Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 532369, 26 pages
doi:10.1155/2012/532369
Research Article
The Local and Global Existence of Solutions for
a Generalized Camassa-Holm Equation
Nan Li, Shaoyong Lai, Shuang Li, and Meng Wu
Department of Applied Mathematics, Southwestern University of Finance and Economics,
Chengdu 610074, China
Correspondence should be addressed to Shaoyong Lai, laishaoy@swufe.edu.cn
Received 13 October 2011; Accepted 13 January 2012
Academic Editor: Sergey Piskarev
Copyright q 2012 Nan Li et al. 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.
A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the
pseudoparabolic regularization technique, its local well posedness in Sobolev space H S R with
s > 3/2 is established via a limiting procedure. Provided that the initial value u0 satisfies the sign
condition and u0 ∈ H s R s > 3/2, it is shown that there exists a unique global solution for the
equation in space C0, ∞; H s R ∩ C1 0, ∞; H s−1 R.
1. Introduction
Camassa and Holm 1 employed the Hamiltonian method to derive a completely integrable
shallow water wave model
ut − utxx 2kux 3uux 2ux uxx uuxxx ,
1.1
which was alternatively established as a water wave equation in 2–4. Equation 1.1 also
models wave current interaction 5, while Dai 6 derived it as a model in elasticity see
7. In addition, it was pointed out in Lakshmanan 8 that the Camassa-Holm equation
1.1 could be relevant to the modeling of tsunami waves see Constantin and Johnson 9.
After the birth of the Camassa-Holm equation 1.1, many works have been carried
out to probe its dynamic properties. For k 0, 1.1 has travelling wave solutions of the
form ce−|x−ct| , called peakons, which describes an essential feature of the travelling waves
of largest amplitude see 10–14. For k > 0, its solitary waves are stable solitons 15. It
is shown in 16–18 that the inverse spectral or scattering approach is a powerful tool to
handle the Camassa-Holm equation and analyze its dynamics. It is worthwhile to mention
2
Abstract and Applied Analysis
that 1.1 gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group
19–21, and this geometric illustration leads to a proof that the least action principle holds.
Xin and Zhang 22 proved the global existence of the weak solution in the energy space
H 1 R without any sign conditions on the initial value, and the uniqueness of this weak
solution is obtained under some assumptions on the solution 23. Coclite et al. 24 extended
the analysis presented in 22, 23 and obtained many useful dynamic properties to other
partial differential equations see 25–28 for an alternative approach. Li and Olver 29
established the local well posedness in the Sobolev space H s R with s > 3/2 for 1.1 and
gave conditions on the initial data that lead to finite time blowup of certain solutions. It is
shown in Constantin and Escher 30 that the blowup occurs in the form of breaking waves,
namely, the solution remains bounded but its slope becomes unbounded in finite time. For
other methods to handle the problems relating to various dynamic properties of the CamassaHolm equation and other shallow water equations, the reader is referred to 31–39 and the
references therein.
Motivated by the work in Hakkaev and Kirchev 33 to investigate the generalization
forms of the Camassa-Holm equation with high-order nonlinear terms, we study the
following generalized Camassa-Holm equation:
ut − utxx kum ux m 3um1 ux m 2um ux uxx um1 uxxx ,
1.2
where m ≥ 0 is a natural number and k ≥ 0. Obviously, 1.2 reduces to 1.1 if we set m 0.
As the Camassa-Holm equation 1.1 has been discussed by many mathematicians, we let the
natural number m ≥ 1 in this paper.
The objective of this paper is to study 1.2. Its local well posedness of solutions
in the Sobolev space H s R with s > 3/2 is developed by using the pseudoparabolic
regularization method. Provided that 1 − ∂2x u0 k/2m 1 ≥ 0 and u0 ∈ H s s >
3/2, the existence and uniqueness of the global solutions are established in space
C0, ∞; H s R C1 0, ∞; H s−1 R. It should be mentioned that the existence and
uniqueness of global strong solutions for the nonlinear generalized Camassa-Holm models
like 1.2 have never been investigated in the literatures.
2. Main Results
The space of all infinitely differentiable functions φt, x with compact support in 0, ∞ × R
is denoted byC0∞ . Lp Lp R1 ≤ p < ∞ is the space of all measurable functions h such
p
that hLp R |ht, x|p dx < ∞. We define L∞ L∞ R with the standard norm hL∞ infme0 supx∈R\e |ht, x|. For any real number s, H s H s R denotes the Sobolev space with
the norm defined by
hH s 2 1/2
s 1 |ξ| ht, ξ dξ
< ∞,
2
2.1
R
ξ e−ixξ ht, xdx.
where ht,
R
For T > 0 and nonnegative number s, C0, T ; H s R denotes the Frechet space of all
1/2
continuous H s -valued functions on 0, T . We set Λ 1 − ∂2x . For simplicity, throughout
this paper, we let c denote any positive constant which is independent of parameter ε.
Abstract and Applied Analysis
3
We consider the Cauchy problem of 1.2, which has the equivalent form
ut − utxx −
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
− m 1∂x um u2x um ux uxx ,
k ≥ 0, m ≥ 1,
2.2
u0, x u0 x.
Now, we give our main results for problem 2.2.
Theorem 2.1. Suppose that the initial function u0 x belongs to the Sobolev space H s R with s >
3/2. Then there is a T > 0, which depends on u0 H s , such that there exists a unique solution ut, x
of the problem 2.2 and
ut, x ∈ C0, T ; H s R
C1 0, T ; H s−1 R .
2.3
Theorem 2.2. Let u0 x ∈ H s , s > 3/2 and 1 − ∂2x u0 k/2m 1 ≥ 0 for all x ∈ R. Then
problem 2.2 has a unique solution satisfying that
ut, x ∈ C0, ∞; H s R
C1 0, ∞; H s−1 R .
2.4
3. Local Well-Posedness
In order to prove Theorem 2.1, we consider the associated regularized problem
ut − utxx εutxxxx −
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
− m 1∂x um u2x um ux uxx ,
3.1
u0, x u0 x,
where the parameter ε satisfies 0 < ε < 1/4.
Lemma 3.1. Let r and q be real numbers such that −r < q ≤ r. Then
uvH q ≤ cuH r vH q ,
uvH rq−1/2 ≤ cuH r vH q ,
1
,
2
1
if r < .
2
if r >
3.2
This lemma can be found in 34, 40.
Lemma 3.2. Let u0 x ∈ H s R with s > 3/2. Then the Cauchy problem 3.1 has a unique solution
ut, x ∈ C0, T ; H s R where T > 0 depends on u0 H s R . If s ≥ 2, the solution u ∈
C0, ∞; H s exists for all time.
4
Abstract and Applied Analysis
−1
Proof. Assuming that D 1 − ∂2x ε∂4x , we know that D : H s → H s4 is a bounded linear
operator. Applying the operator D on both sides of the first equation of system 3.1 and then
integrating the resultant equation with respect to t over the interval 0, t lead to
ut, x u0 x t
D −
0
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx dt.
3.3
Suppose that both u and v are in the closed ball BM0 0 of radius M0 about the zero function in
C0, T ; H s R and A is the operator in the right-hand side of 3.3. For any fixed t ∈ 0, T ,
we get the following:
t
k m1 m 3 m2 1
u
u
∂3x um2
−
D −
0
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx dt
t k m1 m 3 m2 1
v
v
∂3x vm2
− D −
−
x
x
m1
m2
m2
0
−m 1∂x vm vx2 vm vx vxx dt
s
H
≤ T C1 sup um1 − vm1 s sup um2 − vm2 s
H
0≤t≤T
H
0≤t≤T
sup D∂x um u2x − vm vx2 sup Du ux uxx − v vx vxx H s
m
Hs
0≤t≤T
3.4
m
,
0≤t≤T
where C1 may depend on ε. The algebraic property of H s0 R with s0 > 1/2 derives
m2
− vm2 u
Hs
u − v um1 um v · · · uvm vm1 Hs
≤ u − vH s
m1
m1−j
uH s
3.5
j
vH s
j0
≤ M0m1 u − vH s ,
m1
− vm1 s ≤ M0m u − vH s ,
u
H
m 2
m 2 D∂x u ux − v vx s ≤ D∂x um u2x − vx2 s D∂x vx2 um − vm s
H
H
H
m 2
2 m
2 m ≤ C u ux − vx s−1 vx u − v s−1
H
≤ CM0m1 u − vH s .
H
3.6
Abstract and Applied Analysis
5
Using the first inequality of Lemma 3.1, we have
Du ux uxx − v vx vxx H s
m
m
1 m 2 m
D u ux − v vx2
x
x H s
2
1 ≤
s D vx2 um − vm s
D um u2x − vx2
x
x
2 H
H
3.7
m 2
2
2
m
m ≤ C u ux − vx s−2 vx u − v s−2
x H
H
x
≤ C um H s u2x − vx2 s−1 vx2 s−1 um − vm H s
H
H
≤ CM0m1 u − vH s ,
where C may depend on ε. From 3.5–3.7, we obtain that
Au − AvH s ≤ θu − vH s ,
3.8
where θ T C2 M0m M0m1 and C2 is independent of 0 < t < T . Choosing T sufficiently
small such that θ < 1, we know that A is a contraction. Applying the above inequality yields
that
AuH s ≤ u0 H s θuH s .
3.9
Choosing T sufficiently small such that θM0 u0 H s < M0 , we deduce that A maps BM0 0
to itself. It follows from the contraction-mapping principle that the mapping A has a unique
fixed-point u in BM0 0.
For s ≥ 2, using the first equation of system 3.1 derives
d
dt
R
u2 u2x εu2xx dx 0,
3.10
from which we have the conservation law
R
u2 u2x εu2xx dx u20 u20x εu20xx dx.
3.11
R
The proof of the global existence result is a routine argument by using 3.11 see Xin and
Zhang 22.
Lemma 3.3 Kato and Ponce 41. If r ≥ 0, then H r
L∞ is an algebra. Moreover
uvr ≤ cuL∞ vr ur vL∞ ,
where c is a constant depending only on r.
3.12
6
Abstract and Applied Analysis
Lemma 3.4 Kato and Ponce 41. Let r > 0. If u ∈ H r
Λr , uvL2 ≤ c ∂x uL∞ Λr−1 v
L2
W 1,∞ and v ∈ H r−1
L∞ , then
Λr uL2 vL∞ .
3.13
Lemma 3.5. Let s ≥ 2, and the function ut, x is a solution of problem 3.1 and the initial data
u0 x ∈ H s . Then the following inequality holds:
u2H 1
≤
u2 u2x εu2xx dx
R R
u20
u20x
εu20xx
3.14
dx.
For q ∈ 0, s − 1, there is a constant c independent of ε such that
Λ
q1
2
u dx ≤
R
Λ
R
c
t
0
q1
u0
2
u2H q1
2
εΛ u0xx dx
q
um−1
L∞
um
L∞
ux L∞ 2
um−1
L∞ ux L∞
3.15
dτ.
For q ∈ 0, s − 1, there is a constant c independent of ε such that
1 − 2εut H q ≤ cuH q1
m
m
m−1
2
um−1
L∞ uL∞ uH 1 uL∞ ux L∞ uL∞ ux L∞ .
3.16
Proof. The inequality u2H 1 ≤ R u2 u2x dx and 3.11 derives 3.14.
Using ∂2x −Λ2 1 and the Parseval equality gives rise to
Λq1 u Λq1 um1 ux dx
Λq uΛq ∂3x um2 dx −m 2
R
R
m 2 Λq uΛq um1 ux dx.
3.17
R
For q ∈ 0, s − 1, applying Λq uΛq to both sides of the first equation of system 3.1
and integrating with respect to x by parts, we have the identity
Λq u2 Λq ux 2 εΛq uxx 2 dx
R
k
−
Λq uΛq um1 dx − m 2 Λq uΛq um1 ux dx
x
m1 R
R
Λq1 u Λq1 um1 ux dx m 1 Λq ux Λq um u2x dx
−
R
R
Λq uΛq um ux uxx dx.
1 d
2 dt
R
3.18
Abstract and Applied Analysis
7
We will estimate the terms on the right-hand side of 3.18 separately. For the second term,
by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have
Λq uΛq um1 ux dx Λq u Λq um1 ux − um1 Λq ux dx Λq uum1 Λq ux dx
R
R
R
m
m
≤ cuH q m 1uL∞ ux L∞ uH q ux L∞ uL∞ uH q
m1
2
q
um
L∞ ux L∞ Λ uL2
2
≤ cu2H q um
L∞ ux L∞ .
3.19
Similarly, for the first term in 3.18, we have
Λq uΛq um ux dx ≤ cu2 q um−1
H
L∞ ux L∞ .
3.20
R
Using the above estimate to the third term yields that
q1
q1
m1
Λ u Λ
u ux dx ≤ cu2H q1 um
L∞ ux L∞ .
3.21
R
For the fourth term, using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain that
q
m 2 Λq ux Λq um u2x dx ≤ Λq ux 2 u
u
Λ
x 2
L
L
R
≤ cuH q1 um ux L∞ ux H q ux L∞ um ux H q 3.22
≤ cu2H q1 ux L∞ um
L∞ ,
in which we have used um ux H q ≤ cum1 x H q ≤ cum
L∞ uH q1 .
For the last term in 3.18, using um u2x x um u2x x − um x u2x results in
q
q
m
2
Λq uΛq um ux uxx dx ≤ Λq ux Λq um u2x dx Λ
uΛ
u
u
x x dx
R
R
R
3.23
K1 K2 .
For K1 , it follows from 3.22 that
K1 ≤ cu2H q1 ux L∞ um
L∞ .
3.24
8
Abstract and Applied Analysis
For K2 , applying Lemma 3.3 derives
K2 ≤ cuH q um x u2x q
H
≤ cuH q um x L∞ u2x q um x H q u2x ∞
H
L
2
m−1
2
≤ cuH q1 uL∞ ux L∞ .
3.25
It follows from 3.19–3.25 that there exists a constant c depending only on m such that
1 d
2 dt
R
m−1
2
Λq u2 Λq ux 2 εΛq uxx 2 dx ≤ cu2H q1 ux L∞ um
L∞ uL∞ ux L∞ .
3.26
Integrating both sides of the above inequality with respect to t results in 3.15.
−1
To estimate the norm of ut , we apply the operator 1 − ∂2x to both sides of the first
equation of system 3.1 to obtain the equation
1 − εut − εutxx 1 −
∂2x
−1 m 3 m2 u
x
m2
∂3x um2 − m 1∂x um u2x um ux uxx .
−εut − kum ux −
1
m2
3.27
Applying Λq ut Λq to both sides of 3.27 for q ∈ 0, s − 1 gives rise to
1 − ε Λ ut dx ε Λq uxt 2 dx
R
R
m 3 m2
1
k
q
q−2
m1
2 m2
m 2
u
u
∂ u
Λ ut Λ
−
− m 1u ux
−εut ∂x −
m1
m2
m2 x
R
um ux uxx dτ.
q
2
3.28
For the right hand of 3.28, we have
Λq ut Λq−2 −εut dx ≤ εut 2 q ,
H
R
−1
Λq ut 1 − ∂2x Λq ∂x − k um1 − m 3 um2 − m 1um u2x dx
m1
m2
R
q−1 m3
k
m ξ−η u
≤ cut H q
u
1 ξ2
−
η −
η
um1 ξ − η u
m
1
m
2
R
R
2 1/2
m
−m 1u ux ξ − η u
x η dη
m
≤ cut H q uH 1 uH q1 um−1
L∞ uL∞ .
3.29
Abstract and Applied Analysis
9
Since
−1
2
q 2
m1
Λ ut 1 − ∂x Λ ∂x u ux dx − Λq ut Λq um1 ux dx
−1 Λq ut 1 − ∂2x Λq um1 ux dx,
q
3.30
using Lemma 3.3, um1 ux H q ≤ cum2 x H q ≤ cm 2um1
L∞ uH q1 and uL∞ ≤ uH 1 ,
we have
m1
Λq ut Λq um1 ux dx ≤ cut q ux q
H u
H
≤ cut H q um
L∞ uH 1 uH q1 ,
−1
Λq ut 1 − ∂2x Λq um1 ux dx ≤ cut q um∞ u 1 u q1 .
H
H
H
L
3.31
Using the Cauchy-Schwartz inequality and Lemmas 3.1 and 3.3 yields that
−1
Λq ut 1 − ∂2x Λq um ux uxx dx ≤ cut q um ux uxx q−2
H
H
R
≤ cut H q um u2x q−2
x H
≤ cut H q um u2x
− um x u2x q−2
x
H
m 2
m
2
≤ cut H q u ux q−1 u x ux q−2
H H
≤ cut H q um u2x q um x u2x q
H
H
m
2
,
≤ cut H q uH q1 uL∞ ux L∞ um−1
u
∞
∞
x
L
L
3.32
in which we have used 3.25.
Applying 3.29–3.32 into 3.28 yields the inequality
1 − 2εut H q ≤ cuH q1
m
u
um−1
uH 1
∞
∞
L
L
m−1
2
um
L∞ ux L∞ uL∞ ux L∞
3.33
for a constant c > 0. This completes the proof of Lemma 3.5.
Remark 3.6. In fact, letting ε 0 in problem 3.1, 3.14, 3.15, and 3.16 are still valid.
10
Abstract and Applied Analysis
uε0
Setting φε x ε−1/4 φε−1/4 x with 0 < ε < 1/4 and uε0 φε u0 , we know that
∈ C∞ for any u0 ∈ H s , s > 0. From Lemma 3.2, it derives that the Cauchy problem
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
− m 1∂x um u2x um ux uxx ,
ut − utxx εutxxxx −
3.34
u0, x uε0 x, x ∈ R
has a unique solution uε t, x ∈ C∞ 0, ∞; H ∞ .
Furthermore, we have the following.
Lemma 3.7. For s > 0, u0 ∈ H s , it holds that
uε0x L∞ ≤ cu0x L∞ ,
3.35
uε0 H q ≤ c
3.36
if q ≤ s,
uε0 H q ≤ cεs−q/4
uε0 − u0 H q ≤ cεs−q/4
3.37
if q > s,
if q ≤ s,
3.38
uε0 − u0 H s o1,
3.39
where c is a constant independent of ε.
The proof of Lemma 3.7 can be found in 38.
Remark 3.8. For s ≥ 1, using uε L∞ ≤ cuε H 1/2 ≤ cuε H 1 , uε 2H 1 ≤ c
3.36, and 3.37, we know that
u2ε0 u2ε0x εu2ε0xx dx
uε 2L∞ ≤ cuε H 1 ≤ c
R
≤ c uε0 2H 1 εuε0 2H 2
≤ c c cε × εs−2/2
R
u2ε u2εx dx, 3.14,
3.40
≤ c0 ,
where c0 is independent of ε.
Lemma 3.9. If u0 x ∈ H s R with s ≥ 1 such that u0x L∞ < ∞. Let uε0 be defined as in system
3.34. Then there exist two positive constants T and c, which are independent of ε, such that the
solution uε of problem 3.34 satisfies uεx L∞ ≤ c for any t ∈ 0, T .
Abstract and Applied Analysis
11
Proof. Using notation u uε and differentiating both sides of the first equation of problem
3.34 or 3.27 with respect to x give rise to
1 m 2
1
∂2 um2 − m u ux
m2
2
k
k
1 m 2
m1
m2
−2
m1
m2
u
u
u ux
u
− Λ εutx u
m
m1
m1
2
1 ∂x um x u2x .
2
1 − εutx − εutxxx 3.41
Letting p > 0 be an integer and multiplying the above equation by ux 2p1 and then
integrating the resulting equation with respect to x yield the equality
p−m
1−ε d
ux 2p2 dx − ε ux 2p1 utxxx dx ux 2p3 um dx
2p 2 dt R
2p
2
R
R
k
2p1
m1
m2
u
u
dx
ux m1
R
k
1 m 2 1 m
um1 um2 m −
u ux ∂x u x u2x dx.
ux 2p1 Λ−2 εutx m1
2
2
R
3.42
Applying the Hölder’s inequality yields that
1−ε d
2p 2 dt
ux 2p2
1/2p2 1/2p2
m1 2p2
2p2
dx ≤ ε
dx
dx
|utxxx |
u R
R
R
1/2p2 1/2p2 m2 2p2
2p2
dx
dx
|G|
u R
|ux |2p2 dx
×
2p1/2p2
R
R
p−m
m
|ux |2p2 dx
ux L∞ uL∞
2p 2 R
3.43
or
d
1 − ε
dt
ux R
2p2
1/2p2
dx
1/2p2 1/2p2
m1 2p2
≤ ε
dx
|utxxx |2p2 dx
u R
R
1/2p2 1/2p2 m2 2p2
2p2
dx
dx
|G|
u R
R
1/2p2
p−m
2p2
ux ∞ um∞
dx
,
|u
|
x
L
L
2p 2 R
3.44
12
Abstract and Applied Analysis
where
G Λ−2 εutx k
1 m 2 1 m
um1 um2 m u ux ∂x u x u2x .
m1
2
2
3.45
Since fLp → fL∞ as p → ∞ for any f ∈ L∞ L2 , integrating both sides of 3.44 with
respect to t and taking the limit as p → ∞ result in the estimate
1 − εux L∞ ≤ 1 − εu0x L∞
t
1
3.46
m2
m
2
εutxxx L∞ c um1
dτ.
u
G
u
u
∞
∞
∞
∞
∞
x
L
L
L
L
L
2
0
Using the algebraic property of H s0 R with s0 > 1/2 and 3.40 yields that
3.47
m2
um2
L∞ ≤ cuH 1 ≤ c,
GL∞ ≤ cGH 1/2
−2
1 m 2 1 m
m1
m2
2
Λ
u
∂
c
u
u
m
u
u
εu
u
tx
x
x
x x 2
2
1/2
H
−2 −2 m 2 −2
m
2 ≤ c Λ uxt 1/2 Λ u ux 1/2 Λ ∂x u x ux 1/2 c
H
H
H
≤ c ut L2 um u2x 0 um x u2x 0 c
H
H
m
≤ c ut L2 uL∞ ux L∞ uH 1 ux 2L∞ um−1
L∞ uH 1 c
≤ c ut L2 ux L∞ ux 2L∞ c
≤ c 1 ux 2L∞ ,
3.48
where we have used 3.16 and 3.40. Using 3.48, we have
t
0
GL∞ dτ ≤ c
t
0
1 ux 2L∞ dτ,
3.49
where c is a constant independent of ε. Moreover, for any fixed r ∈ 1/2, 1, there exists a
constant cr such that utxxx L∞ ≤ cr utxxx H r ≤ cr ut H r3 . Using 3.16 and 3.40 yields that
utxxx L∞ ≤ cuH r4 1 ux 2L∞ .
3.50
Making use of the Gronwall’s inequality to 3.15 with q r 3, u uε and 3.40 gives rise
to
t
2
2 2
2
r4
r3
Λ u0 ε Λ u0xx
1 ux L∞ dτ .
3.51
exp c
uH r4 ≤
R
0
Abstract and Applied Analysis
13
From 3.36, 3.37, 3.50, and 3.51, one has
utxxx L∞ ≤ cε
s−r−4/4
1
ux 2L∞
t
2
exp c
1 ux L∞ dτ .
3.52
0
For ε < 1/4, it follows from 3.46, 3.49, and 3.52 that
ux L∞ ≤ u0x L∞
t
τ
2
2
2
s−r/4
1 ux L∞ exp c
1 ux L∞ dς 1 ux L∞ dτ.
c
ε
0
3.53
0
It follows from the contraction mapping principle that there is a T > 0 such that the
equation
WL∞ u0x L∞
t τ
2
2
2
1 WL∞ dς 1 WL∞ dτ
1 WL∞ exp c
c
0
3.54
0
has a unique solution W ∈ C0, T . Using the Theorem presented at page 51 in Li and Olver
29 or Theorem II in section I.1 presented in 42 yields that there are constants T > 0 and
c > 0, which are independent of ε, such that ux L∞ ≤ Wt for arbitrary t ∈ 0, T , which
leads to the conclusion of Lemma 3.9.
Lemma 3.10 Li and Olver 29. If u and f are functions in H q1 ∩ {ux L∞ < ∞}, then
⎧
1
⎪
2
⎪
f
,
1
,
,
q
∈
c
u
⎪
q
H
⎪
⎨ q H q1
2
Λq uΛq uf dx ≤
⎪cq f H q1 uH q uL∞
x
⎪
R
⎪
⎪
⎩
fx L∞ u2H q f H q uH q ux L∞ , q ∈ 0, ∞.
3.55
Lemma 3.11. For u, v ∈ H s R with s > 3/2, w u − v, q > 1/2, and a natural number n, it holds
that
Λs wΛs un1 − vn1 dx ≤ c w s w q v s1 w2 s .
H
H
H
H
x
R
The proof of this Lemma can be found in 38.
3.56
14
Abstract and Applied Analysis
Lemma 3.12. For problem 3.34, s > 3/2 and u0 ∈ H s R, there exist two positive constants c and
M, which are independent of ε, such that the following inequalities hold for any sufficiently small ε
and t ∈ 0, T uε H s ≤ Mect ,
uε H sk1 ≤ ε−k1 /4 Mect ,
3.57
3.58
k1 > 0,
uεt H sk1 ≤ ε−k1 1/4 Mect ,
3.59
k1 > −1.
Proof. If s > 3/2, u0 ∈ H s , we obtain that
u0 ∈ H s1
with 1 ≤ s1 ≤
3
,
2
3.60
u0x L∞ ≤ cu0x H 1/2 ≤ cu0 H s ≤ c.
From Lemma 3.9, we know that there exist two constants T and c both independent of ε
such that
uεx L∞ ≤ c
for any t ∈ 0, T .
3.61
Applying the inequality 3.15 with q 1 s and the bounded property of solution u
see 3.40 and 3.60, we have
2
Λ uε dx ≤
s
R
R
Ac
t
0
where
A
R
2
Λ uε0 ε Λ
s
s−1
uε0xx
2 dx c
t
0
uε 2H s dτ,
2 dx ≤ uε0 2H s uε0 2H s1
Λs uε0 2 ε Λs−1 uε0xx
≤ c cεε
−1/2
3.62
uε 2H s dτ,
3.63
≤ 2c,
in which we have used 3.36 and 3.37.
From 3.61 and 3.62 and using the Gronwall’s inequality, we get the following:
uε H s ≤ 2cect ,
3.64
from which we know that 3.57 holds.
In a similar manner, for q 1 s k1 and k1 > 0, applying 3.40 and 3.60 to 3.15,
we have
t
2
−k1 /2
−k1 1/2
3.65
≤
cε
cε
ε
c
uε 2H sk1 dτ,
uε H sk1
0
which results in 3.58 by using Gronwall’s inequality.
Abstract and Applied Analysis
15
From 3.16, for q s k1 , we have
1 − 2εuεt H sk1 ≤ cuε H sk1 1 ,
3.66
which leads to 3.59 by 3.58.
Lemma 3.13. If 1/2 < q < min{1, s − 1} and s > 3/2, then for any functions w, f defined on R, it
holds that
Λq wΛq−2 wf dx ≤ cw2 q f q ,
H
H
x
3.67
Λq wΛq−2 wx fx dx ≤ cw2 q f s .
H
H
x
3.68
R
R
The proof of this lemma can be found in 38.
Our next step is to demonstrate that uε is a Cauchy sequence. Let uε and uδ be solutions
of problem 3.34, corresponding to the parameters ε and δ, respectively, with 0 < ε < δ < 1/4,
and let w uε − uδ . Then w satisfies the problem
1 − εwt − εwxxt δ − εuδt uδxxt −1 k
m1
m2
m2
−
∂
u
∂x um1
1 − ∂2x
−
u
−
u
− εwt δ − εuδt −
x
ε
ε
δ
δ
m1
∂x w ∂x um1
∂x uδ
− ∂x ∂x um1
− um1
ε
ε
ε
1
m
m2
m2
um
∂
u
,
u
u
−
u
u
u
−
u
−
εx
εxx
δx
δxx
x
ε
ε
δ
δ
m2
wx, 0 w0 x uε0 x − uδ0 x.
3.69
3.70
Lemma 3.14. For s > 3/2, u0 ∈ H s R, there exists T > 0 such that the solution uε of 3.34 is
a Cauchy sequence in C0, T ; H s R C1 0, T ; H s−1 R.
Proof. For q with 1/2 < q < min{1, s − 1}, multiplying both sides of 3.69 by Λq wΛq and then
integrating with respect to x give rise to
1 d
2 dt
1 − εΛq w2 εΛq wx 2 dx
R
ε − δ Λq wΛq uδt Λq uδxxt dx
R
q
− ε Λ wΛq−2 wt dx δ − ε Λq wΛq−2 uδt dx
R
R
1
− um2
dx
−
Λq wΛq um2
ε
δ
x
m2 R
k
−
Λq wΛq−2 um1
− um1
dx
ε
δ
x
m1 R
16
Abstract and Applied Analysis
−
R
−
R
−
R
R
m2
Λq wΛq−2 um2
−
u
dx
ε
δ
x
∂x w dx
Λq wΛq−2 ∂x um1
ε
x
∂x uδ dx
Λq wΛq−2 ∂x um1
− um1
ε
δ
x
m
Λq wΛq−2 um
ε uεx uεxx − uδ uδx uδxx dx.
3.71
It follows from the Schwarz inequality that
d 1 − εΛq w2 εΛq wx 2 dx
dt
!
≤ c Λq wL2 δ − εΛq uδt L2 Λq uδxxt L2 εΛq−2 wt 2 δ − εΛq−2 uδt 2
L
L
q
q
m2
m2
q
q−2
m1
m1
Λ wΛ uε − uδ
u ε − uδ
dx Λ wΛ
dx
x
x
R
m2
q
q−2
m1
Λq wΛq−2 um2
∂
u
∂
−
u
dx
Λ
wΛ
w
dx
x
x
ε
ε
δ
x
x
R
Λq wΛq−2 ∂x um1
∂x uδ dx
− um1
ε
δ
x
R
"
q
q−2
m
m
Λ wΛ
uε uεx uεxx − uδ uδx uδxx dx .
3.72
R
Using the first inequality in Lemma 3.10, we have
m2
q
q
Λq wΛq um2
−
u
dx
Λ
wΛ
dx
wg
m1 x
ε
δ
x
R
R
≤ cw2H q gm1 H q1 ,
3.73
#
m1−j j
where gm1 m1
uδ . For the last three terms in 3.72, using Lemmas 3.1 and 3.13,
j0 uε
1/2 < q < min{1, s − 1}, s > 3/2, the algebra property of H s0 with s0 > 1/2 and 3.40, we
have
≤ cw2 q uε m1
Λq wΛq−2 ∂x um1
∂
w
dx
3.74
x
ε
H
Hs ,
x
R
m1
Λq wΛq−2 ∂x um1
∂x uδ dx
− uδ
ε
x
R
m1
m1 ≤ cwH q uδ H s uε − uδ q
H
≤ cw2H q uδ H s ,
Abstract and Applied Analysis
m
Λq wΛq−2 um
ε uεx uεxx − uδ uδx uδxx dx
R
2 m
m
2
2
≤ cwH q um
u
u
−
u
u
−
u
ε
εx
εx
δx x q−2
δ
δ
x
H
2 m 2
m
u
u
≤ cwH q um
−
u
−
q−1 uδ εx u2δx q−2
ε
εx
δ
x H
x H
m 2 2
m
m ≤ cwH q uε − uδ H q uεx q−1 uδ H s uεx − u2δx q−2
x H
x H
s uεx uδx q w q
≤ cwH q wH q gm−1 H q u2H s um
H
H
δ H
s uεx uδx q .
≤ cw2H q gm−1 H q u2H s um
H
δ H
17
3.75
Using 3.67, we derives that the inequality
m2
q
q−2
Λq wΛq−2 um2
− uδ
dx Λ wΛ
wgm1 x dx
ε
x
R
R
≤ cgm1 H q w2H q
holds for some constant c, where gm1 #m1
3.76
m1−j j
uδ .
Using the algebra property of H q
with q > 1/2, q 1 < s and Lemma 3.12, we have gm H q1 ≤ c for t ∈ 0, T$ . Then it follows
from 3.57–3.59 and 3.73–3.76 that there is a constant c depending on T$ such that the
estimate
d
dt
j0
uε
1 − εΛq w2 εΛq wx 2 dx ≤ c δγ wH q w2H q
3.77
R
holds for any t ∈ 0, T$ , where γ 1 if s ≥ 3 q and γ 1 s − q/4 if s < 3 q. Integrating
3.77 with respect to t, one obtains the estimate
1
1
2
Λq w2 dx
wH q 2
2 R
≤
1 − εΛq w2 εΛq w2 dx
3.78
R
t
2
2
q
q
δγ wH q w2H q dτ.
≤
Λ w0 εΛ w0x dx c
R
0
Applying the Gronswall inequality, 3.37 and 3.39 yields that
uH q ≤ cδs−q/4 ect δγ ect − 1 ,
for any t ∈ 0, T$ .
3.79
18
Abstract and Applied Analysis
Multiplying both sides of 3.69 by Λs wΛs and integrating the resultant equation with
respect to x, one obtains that
1 d
2 dt
1 − εΛs w2 εΛs wx 2 dx
R
ε − δ Λs wΛs uδt Λs uδxxt dx
R
s
s−2
− ε Λ wΛ wt dx δ − ε Λs wΛs−2 uδt dx
R
R
k
−
− um1
dx
Λs wΛs um1
ε
δ
x
m1 R
1
− um2
dx
−
Λs wΛs um2
ε
δ
x
m2 R
Λs wΛs−2 um2
− um2
dx
−
ε
δ
x
R
∂
Λs wΛs−2 ∂x um1
w
dx
−
x
ε
x
R
∂
Λs wΛs−2 ∂x um1
− um1
u
dx
−
x
δ
ε
δ
x
R
m
Λs wΛs−2 um
ε uεx uεxx − uδ uδx uδxx dx.
3.80
R
From Lemma 3.13, we have
m2
≤ c3 gm1 s w2 s .
Λs wΛs−2 um2
−
u
dx
ε
H
δ
H
x
3.81
R
From Lemma 3.11, it holds that
m2
≤ c w s w q uδ s1 w2 s .
Λs wΛs um2
−
u
dx
ε
H
H
H
H
δ
x
3.82
R
Using the Cauchy-Schwartz inequality and the algebra property of H s0 with s0 > 1/2, for
s > 3/2, we have
q
s−2
m1
Λs wΛs−2 ∂x um1
∂
∂
u
∂
w
dx
Λ
wΛ
w
dx
x
x
x
ε
ε
x
x
R
R
≤ cΛs wL2 Λs−2 ∂x um1
∂x w 2
ε
x L
∂
≤ cwH q ∂x um1
w
s−1
x
ε
H
2
≤ cuε m1
H s wH s
Λs wΛs−2 ∂x um1
∂x uδ dx ≤ cwH s Λs−2 ∂x um1
∂x uδ 2
− um1
− um1
ε
ε
δ
δ
x
x
L
R
2
≤ cuδ H s gm H s wH s ,
3.83
Abstract and Applied Analysis
19
m
Λs wΛs−2 um
ε uεx uεxx − uδ uδx uδxx dx
R
2 m 2
m
2
≤ cwH s um
u
u
−
u
−
u
u
ε
εx
εx
δx x
δ
δ
s−2
s−2
x
H H
m m
m
2
2
≤ cwH s uε − uδ H s uεx s−2 uδ H s uεx − u2δx 3.84
x H s−2
x H
≤ cw2H s ,
in which we have used Lemma 3.1 and the bounded property of uε H s and uδ H s see
Lemma 3.12. It follows from 3.80–3.84 and 3.57–3.59 and 3.79 that there exists a
constant c depending on m such that
d
dt
1 − εΛs w2 εΛs wx 2 dx
R
≤ 2δ uδt H s uδxxt H s Λs−2 wt 2 Λs−2 uδt wH s
L
c w2H s wH q wH s uδ H s1
≤ c δγ1 wH s w2H s ,
3.85
where γ1 min1/4, s − q − 1/4 > 0. Integrating 3.85 with respect to t leads to the estimate
1
w2H s ≤
2
≤
R
R
1 − εΛs w2 εΛs wx 2 dx
t
δγ1 wH s w2H s dτ.
Λs w0 2 εΛs w0x 2 dx c
3.86
0
It follows from the Gronwall inequality and 3.86 that
1/2
2
ect δγ1 ect − 1
Λs w0 2 εΛs w0x 2 dx
R
≤ c1 w0 H s δ3/4 ect δγ1 ect − 1 ,
wH s ≤
3.87
where c1 is independent of ε and δ.
Then 3.39 and the above inequality show that
wH s −→ 0 as ε −→ 0, δ → 0.
3.88
20
Abstract and Applied Analysis
Next, we consider the convergence of the sequence {uεt }. Multiplying both sides of 3.69 by
Λs−1 wt Λs−1 and integrating the resultant equation with respect to x, we obtain
1
Λs−1 wt Λs−1 um2
− um2
dx
1 − εwt 2H s−1 ε
δ
x
m2 R
−ε Λs−1 wt Λs−1 wxxt δ − ε Λs−1 wt Λs−1 uδt uδxxt dx
R
k
s−1
s−3
m2
m2
∂x um1
Λ wt Λ
−
∂
u
− um1
−
u
−εwt δ − εuδt −
x
ε
ε
δ
δ
m1
R
− ∂x ∂x um1
∂x w ∂x um1
∂x uδ
− um1
ε
ε
ε
m
um
ε uεx uεxx − uδ uδx uδxx dx.
3.89
It follows from 3.57–3.60 and the Schwartz inequality that there is a constant c
depending on T$ and m such that
1 − εwt 2H s−1 ≤ c δ1/2 wH s ws−1 wt H s−1 εwt 2H s−1 .
3.90
1
wt 2H s−1 ≤ 1 − 2εwt 2H s−1
2
≤ c δ1/2 wH s wH s−1 wt H s−1 ,
3.91
1
wt H s−1 ≤ c δ1/2 wH s wH s−1 .
2
3.92
Hence,
which results in
It follows from 3.79 and 3.88 that wt → 0 as ε, δ → 0 in the H s−1 norm. This
implies that uε is a Cauchy sequence in the spaces C0, T ; H s R and C0, T ; H s−1 R,
respectively. The proof is completed.
Proof of Theorem 2.1. We consider the problem
−1 −
1 − εut − εutxx 1 − ∂2x
m 3 m2 1
k m1 u
u
∂3x um2
−
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx ,
u0, x uε0 x.
3.93
Abstract and Applied Analysis
21
Letting ut, x be the limit of the sequence uε and taking the limit in problem 3.93 as ε → 0,
from Lemma 3.14, it is easy to see that u is a solution of the problem
−1 ut 1 − ∂2x
−
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx ,
3.94
u0, x u0 x,
and hence u is a solution of problem 3.94 in the sense of distribution. In particular, if s ≥ 4,
u is also a classical solution. Let u and v be two solutions of 3.94 corresponding to the same
initial data u0 such that u, v ∈ C0, T ; H s R. Then w u − v satisfies the Cauchy problem
−1 ! wt 1 − ∂2x
∂x −
m3
1
k
wgm −
wgm1 ∂2x wgm1
m1
m2
m2
−∂x um1 ∂x w − ∂x um1 − vm1 ∂x v
"
m
m
u ux uxx − v vx vxx ,
3.95
w0, x 0.
For any 1/2 < q < min{1, s−1}, applying the operator Λq wΛq to both sides of equation
3.95 and integrating the resultant equation with respect to x, we obtain the equality
1 d
w2H q 2 dt
! Λq wΛq−2 ∂x −
R
m3
1
k
wgm −
wgm1 ∂2x wgm1
m1
m2
m2
m1
m1
m1
−∂x u
∂x w − ∂x u
∂x v
−v
"
um ux uxx − vm vx vxx dx.
3.96
By the similar estimates presented in Lemma 3.14, we have
d
w2H q ≤ c$w2H q .
dt
3.97
Using the Gronwall inequality leads to the conclusion that
wH q ≤ 0 × ec$t 0
for t ∈ 0, T$ . This completes the proof.
3.98
22
Abstract and Applied Analysis
4. Global Existence of Strong Solutions
We study the differential equation
pt um1 t, p ,
t ∈ 0, T ,
4.1
p0, x x.
Motivated by the Lagrangian viewpoint in fluid mechanics, by which one looks at the
motion of individual fluid particles see 43, we state the following Lemma.
Lemma 4.1. Let u0 ∈ H s , s ≥ 3 and let T > 0 be the maximal existence time of the solution to
problem 2.2. Then problem 4.1 has a unique solution p ∈ C1 0, T × R. Moreover, the map pt, .
is an increasing diffeomorphism of R with px t, x > 0 for t, x ∈ 0, T × R.
Proof. From Theorem 2.1, we have ut, x ∈ C0, T ; H s R C1 0, T ; H s−1 R and
H s R ∈ C1 R, where the Sobolev imbedding theorem is used. Thus, we conclude that
both functions ut, x and ux t, x are bounded, Lipschitz in space and C1 in time. Using the
existence and uniqueness theorem of ordinary differential equations derives that problem
4.1 has a unique solution p ∈ C1 0, T × R.
Differentiating 4.1 with respect to x yields that
d
px m 1um ux t, p px , t ∈ 0, T , b /
0,
dt
px 0, x 1,
4.2
which leads to
px t, x exp
t
m 1u ux τ, pτ, x dτ .
m
4.3
0
For every T < T , using the Sobolev imbedding theorem yields that
sup
|ux τ, x| < ∞.
τ,x∈0,T ×R
4.4
It is inferred that there exists a constant K0 > 0 such that px t, x ≥ e−K0 t for t, x ∈
0, T × R. It completes the proof.
The next Lemma is reminiscent of a strong invariance property of the Camassa-Holm
equation the conservation of momentum 44, 45.
Lemma 4.2. Let u0 ∈ H s with s ≥ 3, and let T > 0 be the maximal existence time of the problem
2.2, it holds that
t
m
y t, pt, x px2 t, x y0 xe 0 mu ux dτ ,
where t, x ∈ 0, T × R and y : u − uxx k/2m 1.
4.5
Abstract and Applied Analysis
23
Proof. We have
d y t, pt, x px2 t, x yt px2 2ypx pxt yx pt px2
dt
yt px2 2ym 1um ux px2 um1 yx px2
yt kum ux m 2um ux y yx um1 px2 mum ux ypx2
ut − utxx kum ux m 2um ux u − uxx um1 ux − uxxx px2
mum ux ypx2
mum ux ypx2 .
4.6
Using px 0, x 1 and solving the above equation, we complete the proof of this lemma.
Lemma 4.3. If u0 ∈ H s , s ≥ 3/2, such that 1 − ∂2x u0 k/2m 1 ≥ 0, then the solution of
problem 2.2 satisfies the following:
ux L∞ ≤ uL∞ k
≤ c.
2m 1
4.7
Proof. Using u0 − u0xx k/2m 1 ≥ 0, it follows from Lemma 4.2 that u − uxx k/2m 1 ≥ 0.
Letting Y1 u − uxx , we have
1
u e−x
2
x
1
e Y1 t, η dη ex
2
−∞
η
∞
e−η Y1 t, η dη,
4.8
x
from which we obtain that
∞
∞
1 −x x η x
−η
x
e Y1 t, η dη e
e Y1 t, η dη e
e−η Y1 t, η dη
∂x ut, x − e
2
−∞
x
x
∞
−ut, x ex
e−η Y1 t, η dη
x
∞
∞
k
k
x
−η
x
dη −
e
e
e−η dη
Y1 t, η −ut, x e
2m 1
2m 1
x
x
∞
k
−ut, x ex
e−η y t, η dη −
2m
1
x
k
≥ −ut, x −
.
2m 1
On the other hand, we have
∞
x
1 −x x η x
−η
−x
∂x ut, x e
e Y1 t, η dη e
e Y1 t, η dη − e
eη Y1 t, η dη
2
−∞
x
−∞
x
ut, x − e−x
eη Y1 t, η dη
−∞
4.9
24
Abstract and Applied Analysis
x
x
k
k
dη e−x
eη Y1 t, η eη dη
ut, x − e−x
2m 1
2m 1
−∞
−∞
x
k
ut, x − e−x
eη y t, η dη 2m 1
−∞
k
.
≤ ut, x 2m 1
4.10
The inequalities 3.40, 4.9, and 4.10 derive that 4.7 is valid.
Proof of Theorem 2.2. Noting Remarks 3.6 and 3.8, uH 1 ≤ c and taking q 1 s in inequality
3.15, we have
u2H s ≤ u0 2H s c
t
0
u2H s ux L∞ ux 2L∞ dτ,
4.11
from which we obtain that
uH s ≤ u0 H s ec
t
2
0 ux L∞ ux L∞ dτ
.
4.12
Applying Lemma 4.3 derives
uH s ≤ u0 H s ecc t .
2
4.13
From Theorem 2.1 and 4.13, we know that the result of Theorem 2.2 holds.
Acknowledgment
This work is supported by the Applied and Basic Project of Sichuan Province 2012JY0020.
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