Jia and Kang Boundary Value Problems ( 2016) 2016:218
DOI
10.1186/s13661-016-0726-0
R E S E A R C H Open Access
An asymptotic property of the
Camassa-Holm equation on the half-line
Jia Jia and Shunguang Kang
*
*
Correspondence:
kangshunguang@163.com
College of Information Engineering,
Tarim University, Alar, 843300,
P.R. China
Abstract
The paper addresses the asymptotic properties of Camassa-Holm equation on the
half-line. That is, using the method of asymptotic density, under the assumption that
it is unique, the paper proves that the positive momentum density of the
Camassa-Holm equation i s a combination of Dirac measures supported on the
positive axis. This means that as time goes to infinity, the momentum density
concentrates in small intervals moving right with different constant speeds.
MSC: 35Q53; 37K40
Keywords: Camassa-Holm equation; asymptotic density; momentum density;
asymptotic property; Dirac measures
1 Introduction
In this paper we consider the following initial boundary value problem of the C amassa-
Holm equation on the half line:
⎧
⎪
⎨
⎪
⎩
u
t
– u
txx
+uu
x
=u
x
u
xx
+ uu
xxx
, t >,x ∈ R
+
,
u(, x)=u
(x), x ∈ R
+
,
u
(k)
x
(t,)=u
(k–)
x
(t,)=···= u(t,)=, t ≥ ,
(.)
where u
(k)
() = u
(k–)
() = ··· = u
(). Let h = u – u
xx
be the momentum density. The
Camassa-Holm equation on the half line in momentum density form is
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
h
t
+ uh
x
+u
x
h =, t >,x ∈ R
+
,
h(t, x)=(–∂
x
)u(t, x), t >,x ∈ R
+
,
h(x,)=h
(x)=(–∂
x
)u
, x ∈ R
+
,
∂
k–
x
h(t,)=···= ∂
x
h(t,)=h(t,)=, t ≥ .
(.)
The Camassa-Holm equation is a model for the unidirectional propagation of shallow
water w aves over a flat bottom. It has a bi-Hamiltonian structure [
] and is completely
integrable [
, ]. Its solitary waves are peaked []. The convergence of the solution of the
Camassa-Holm equation to the distributional solution of the Burgers one and the solution
of the dispersive equation converging to the unique entropy solution of a scal ar conser-
vation law are proven [
, ]. In [], numerical studies illustrate that some nonnegative
initial condition evolves into a train of peakons moving with different velocities. On the
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-
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indicate if changes were made.
Jia and Kang Boundary Value Problems ( 2016) 2016:218 Page 2 of 14
theoretical side of the topic, there is research on the stability of p eakons, which says that
at least for initial values close to peakons, it will stay close to the peakons. The shapes
of peakons and multipeakon/antipeakons are stable under small perturbations, making
them recognizable physically [
–]. Constantin and Strauss [] proved the stability of a
single peakon among C([; T]; H
(R)) solutions. El Dika and Molinet proved the stability
of multi-p eakons [
] and multi-anti-p eakons-peakons [] among a slightly more regular
class of solutions.
In this article, we u se the method of asymptotic density of the momentum to show that
under the assumption of its u niqueness, at the momentum level, the solution approaches
a train of peakons moving right with different speeds. This approach was introduced by
Chen and Frid [
] to study the asymptotic behavior of the entropy solutions of conserva-
tion laws. It has been used to discuss the asymptotic behavior of the vorticity of the two di-
mensional incompressible Euler equation by Iftimie, Lopes, and Nussenzveig [
, ]. Ref-
erence [
] is an exposition. Notice that the vorticity in the incompressible Euler equations
and the momentum density in the Camassa-Holm equation are similar. They satisfy simi-
lar first order nonlinear nonlocal equations and give the velocity through similar integrals.
There are also differences. The D Euler flow preserves volume and the vorticity is trans-
por ted along the particle trajectories, but the same do not hold for the Camassa-Holm
flow and its momentum density. We have studied the asymptotic property for a global
strong solutions of the Camassa-Holm equation by the approach in [
], and the asymp-
toticpropertyofthesolutionsoftheDegasperis-Procesiequationisstudied[
]. Results
on the half-line may not automatically follow those on the whole line. For example, if the
momentum density is non-positive, the solution on R approaches a left moving peakon
train whereas on R
+
, solutions with non-po sitive initial momentum densities must blow
up in finite time (see [
]) and we cannot di scuss their asymptotic prop erties. New, we pro-
ceed to a study of the momentum density of the Camassa-Holm equation on the half line.
In this paper, we will investigate initial boundary value problems of the Camassa-Holm
equation with initial data u
∈ H
s
(R
+
) ∩ H
(R
+
), s >
,whereR
+
=[,∞). Let k ∈ N \{},
and for k +
< s ≤ k +,weset
G
s
(R
+
)=
u ∈ H
s
(R
+
) | u
(k)
() = u
(k–)
() = ···= u()
.
Theorem . ([
], Theorem .) Let u
∈ G
s
(R
+
), with k ∈ N \{}, and k +
< s ≤ k +.
Assume that h
(x):=u
(x)–u
,xx
(x) ≥ for all x ∈ R
+
. Then there exists a global solution
u(t, x) to (
.) such that, for all T >,
u = u(·, u
) ∈ C
[, T); G
s
(R
+
)
∩ C
[, T); G
s–
(R
+
)
.
Moreover, the solution depends continuously on the initial data, i.e., the mapping u
−→
u(·, u
):G
s
(R
+
) −→ C([, T); G
s
(R
+
)) ∩ C
([, T); G
s–
(R
+
)) is continuous.
Definition . Let u be a global solution of (
.), and the initial momentum density h
≥
is compactly supported. For t >,let
˜
h(t, y):=th(t, ty)(.)
be the scaled momentum density of u.
Jia and Kang Boundary Value Problems ( 2016) 2016:218 Page 3 of 14
Definition . Let [a, b ] ⊂ R
+
be a finite interval, and supp
˜
h(t, ·) ⊂ [a, b] for all t ≥ .
Suppose there is a sequence t
k
→∞as k →∞, and a positive Radon measure μ ∈ M[a, b]
such that
˜
h(t
k
, ·) ⇀μ,ask →∞.
Then we call μ an asymptotic density associated with the initial momentum density h
.
Remark .
(a) M[a, b] is the space of regular Borel measures on [a, b].
(b) The convergence is the weak-∗ convergence in
M[a, b], i.e. for all ψ ∈ C[a, b],
b
a
˜
h(t, y)ψ(y)dy →μ, ψ.
(c) The asymptotic densities associated with h
may not be unique.
The following is the main result of this article.
Theorem . Let u be a global solution of (
.), and h
(·)=h(, ·) ≥ has compact support.
For t ≥ , suppose that
˜
h(t, ·) has a unique asymptotic density μ associated with h
. Then
there exist finitely or countably infinitely many m
i
, α
i
∈ [, ∞) such that
μ =
∞
i=
m
i
δ
α
i
,(.)
where δ
α
i
is the delta function supported at α
i
, and
(a) α
i
≤
m
i
, for all i.
(b) α
i
∈ [, M), for any i, where M = u
L
∞
(R
+
×R
+
)
.
(c) α
i
→ as i →∞and α
i
= α
j
if i = j.
(d) m
i
>and
∞
i=
m
i
= h
L
(R
+
)
.
In other words, the momentum densities of such global solutions concentrates in slumps
moving right approaching different speeds.
2Preliminaries
Consider the following differential equation:
dq(t,x)
dt
= u(t, q(t, x)), t >,x ∈ R
+
,
q(, x)=x, x ∈ R
+
.
(.)
Applying classical results in the theory of ordinary differential equations, one can obtain
the following results on q.
Lemma . Let u ∈ C([, T); G
s
(R
+
)) ∩ C
([, T); G
s–
(R
+
)) be a nonnegative solution of
(
.) for all T >,Then the ( .) has a unique solution q ∈ C
([, T) × R
+
, R
+
). Moreover,
the map q(t, ·) is an increasing diffeomorphism of R
+
. And
h
t, q(t, x)
q
x
(t, x)=h
(x), (t, x) ∈
[, T) × R
+
.
Jia and Kang Boundary Value Problems ( 2016) 2016:218 Page 4 of 14
Remark .
(a) If supp h
⊂ [a, b],then,forallt ∈ (, T), supp h(t, ·) ⊂ [q(t, a), q(t, b)].
(b) If h
≥ ,then,fort ∈ [, T), h(t, ·) ≥ .
Lemma . Let u be a global solution of (
.). Suppose that h
∈ L
(R
+
) and h
≥ . Then,
for t ≥ , we have
˜
h(t, ·)
L
(R
+
)
=
h(t, ·)
L
(R
+
)
=
h
(t, ·)
L
(R
+
)
. (.)
Proof From h
≥ andRemark
.(b) h(t, ·) ≥ , for t > , the first equality of (.)results
from a change of variable. The seconde equality can be got from (
.) using integration by
parts . That is ,
d
dt
R
+
h(t, x)dx =
R
+
(–uh
x
–u
x
h)dx =
R
+
–u
x
h dx =.
Proposition . Let u is a global solution of (
.), and h is the momentum density of u.
Then
u(t, x)=
R
+
e
–|x–y|
–e
–|x+y|
h(t, y)dy.(.)
Proof Let
h(t, x)=
⎧
⎨
⎩
h(t, x), x ≥ ,
–h(t,–x), x <,
u(t, x)=
⎧
⎨
⎩
u(t, x), x ≥ ,
–u(t,–x), x <.
For convenience, the following proof omits the t.Whenx <,
u
′′
(x)=
d
dx
u(x)=
d
dx
–u(–x)
=–u
′′
(–x).
Then
u(x)–u
′′
(x)=–u(–x)+u
′′
(–x)=–
u(–x)–u
′′
(–x)
=–h(–x)=h(x).
So
h(t, x) is the momentum density of u(t, x) on the whole line. Then
u(x)=
R
e
–|x–y|
h(y)dy =
R
+
e
–|x–y|
h(y)dy +
R
–
e
–|x–y|
h(y)dy
= I + II.
Because
I =
R
+
e
–|x–y|
h(y)dy =
R
+
e
–|x–y|
h(y)dy
Jia and Kang Boundary Value Problems ( 2016) 2016:218 Page 5 of 14
and
II =
R
–
e
–|x–y|
h(y)dy =
R
–
e
–|x–y|
–h(–y)
dy
=–
R
+
e
–|x+z|
h(z)dz,
we have
u(x)=I + II =
R
+
e
–|x–y|
– e
–|x+y|
h(t, y)dy.
We record here two formulas frequently used later. From (
.), we get
u(t, tx)=
R
+
e
–|tx–y|
– e
–|tx+y|
h(t, y)dy =
R
+
e
–t|x–z|
– e
–t|x+z|
h(t, z)dz.(.)
Differentiate (
.) with respect to the spatial variable to get
u
x
(t, x)=
R
+
sgn(y – x)e
–|x–y|
+ e
–|x+y|
h(t, y)dy
=
R
+
sgn(tz – x)e
–|x–tz|
+ e
–|x+tz|
h(t, z)dz.
Hence
u
x
(t, tx)=
R
+
sgn(z – x)e
–t|x–z|
+ e
–t|x+z|
h(t, z)dz.(.)
Lemma . Let u be a global solution of (
.). Suppose that h
≥ and supp(h
) ⊂ [a, b] ⊂
R
+
. Then, for t ≥ ,
supp
˜
h(t, ·) ⊂
a
t
,
b
t
+ M
.(.)
Here M := u
L
∞
(R
+
×R
+
)
. There exists d > M, such that, for t ≥ , supp
˜
h(t, ·) ⊂ [, d].
Proof Notice that u
∈ G
s
(R
+
), with k ∈ N \{},andk +
< s ≤ k +,impliesthat
h
∈ H
s–
(R
+
) ⊂ L
∞
(R
+
), and that the h
has compact support implies that h
∈ L
(R
+
).
From (
.)andtheRemark.(b), h(t, ·) ≥ for all t ≥ . Equation (.) and Lemma .
imply that, for all (t, x) ∈ [, ∞) × R
+
,
≤ u(t, x) ≤
h(t, ·)
L
(R
+
)
=
h
L
(R
+
)
< ∞.(.)
Let M := u
L
∞
(R
+
×R
+
)
.BytheRemark
.(a), supp h(t, ·) ⊂ [a, b + Mt], then
supp
˜
h(t, ·) ⊂
a
t
,
b
t
+ M
.