Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 327572, 15 pages
doi:10.1155/2012/327572
Research Article
On the Global Well-Posedness of the Viscous
Two-Component Camassa-Holm System
Xiuming Li
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
Correspondence should be addressed to Xiuming Li, lixiumingxu@gmail.com
Received 3 January 2012; Accepted 15 February 2012
Academic Editor: Valery Covachev
Copyright q 2012 Xiuming Li. 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.
We establish the local well-posedness for the viscous two-component Camassa-Holm system.
Moreover, applying the energy identity, we obtain a global existence result for the system with
u0 , η0 ∈ H 1 R × L2 R.
1. Introduction
We are interested in the global well-pose dness of the initial value problem associated to the
viscous version of the two-component Camassa-Holm shallow water system 1–3, namely,
mt umx 2ux m − Aux ρρx 0,
m u − uxx, ,
ρt uρ x 0,
t > 0, x ∈ R,
t > 0, x ∈ R,
1.1
t > 0, x ∈ R,
where the variable ut, x represents the horizontal velocity of the fluid or the radial stretch
related to a prestressed state, and ρt, x is related to the free surface elevation from
equilibrium or scalar density with the boundary assumptions, u → 0 and ρ → 1 as |x| → ∞.
The parameter A > 0 characterizes a linear underlying shear flow, so that 1.1 models wavecurrent interactions 4–6. All of those are measured in dimensionless units.
2
Abstract and Applied Analysis
−1
Set px : 1/2e−|x| , x ∈ R. Then 1 − ∂2x f p ∗ f for all f ∈ L2 R, where ∗ denotes
the spatial convolution. Let η ρ−1, 1.1 can be rewritten as a quasilinear nonlocal evolution
system of the type
−1 1
1
u2 u2x − Au η2 η ,
ut uux −∂x 1 − ∂2x
2
2
ηt uηx ηux ux 0,
t > 0, x ∈ R,
1.2
t > 0, x ∈ R.
The system 1.1 without vorticity, that is, A 0, was also rigorously justified
by Constantin and Ivanov 1 to approximate the governing equations for shallow water
waves. The multipeakon solutions of the same system have been constructed by Popivanov
and Slavova 7, and the corresponding integral surface is partially ruled. Chen et al. 8
established a reciprocal transformation between the two-component Camassa-Holm system
and the first negative flow of AKNS hierarchy. More recently, Holm et al. 9 proposed
a modified two-component Camassa-Holm system which possesses singular solutions in
component ρ. Mathematical properties of 1.1 with A 0 have been also studied further
in many works. For example, Escher et al. 10 investigated local well-posedness for the
two-component Camassa-Holm system with initial data u0 , ρ0 ∈ H s × H s−1 with s ≥ 2
and derived some precise blow-up scenarios for strong solutions to the system. Constantin
and Ivanov 1 provided some conditions of wave breaking and small global solutions. Gui
and Liu 11 recently obtained results of local well-posedness in the Besov spaces and wave
breaking for certain initial profiles. More recently, Gui and Liu 12 studied global existence
and wave-breaking criteria for the system 1.2 with initial data u0 , ρ0 − 1 ∈ H s × H s−1 with
s > 3/2.
In this paper, we consider the global well-posedness of the viscous two-component
Camassa-Holm system
−1 1
1
u2 u2x − Au η2 η ,
ut uux − uxx −∂x 1 − ∂2x
2
2
ηt uηx ηux ux 0,
t > 0, x ∈ R,
t > 0, x ∈ R,
u0, x u0 x,
x ∈ R,
η0, x η0 x,
x ∈ R.
1.3
The goal of the present paper is to study global existence of solutions for 1.3 to better
understand the properties of the two-component Camassa-Holm system 1.2. We state the
main result as follows.
Theorem 1.1. For u0 , η0 ∈ H 1 R × L2 R, there exists a unique global solution u, η of 1.3
such that
ut, x ∈ C 0, ∞; L2x R ∩ C 0, ∞; Hx1 R ,
ηt, x ∈ C 0, ∞; L2x R .
1.4
Abstract and Applied Analysis
3
To proof Theorem 1.1, we will first establish global well-posedness of the following
regularized two-component system with ε > 0 given:
−1 1
1
u2 u2x − Au η2 η ,
ut uux − uxx −∂x 1 − ∂2x
2
2
ηt − εηxx uηx ηux ux 0,
t > 0, x ∈ R,
t > 0, x ∈ R,
u0, x u0 x,
x ∈ R,
η0, x η0 x,
x ∈ R,
1.5
that is,
mt − mxx umx 2ux m − Aux ηηx ηx 0,
ηt − εηxx uηx ηux ux 0,
m u − uxx ,
u0, x u0 x,
t > 0, x ∈ R,
t > 0, x ∈ R,
t ≥ 0, x ∈ R,
η0, x η0 x,
1.6
x ∈ R.
Due to the Duhamel’s principle, we can also rewrite 1.6 as an integral equation
2
ut, x et∂x u0 t
2
et−τ∂x f u, ∂x u, η dτ,
0
ηt, x e
εt∂2x
η0 t
e
εt−τ∂2x
1.7
g u, ∂x u, η, ∂x η dτ,
0
where gu, ∂x u, η, ∂x η −u∂x η − η∂x u − ∂x u,
−1 1
1
1 u2 ∂x u2 − Au η2 η − ∂x u2 ,
f u, ∂x u, η −∂x 1 − ∂2x
2
2
2
∨
1.8
∨
et∂x u0 e−4π tξ u
0 ξ , eεt∂x η0 e−4π εtξ η
0 ξ , here and in what follows, we denote the
Fourier or inverse Fourier transform of a function f by f
or f ∨ .
The remainder of the paper is organized as follows. In Section 2, we will set up
and introduce some estimates for the nonlinear part of 1.5. In Section 3, we will get
the local well-posedness of 1.3 by constructing the global well-posedness of 1.5 using
the contraction argument and energy identity. The last section is devoted to the proof of
Theorem 1.1.
2
2
2
2
2. Preliminaries
We will list some lemmas needed in Section 3. First, we state the following lemma which
−1
consists of the crucial inequality involving the operator ∂x 1 − ∂2x .
4
Abstract and Applied Analysis
Lemma 2.1 see 13. For g, h ∈ L2 R,
−1 ∂x 1 − ∂2x
gh L2x
≤ cg L2x h
L2x ,
2.1
or more generally
−1 |∂x |s 1 − ∂2x
gh L2x
≤ cg L2x h
L2x ,
2.2
for all s < 3/2.
The next two lemmas are regarding the nonlinear part of 1.5.
Lemma 2.2. Consider the following:
t
t−τ∂2x
f u, ∂x u, η τ, xdτ e
0
2
L∞
T Lx
≤ C u
2L2 L2 ∂x u
2L2 L2
x
x
T
T
2
ηL2 L2x T 1/2 u
L2 L2x ηL2 L2x ,
T
T
T
t
2
eεt−τ∂x g u, ∂x u, η, ∂x η τ, xdτ 0
2
L∞
T Lx
≤ C ∂x u
2L2 L2
x
T
2.3
2.4
2
∂x ηL2 L2x T 1/2 ∂x u
L2 L2x .
T
T
Proof. Let us prove 2.3 firstly. Thanks to Lemma 2.1, the Sobolev embedding theorem
Hx1 R → L∞
x R, and the Hölder’s inequality, we have
t
t−τ∂2x
f u, ∂x u, η τ, xdτ e
0
2
L∞
T Lx
≤ sup
t∈0,T ≤
T
0
t t−τ∂2x f u, ∂x u, η τ, x 2 dτ
e
u
2L2x dt T
0
Lx
0
1
2
η 2 dt Lx
T
0
T
0
∂x u
2L2x dt A
u∂x u
L2x dt,
T
0
u
L2x dt 1
2
T
0
2.5
2
η 2 dt
Lx
Abstract and Applied Analysis
5
which yields that
t
t−τ∂2x
f u, ∂x u, η τ, xdτ e
0
L2T L2x
≤
T
0
u
2L2x dt
T 1/2
T
0
1
2
T
0
∂x u
2L2x dt
2
η 2 dt
1/2
Lx
AT
1/2
T
0
T
0
1/2
u
2L2x dt
1
2
T
2
η 2 dt
Lx
0
2.6
u
Hx1 ∂x u
L2x dt
2
≤ C u
2L2 L2 ∂x u
2L2 L2 ηL2 L2x T 1/2 u
L2 L2x ηL2 L2x .
T
x
x
T
T
T
T
Similarly, we can get
t
2
eεt−τ∂x g u, ∂x u, η, ∂x η τ, xdτ 0
2
L∞
T Lx
≤ sup
t∈0,T ≤
T
0
≤
T
0
t εt−τ∂2x g u, ∂x u, η, ∂x η τ, x 2 dτ
e
Lx
0
u∂x η 2 dt Lx
T
0
η∂x u 2 dt Lx
T
u
L∞x ∂x ηL2x dt 0
T
0
2.7
∂x u
L2x dt
η ∞ ∂x u
2 dt T 1/2
Lx
Lx
T
0
1/2
∂x u
2L2x dt
,
and then
t
εt−τ∂2x
g u, ∂x u, η, ∂x η τ, x dτ e
0
2
L∞
T Lx
≤
T
0
≤
T
0
≤2
u
Hx1 ∂x ηL2x dt T
0
∂x u
2L2x dt 2
which implies 2.4.
T
0
Hx1 ∂x
0
∂x u
L2x ∂x ηL2x dt T
η T
0
u
L2x dt T
1/2
T
0
∂x η 2 ∂x u
2 dt T 1/2
Lx
Lx
∂x η2 2 dt T 1/2
Lx
T
0
1/2
∂x u
2L2x dt
T
0
1/2
∂x u
2L2x dt
1/2
∂x u
2L2x dt
,
2.8
6
Abstract and Applied Analysis
Lemma 2.3. Consider the following:
t
t−τ∂2x
f u, ∂x u, η τ, xdτ e
0
L2T L2x
≤ CT 1/2 u
2L2 L2 ∂x u
2L2 L2
x
x
T
T
2
η 2
LT L2x
T 1/2 u
L2 L2x
T
η L2T L2x
2.9
,
t
2
eεt−τ∂x g u, ∂x u, η, ∂x η τ, xdτ 0
2.10
L2T L2x
≤ CT 1/2 ∂x u
2L2 L2
x
T
2
∂x ηL2 L2x T 1/2 ∂x u
L2 L2x .
T
T
Proof. We mainly prove 2.9. For this, we have that
t
t−τ∂2x
f u, ∂x u, η τ, xdτ e
0
⎛
L2T L2x
2 T t
2
⎜
t−τ∂x
≤⎝ e
fτ,
xdτ
dx
0 R
0
≤T
t
t−τ∂2x
fτ, xdτ e
0
1/2 ⎞1/2
⎟
dt⎠
L∞
T
.
2
L∞
T Lx
2.11
Therefore, applying Lemma 2.2, we can easily obtain 2.9.
Similarly, we can also obtain 2.10.
Let us state the following lemma, which was obtained in 13 up to a slight modification.
Lemma 2.4. For any u0 ∈ H 1 R and δ > 0, there exists T1 T1 u0 > 0 such that
2
∂x et∂x u0 L2T L2x
T 1/2
2
1
t∂2x
≤ δ.
∂x e u0 dx dt
0
R
2.12
For any η0 ∈ L2 R, ε > 0, and δ > 0, there exists T2 T2 η0 , ε > 0 such that
2
∂x eεt∂x η0 L2T L2x
T 1/2
2
2
εt∂2x
≤ δ.
∂x e η0 dx dt
0
R
2.13
Abstract and Applied Analysis
7
2
Next, we consider the nonlinear part of 1.7. When written as v u − et∂x u0 , μ εt∂2x
η − e η0 , then we have
∂t v ∂2x v f u, ∂x u, η ,
∂t μ ε∂2x μ g u, ∂x u, η, ∂x η ,
2.14
vx, 0 0,
μx, 0 0,
that is,
vx, t t
2
et−τ∂x f u, ∂x u, η τ, xdτ,
0
μx, t t
2.15
2
eεt−τ∂x g u, ∂x u, η, ∂x η τ, xdτ.
0
First, we have the following basic estimates.
Lemma 2.5. Consider the following:
f L1T L2x
2
≤ C u
2L2 L2 ∂x u
2L2 L2 ηL2 L2x T 1/2 u
L2 L2x ηL2 L2x ,
T
x
x
T
T
T
T
g 1 2 ≤ C u
2 2 2 ∂x u
2 2 2 η2 2 2 ∂x η2 2 2 T 1/2 ∂x u
2 2 .
L
L
L Lx
L Lx
L Lx
L L
L L
x
T
T
x
T
x
T
T
T
2.16
2.17
Proof. First, let us prove 2.16,
−1 1
1 2
2
2
f 1 2 1 ∂x u2 ∂x 1 − ∂2x
η
u
−
Au
η
u
∂
x
2
LT Lx
2
2
L1T L2x
≤
T −1 1
1
1 2
2
2
dt
∂x u2 ∂x 1 − ∂2x
η
u
−
Au
η
u
∂
x
2
2
2
2
0
Lx
≤
1
2
T T −1
2 −1 2 ∂x 1 − ∂2x ∂x u2 dt 1
−
∂
u
∂
2 dt
x
x
2
Lx
Lx
0
0
T −1
−1 1 T
2
2
η2 2 dt
∂x 1 − ∂x Au 2 dt 2
∂x 1 − ∂x
Lx
Lx
0
0
T
0
u∂x u
L2x dt T −1 ∂x 1 − ∂2x
η dt,
0
L2x
2.18
8
Abstract and Applied Analysis
which implies
f L1T L2x
T
1
≤
2
∂x u
2L2x dt
0
T
1
2
0
T
0
2
η 2 dt Lx
u
2L2x dt
T
T
0
u
L∞x ∂x u
L2x dt A
T
0
u
L2x dt
2.19
η 2 dt,
Lx
0
then we can get that
f 1 2 ≤ 1
LT Lx
2
T
0
∂x u
2L2x dt
C
T
0
u
2L2x dt
T
0
u
Hx1 ∂x u
L2x dt
T
T
1/2
1/2
2
2
1 T
2
1/2
1/2
η L2x dt T
η L2x dt
AT
u
L2x dt
2 0
0
0
2
≤ C u
2L2 L2 ∂x u
2L2 L2 ηL2 L2x T 1/2 u
L2 L2x ηL2 L2x ,
T
x
x
T
T
T
2.20
T
where we applied Lemma 2.1, Sobolev embedding theorem Hx1 R → L∞
x R, and Hölder’s
inequality. This proves 2.16.
Next, we prove 2.17,
g 1 2 −u∂x η − η∂x u − ∂x u 1 2
L Lx
L Lx
T
T
≤
T
0
≤
T
0
≤
T
0
u∂x η η∂x u ∂x u 2 dt
Lx
u∂x η 2 dt Lx
T
η∂x u 2 dt Lx
0
u
L∞x ∂x ηL2x dt T
0
T
0
2.21
∂x u
L2x dt
η ∞ ∂x u
2 dt Lx
Lx
T
0
∂x u
L2x dt,
which yields that
g 1 2 ≤
L Lx
T
T
0
≤C
u
Hx1 ∂x ηL2x dt T
0
u
2L2x dt
T
0
T
0
η 1 ∂x u
2 dt Lx
Hx
∂x u
2L2x dt
T
0
T
0
2
η 2 dt Lx
∂x u
L2x dt
T
0
∂x η2 2 dt Lx
T
0
2
2
≤ C u
2L2 L2 ∂x u
2L2 L2 ηL2 L2x ∂x ηL2 L2x T 1/2 ∂x u
L2 L2x ,
T
x
x
T
and this ends the proof of 2.17.
T
T
T
∂x u
L2x dt
2.22
Abstract and Applied Analysis
9
Then we have some estimates for ∂x v and ∂x μ.
Lemma 2.6. Consider the following:
2
∂x v
L2T L2x ≤ C u
2L2 L2x ∂x u
2L2 L2x ηL2 L2x T 1/2 u
L2T L2x ηL2 L2x ,
T
T
T
2.23
T
∂x μ 2 2 ≤ Cε−1/2 u
2 2 2 ∂x u
2 2 2 η2 2 2 ∂x η2 2 2 T 1/2 ∂x u
2 2 .
L Lx
L Lx
L Lx
L Lx
L L
L L
T
T
x
T
x
T
T
T
2.24
Proof. We mainly prove 2.24. We have that
μ
2
L∞
T Lx
t
εt−τ∂2x
e
gu, ∂x u, η, ∂x ητ, xdτ 0
2
L∞
T Lx
≤
T
2.25
g 2 dt g 1 2 .
Lx
L Lx
T
0
Multiply μ to the second equation of 2.14 and integrate with respect to x over R.
After integration by parts, we have
d
dt
1
2
1
μ2 T dx −
2
R
R
μ2
dx ε
2
R
μ2 0dx ε
T 0
R
R
2
∂x μ dx 2
∂x μ dx dt R
μgdx,
2.26
T 0
R
μgdx dt,
for any ε > 0, which implies
T ε
0
R
2
∂x μ dx dt ≤
T R
0
∂x μ2 2
LT L2x
≤
1
ε
T
0
μgdx dt,
μ 2 g 2 dt ≤ 1 μ ∞ 2 g 1 2 .
Lx
Lx
L
L
LT Lx
x
T
ε
2.27
By 2.25, together with 2.27, we have that
∂x μ 2 2 ≤ 1 g 1 2 .
LT Lx
LT Lx
1/2
ε
By Lemma 2.5, 2.24 follows.
Similarly, we can also obtain 2.23.
2.28
10
Abstract and Applied Analysis
3. Local Well-Posedness
Let z :
u
η
, Az :
2
et∂x u0
2
eεt∂x η0
t
⎛
⎜
⎜
Bz : ⎜ t
⎝
,
⎞
2
et−τ∂x f u, ∂x u, η τ, xdτ
0
2
eεt−τ∂x g u, ∂x u, η, ∂x η τ, xdτ
0
⎟
⎟
⎟,
⎠
D1 : C 0, T ; L2x R ∩ C 0, T ; Hx1 R ,
3.1
D2 : C 0, T ; L2x R ,
XaT : z ∈ D1 × D2 : z z − Az
L∞ L2x ∂x z
L2 L2x z
L2 L2x ≤ a ,
T
T
T
and define the mapping Φ : XaT → XaT by
Φz Az Bz.
3.2
Theorem 3.1. For any ε > 0, there exist T Tε > 0 and a > 0 such that ΦXaT ⊆ XaT . In addition,
Φ : XaT → XaT is a contraction mapping.
Proof. We first need to show that the map is well defined for some appropriate a and T . Let
z ∈ XaT , then we have
Φz Φz − Az
L∞ L2x Φz
L2 L2x ∂x Φz
L2 L2x .
T
T
3.3
T
Considering the terms in 3.3 one by one, from Lemma 2.2, the first term in 3.3 can
be estimated as follows:
Φz − Az
L∞T L2x
t
2
et−τ∂x f u, ∂x u, η τ, xdτ 0
2
L∞
T Lx
t
2
eεt−τ∂x g u, ∂x u, η, ∂x η τ, xdτ 0
≤C
2
L∞
T Lx
u
2L2 L2x
T
∂x u
2L2 L2x
T
2
2 ηL2 L2x ∂x ηL2 L2x
CT 1/2 u
L2 L2x ∂x u
L2 L2x
T
T
≤ Cz2 CT 1/2 z.
T
ηL2 L2x
T
T
3.4
Abstract and Applied Analysis
11
From Lemma 2.3, the second term in 3.3 can be estimated as follows:
Φz
L2T L2x
t
2
t∂x
t−τ∂2x
e u0 e
f u, ∂x u, η τ, xdτ 0
L2T L2x
t
2
εt∂x
εt−τ∂2x
e η0 e
g u, ∂x u, η, ∂x η τ, xdτ 0
2 ≤ et∂x u0 L2T L2x
t
t−τ∂2x
e
f u, ∂x u, η τ, xdτ 0
L2T L2x
3.5
L2T L2x
2 eεt∂x η0 L2T L2x
t
εt−τ∂2x
e
g u, ∂x u, η, ∂x η τ, xdτ 0
,
L2T L2x
which implies
Φz
L2T L2x ≤ T
1/2
u0 L2x
t
t−τ∂2x
e
f u, ∂x u, η τ, xdτ 0
L2T L2x
T 1/2 η0 L2x
t
εt−τ∂2x
e
g u, ∂x u, η, ∂x η τ, xdτ 0
3.6
L2T L2x
≤ T 1/2 z0 L2x CT 1/2 z2 CT z.
From Lemma 2.6, the third term in 3.3 can be estimated as follows:
∂x Φz
L2T L2x
t
t∂2x
t−τ∂2x
∂x e u0 ∂x e
f u, ∂x u, η τ, xdτ 0
L2T L2x
t
εt∂2x
εt−τ∂2x
∂x e η0 ∂x e
g u, ∂x u, η, ∂x η τ, xdτ 0
2
≤ ∂x et∂x u0 L2T L2x
2
∂x v
L2 L2x ∂x eεt∂x η0 T
L2T L2x
L2T L2x
∂x μL2 L2x
3.7
T
2
≤ 2δ C u
2L2 L2 ∂x u
2L2 L2 ηL2 L2x T 1/2 u
L2 L2x ηL2 L2x
T
x
x
T
T
T
T
C ε1/2
2
2
u
2L2 L2x ∂x u
2L2 L2x ηL2 L2x ∂x ηL2 L2x T 1/2 ∂x u
L2T L2x
T
T
T
1
1
2
1/2
≤ 2δ C 1 1/2 z CT
1 1/2 z.
ε
ε
T
12
Abstract and Applied Analysis
Combining 3.4–3.7, we have that
1
Φz ≤ 2δ T 1/2 z0 L2x Cz2 CT 1/2 z2 C 1 1/2 z2
ε
1
CT z CT 1/2 2 1/2 z
ε
1
≤ 2δ T 1/2 z0 L2x C 2 T 1/2 1 1/2 a2
ε
1
CT a CT 1/2 2 1/2 a.
ε
3.8
With appropriate values of δ, a, and T , we are able to have that Φz ≤ a, that is,
Φ : XaT → XaT is well defined.
Similar to the above argument, we can show that Φ : XaT → XaT is a contraction
mapping,
Φz1 − Φz2 ≤ C z1 − z2 ,
3.9
where C C T, a, ε, z1 L2T L2x , z2 L2T L2x , ∂x z1 L2T L2x , ∂x z2 L2T L2x can be chosen as 0 < C < 1
with appropriate values of T and a.
Theorem 3.2. For any ε > 0 and u0 , η0 ∈ H 1 R × L2 R, there exist a T T u0 , η0 , ε > 0 and a
unique solution uε , ηε of 1.5 such that
uε x, t ∈ C 0, T ; L2x R ∩ C 0, T ; Hx1 R ,
3.10
ηε x, t ∈ C 0, T ; L2x R .
Proof. Theorem 3.2 is merely Theorem 3.1 with a standard uniqueness argument.
Theorem 3.3. For any ε > 0 and u0 , η0 ∈ H 1 R × L2 R, there exists a unique global solution
uε , ηε of 1.5 such that
uε x, t ∈ C 0, ∞; L2x R ∩ C 0, ∞; Hx1 R ,
ηε x, t ∈ C 0, ∞; L2x R .
3.11
Proof. To prove Theorem 3.3, we need only to establish the a priori energy identity. Multiplying the first equation in 1.6 by u and integrating by parts with respect to x over R, we have
that
1 d
2 dt
R
u 2
u2x
dx R
u2x
u2xx
1
dx −
2
η ux dx 2
R
R
uηx dx 0,
3.12
Abstract and Applied Analysis
13
where we used the relation m u − uxx and R u2 ux dx 0. Multiplying the first equation in
1.6 by η and integrating by parts with respect to x over R, we have that
1 d
2 dt
R
η2 dx ε
R
ηx2 dx 1
2
R
η2 ux dx −
R
uηx dx 0.
3.13
From 3.12 and 3.13, we obtain the energy identity
1 d
2 dt
R
u2 u2x η2 dx u2x u2xx dx ε ηx2 dx 0,
R
R
3.14
which gives rise to the following inequality independent of ε and T :
2 2
sup ut, ·
2H 1 ηt, ·L2 2
ux 2L2 H 1 ≤ u0 2H 1 η0 L2 .
t∈0,T T
3.15
According to Theorem 3.2 and the energy inequality 3.15, one can extend the local solution
to the global one by a standard contradiction argument, which completes the proof of
Theorem 3.3.
From Theorem 3.3, one has the local well-posedness of system 1.3.
Theorem 3.4. For u0 , η0 ∈ H 1 R × L2 R, there exist a T T u0 , η0 > 0 and a unique local
solution u, η of 1.3 such that
ux, t ∈ C 0, T ; L2x R ∩ C 0, T ; Hx1 R ,
ηx, t ∈ C
0, T ; L2x R
3.16
.
Similar to the proof of Theorem 4.1 in 12 up to a slight modification, we may get
the following.
Theorem 3.5. Let z0 u0 , η0 ∈ H s × H s−1 , s ≥ 1, and let z u, η be the corresponding solution
to 1.3. Assume that Tz∗0 > 0 is the maximal time of existence, then
Tz∗0
< ∞ ⇒
Tz∗
0
0
∂x uτ
L∞ dτ ∞.
3.17
14
Abstract and Applied Analysis
4. Proof of Theorem 1.1
Proof of Theorem 1.1. From Theorem 3.4, we have got the local solution of 1.3. On the other
hand, according to the second equation of 1.3, we have
1 d
η2 2 ≤ 2
∂x u
∞ η2 2 ∂x u
2 η 2
Lx
Lx
L
L
Lx
x
x
2 dt
2
≤ 4
∂x u
L∞x ηL2x ∂x u
L2x ηL2x
2
2
1
≤ 4
∂x u
L∞x ηL2x ηL2x ∂x u
2L2
x
4
1
2
4
∂x u
L∞x 1 ηL2x ∂x u
2L2 .
x
4
4.1
An application of Gronwall’s inequality yields
2
η 2 ≤
Lx
t
2
η0 2 1 ∂x u
2 2 2 e 0 14
∂x u
L∞x dτ .
Lx
LT Lx
4
4.2
Similar to the proof of Theorem 3.3, we may get the energy identity
1 d
2 dt
R
u 2
u2x
η
2
d
dx dt
u2x u2xx dx 0,
R
4.3
which implies
2 2
sup ut, ·
2H 1 ηt, ·L2 2
ux 2L2 H 1 ≤ u0 2H 1 η0 L2 .
t∈0,T T
4.4
Due to the Sobolev embedding theorem H 1 R → L∞ R and 4.4, we obtain that for any
T < ∞,
T
0
∂x u
L∞x dt ≤
T
0
∂x u
Hx1 dt ≤ T 1/2 ∂x u
L2T Hx1 < ∞.
4.5
Therefore, from Theorem 3.5, we deduce that the maximal existence time T ∞. This proves
Theorem 1.1.
Acknowledgments
The work of the author is supported in part by the NSF of China under Grant no. 11001111
and no. 11141003. The author would like to thank the referees for constructive suggestions
and comments.
Abstract and Applied Analysis
15
References
1 A. Constantin and R. I. Ivanov, “On an integrable two-component Camassa-Holm shallow water
system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008.
2 P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions
having compact support,” Physical Review E, vol. 53, no. 2, pp. 1900–1906, 1996.
3 R. Ivanov, “Two-component integrable systems modelling shallow water waves: the constant vorticity case,” Wave Motion, vol. 46, no. 6, pp. 389–396, 2009.
4 A. Constantin and W. Strauss, “Exact steady periodic water waves with vorticity,” Communications on
Pure and Applied Mathematics, vol. 57, no. 4, pp. 481–527, 2004.
5 R. S. Johnson, “Nonlinear gravity waves on the surface of an arbitrary shear flow with variable
depth,” in Nonlinear Instability Analysis, vol. 12 of Adv. Fluid Mech., pp. 221–243, Comput. Mech.,
Southampton, UK, 1997.
6 R. S. Johnson, “On solutions of the Burns condition which determines the speed of propagation of
linear long waves on a shear flow with or without a critical layer,” Geophysical and Astrophysical Fluid
Dynamics, vol. 57, no. 1–4, pp. 115–133, 1991.
7 P. Popivanov and A. Slavova, Nonlinear Waves, vol. 4 of Series on Analysis, Applications and Computation,
World Scientific, Hackensack, NJ, USA, 2011.
8 M. Chen, S.-Q. Liu, and Y. Zhang, “A two-component generalization of the Camassa-Holm equation
and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006.
9 D. D. Holm, L. Nraigh, and C. Tronci, “Singular solutions of a modified two-component CamassaHolm equation,” Physical Review E, vol. 85, no. 1, Article ID 016601, 5 pages, 2012.
10 J. Escher, O. Lechtenfeld, and Z. Yin, “Well-posedness and blow-up phenomena for the 2-component
Camassa-Holm equation,” Discrete and Continuous Dynamical Systems. Series A, vol. 19, no. 3, pp. 493–
513, 2007.
11 G. Gui and Y. Liu, “On the Cauchy problem for the two-component Camassa-Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–66, 2011.
12 G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component
Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.
13 W. K. Lim, “Global well-posedness for the viscous Camassa-Holm equation,” Journal of Mathematical
Analysis and Applications, vol. 326, no. 1, pp. 432–442, 2007.
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