I
(
x
0
,
t
)
=
e
-
q
(
x
0
,
t
)
(
u
+
u
x
)
(
q
(
x
0
,
t
)
,
t
)
.
Then, for all
x
∈
ℝ
, the function
t
→
I
(
x
0
,
t
)
is monotonically increasing and
t
→
I
I
(
t
,
x
0
)
is monotonically decreasing.
It is easy to factorize
(24)
(
u
2
-
u
x
2
)
(
q
(
x
0
,
t
)
,
t
)
=
I
(
x
0
,
t
)
I
I
(
x
0
,
t
)
;
from inequality (12) we get
(25)
d
d
t
u
x
(
q
(
x
0
,
t
)
,
t
)
≤
1
2
I
(
x
0
,
t
)
I
I
(
x
0
,
t
)
.
Now let
x
0
be such that
u
0
′
(
x
0
)
<
-
|
u
0
(
x
0
)
|
. Proposition 5 yields, for all
t
∈
[
0
,
T
)
,
(26)
I
(
x
0
,
t
)
≥
I
0
(
x
0
)
>
0
,
I
I
(
x
0
,
t
)
≤
I
I
0
(
x
0
)
<
0
,
where we used
u
0
′
(
x
0
)
<
-
|
u
0
(
x
0
)
|
, then we get
I
0
(
x
0
)
>
0
and
I
I
0
(
x
0
)
<
0
.
Assume, by contradiction,
T
=
∞
; set
A
(
t
)
=
u
x
(
q
(
x
0
,
t
)
,
t
)
; thus we get
(27)
A
′
(
t
)
≤
1
2
I
(
x
0
,
t
)
I
I
(
x
0
,
t
)
≤
1
2
I
0
(
x
0
)
I
I
0
(
x
0
)
<
0
.
Set
β
0
=
(
1
/
2
)
(
u
0
′
2
-
u
0
2
)
(
x
0
)
; then
A
(
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
)
∥
H
1
2
=
∥
u
0
+
γ
0
∥
H
1
2
)
. For
t
≥
t
0
, then
A
(
t
)
≤
A
(
t
0
)
; we obtain
(28)
A
′
(
t
)
≤
1
2
I
(
x
0
,
t
)
I
I
(
x
0
,
t
)
=
1
2
(
u
2
-
u
x
2
)
(
q
(
x
0
,
t
)
,
t
)
≤
1
2
(
1
2
E
1
-
A
(
t
)
2
)
≤
-
1
4
A
(
t
)
2
.
This implies that, for
t
≥
t
0
,
(29)
A
(
t
)
≤
4
A
(
t
0
)
4
-
(
t
-
t
0
)
A
(
t
0
)
.
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:
(30)
u
t
+
u
u
x
+
∂
x
(
G
*
(
u
2
+
1
2
u
x
2
+
δ
2
ρ
2
)
)
=
0
,
t
>
0
,
x
∈
ℝ
,
ρ
t
+
(
ρ
u
)
x
=
0
,
t
>
0
,
x
∈
ℝ
.
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]:
(31)
E
1
=
∫
ℝ
(
u
2
+
u
x
2
+
δ
ρ
2
)
d
x
,
E
2
=
∫
ℝ
(
u
3
+
u
u
x
2
+
δ
u
ρ
2
)
d
x
.
In [6], a blow-up condition is established as
y
0
(
x
0
)
=
0
,
∫
-
∞
x
0
e
ξ
y
0
(
ξ
)
d
ξ
≥
0
and
∫
x
0
∞
e
-
ξ
y
0
(
ξ
)
d
ξ
≤
0
; here
y
0
(
x
0
)
=
(
1
-
∂
x
2
)
u
0
(
x
0
)
. Similar to Theorem 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
(32)
u
0
′
(
x
0
)
<
-
|
u
0
(
x
0
)
|
,
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.