In this article, we study the perturbational method to construct the
non-radially symmetric solutions of the compressible 2-component Camassa-Holm
equations. In detail, we first combine the substitutional method and the
separation method to construct a new class of analytical solutions for that
system. In fact, we perturb the linear velocity: u=c(t)x+b(t), and substitute
it into the system. Then, by comparing the coefficients of the polynomial, we
can deduce the functional differential equations involving
$(c(t),b(t),\rho^{2}(0,t)).$ Additionally, we could apply the Hubble's
transformation c(t)={\dot{a}(3t)}/{a(3t)}, to simplify the ordinary
differential system involving $(a(3t),b(t),\rho ^{2}(0,t))$. After proving the
global or local existences of the corresponding dynamical system, a new class
of analytical solutions is shown. And the corresponding solutions in radial
symmetry are also given. To determine that the solutions exist globally or blow
up, we just use the qualitative properties about the well-known Emden equation:
{array} [c]{c} {d^{2}/{dt^{2}}}a(3t)= {\xi}{a^{1/3}(3t)}, a(0)=a_{0}>0
,\dot{a}(0)=a_{1} {array} . Our solutions obtained by the perturbational
method, fully cover the previous known results in "M.W. Yuen,
\textit{Self-Similar Blowup Solutions to the 2-Component Camassa-Holm
Equations,}J. Math. Phys., \textbf{51} (2010) 093524, 14pp." by the separation
method.Comment: 12 page

Camassa

Chen

Constantin

Deng

Escher

Goldreich

Guan

Hale

Liang

Makino

Manwai Yuen

Walter

Whitham

Yuen

English

arXiv

AbstractIn this article, we study the perturbational method to construct the non-radially symmetric solutions of the compressible 2-component Camassa–Holm equations. In detail, we first combine the substitutional method and the separation method to construct a new class of analytical solutions for that system. In fact, we perturb the linear velocity:(1)u=c(t)x+b(t), and substitute it into the system. Then, by comparing the coefficients of the polynomial, we can deduce the functional differential equations involving (c(t),b(t),ρ2(0,t)). Additionally, we could apply Hubbleʼs transformation c(t)=a˙(3t)a(3t), to simplify the ordinary differential system involving (a(3t),b(t),ρ2(0,t)). After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation. Our solutions obtained by the perturbational method, fully cover Yuenʼs solutions by the separation method

Yuen, Manwai

Elsevier - Publisher Connector 

J. Math. Anal. Appl. 390 (2012) 596–602Contents lists available at ScienceDirect
Journal of Mathematical Analysis and
Applications
www.elsevier.com/locate/jmaa
Note
Perturbational blowup solutions to the 2-component Camassa–Holm
equations
Manwai Yuen
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
a r t i c l e i n f o a b s t r a c t
Article history:
Received 10 March 2011
Available online 11 May 2011
Submitted by J. Xiao
Keywords:
Camassa–Holm equations
Perturbational method
Non-radial symmetry
Construction of solutions
Functional differential equations
Dynamical system
Global existence
Blowup
Radial symmetry
Emden equation
In this article, we study the perturbational method to construct the non-radially symmetric
solutions of the compressible 2-component Camassa–Holm equations. In detail, we ﬁrst
combine the substitutional method and the separation method to construct a new class of
analytical solutions for that system. In fact, we perturb the linear velocity:
u = c(t)x+ b(t), (1)
and substitute it into the system. Then, by comparing the coeﬃcients of the polynomial, we
can deduce the functional differential equations involving (c(t),b(t),ρ2(0, t)). Additionally,
we could apply Hubble’s transformation c(t) = a˙(3t)a(3t) , to simplify the ordinary differential
system involving (a(3t),b(t),ρ2(0, t)). After proving the global or local existences of the
corresponding dynamical system, a new class of analytical solutions is shown. To determine
that the solutions exist globally or blow up, we just use the qualitative properties about
the well-known Emden equation. Our solutions obtained by the perturbational method,
fully cover Yuen’s solutions by the separation method.
© 2011 Elsevier Inc. All rights reserved.
1. Introduction
The 2-component Camassa–Holm equations of shallow water system can be expressed by{
ρt + uρx + ρux = 0, x ∈ R,
mt + 2uxm + umx + σρρx = 0 (2)
with
m = u − α2uxx. (3)
Here u = u(x, t) ∈ R and ρ = ρ(x, t) 0 are the velocity and the density of ﬂuid respectively. The constant σ is equal to 1
or −1. If σ = −1, the gravity acceleration points upwards [2,3,8,10,9]. For σ = 1, the researches regarding the corresponding
models could be referred to [4,6,10,8]. When ρ ≡ 0, the system returns to the Camassa–Holm equation [1]. The searching of
the Camassa–Holm equation can capture breaking waves. Peaked traveling waves is a long-standing open problem [18].
In 2010, Yuen used the separation method to obtain a class of blowup or global solutions of the Camassa–Holm equations
[23] and the Degasperis–Procesi equations [24]. In particular, for the integrable system of the Camassa–Holm equations with
σ = 1, we have the global solutions:
E-mail address: nevetsyuen@hotmail.com.0022-247X/$ – see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2011.05.016
M.W. Yuen / J. Math. Anal. Appl. 390 (2012) 596–602 597⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
ρ(x, t) = max
{
f (η)
a(3t)1/3
,0
}
, u(x, t) = a˙(3t)
a(3t)
x,
a¨(s) − ξ
3a(s)1/3
= 0, a(0) = a0 > 0, a˙(0) = a1,
f (η) = ξ
√
−η
2
ξ
+ (ξα)2
(4)
where η = x
a(s)1/3
with s = 3t; ξ > 0 and α  0 are arbitrary constants [23].
Meanwhile, the isentropic compressible Euler equations can be written in the following form:
{
ρt + ∇ · ρu = 0,
(ρu)t + ∇ · (ρu ⊗ u) + ∇ P = 0. (5)
As usual, ρ = ρ(x, t) and u = u(x, t) ∈ RN are the density and the velocity respectively with x = (x1, x2, . . . , xN ) ∈ RN . For
some ﬁxed K > 0, we have a γ -law on the pressure:
P = P (ρ) = Kργ (6)
with a constant γ  1. For solutions in radially symmetry:
ρ(x, t) = ρ(r, t) and u(x, t) = x
r
V (r, t) =: x
r
V (7)
with the radial r =∑Ni=1 x2i , the compressible Euler equations (5) become⎧⎪⎨
⎪⎩
ρt + Vρr + ρVr + N
r
ρV = 0,
ρ(Vt + V Vr) + ∂
∂r
P = 0.
(8)
Recently, there are some researches concerning the construction of solutions of the compressible Euler and Navier–Stokes
equations by the substitutional method [12,13,19,14]. They assume that the velocity is linear:
u(x, t) = c(t)x (9)
and substitute it into the system to derive the dynamic system about the function c(t). Then they use the standard argument
of phase diagram to derive the blowup or global existence of the ordinary differential equation involving c(t).
On the other hand, the separation method can be governed to seek the radial symmetric solutions by the functional
form:
ρ(r, t) = f (
r
a(t) )
aN(t)
and V (r, t) = a˙(t)
a(t)
r. (10)
(See [7,15,16,5,12,19–22].)
It is natural to consider the more general linear velocity:
u(x, t) = c(t)x+ b(t) (11)
to construct new solutions. In this article, we can ﬁrst combine the two conventional approaches (substitutional method
and separation method) to derive the corresponding solutions for the system. In fact, the main theme of this article is
to substitute the linear velocity (11) into the Camassa–Holm equations (2) and compare the coeﬃcient of the different
polynomial degrees for deducing the functional differential equations involving (c(t),b(t),ρ2(0, t)). Then, we can apply
Hubble’s transformation
c(t) = a˙(3t)
a(3t)
(12)
with a˙(3t) := da(3t)dt to simplify the functional differential system involving (a(3t),b(t),ρ2(0, t)). After proving the local
existences of the corresponding dynamical system, we can show the results below:
598 M.W. Yuen / J. Math. Anal. Appl. 390 (2012) 596–602Theorem 1. For the 2-component Camassa–Holm equations (2), there exists a family of solutions:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
ρ2(x, t) =max
{
ρ2(0, t) − 2
σ
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
x− 3ξ
σa
4
3 (3t)
x2,0
}
,
u(x, t) = a˙(3t)
a(3t)
x+ b(t),
d2
dt2
a(3t) = ξ
a
1
3 (3t)
, a(0) = a0 > 0, a˙(0) = a1,
d2
dt2
b(t) + 6a˙(3t)
a(3t)
d
dt
b(t) + 12ξ
a
4
3 (3t)
b(t) = 0, b(0) = b0, b˙(0) = b1,
d
dt
[
ρ2(0, t)
]+ 2a˙(3t)
a(3t)
ρ2(0, t) = 2b(t)
σ
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
, ρ2(0,0) = α2
(13)
where ξ , a0 , a1 , b0 , b1 and α are arbitrary constants.
To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known
Emden equation (13)3 in Corollary 4 in the end of the article. We remark that the above solutions (13) fully cover Yuen’s
solutions in [23] by the separation method by choosing b0 = b1 = 0.
2. Perturbational method
The proof of Theorem 1 requires the standard manipulation of algebraic computation only:
Proof. The momentum equation (2)2, becomes(
u − α2uxx
)
t + 2ux
(
u − α2uxx
)+ u(u − α2uxx)x + σρρx = 0. (14)
First, we perturb the velocity with this following functional form:
u(x, t) = c(t)x+ b(t) (15)
where b(t) and c(t) are the time functions determined later.
As the velocity u (15) is linear:
uxx = 0, (16)
it can be simpliﬁed to be
ut + 3uux + σρρx = 0, (17)
c˙(t)x+ b˙(t) + 3[c(t)x+ b(t)]c(t) + σ
2
∂
∂x
ρ2 = 0, (18)
σ
2
∂
∂x
ρ2 = −[b˙(t) + 3b(t)c(t)]− [c˙(t) + 3c2(t)]x. (19)
Then, we take integration from [0, x] to have:
σ
2
x∫
0
∂
∂s
ρ2 ds = −[b˙(t) + 3b(t)c(t)]
x∫
0
ds − [c˙(t) + 3c2(t)]
x∫
0
s ds, (20)
σ
2
[
ρ2(x, t) − ρ2(0, t)]= −[b˙(t) + 3b(t)c(t)]x− [c˙(t) + 3c2(t)]
2
x2, (21)
ρ2(x, t) = ρ2(0, t) − 2
σ
[
b˙(t) + 3b(t)c(t)]x− [c˙(t) + 3c2(t)]
σ
x2. (22)
On the other hand, for the 1-dimensional mass equation (2)1, we obtain
ρt +
[
c(t)x+ b(t)]ρx + ρc(t) = 0. (23)
Here, we multiply ρ on both sides to have
1 (
ρ2
)
t +
[c(t)x+ b(t)] (
ρ2
)
x + ρ2c(t) = 0. (24)2 2
M.W. Yuen / J. Math. Anal. Appl. 390 (2012) 596–602 599After that, we can substitute Eq. (22) into Eq. (24):
1
2
(
∂
∂t
[
ρ2(0, t)
]− 2
σ
∂
∂t
[
b˙(t) + 3b(t)c(t)]x− ∂
∂t
[c˙(t) + 3c2(t)]
σ
x2
)
(25)
+ [c(t)x+ b(t)](− 1
σ
[
b˙(t) + 3b(t)c(t)]− 1
σ
[
c˙(t) + 3c2(t)]x) (26)
+ c(t)
[
ρ2(0, t) − 2
σ
[
b˙(t) + 3b(t)c(t)]x− [c˙(t) + 3c2(t)]
σ
x2
]
(27)
= 1
2
∂
∂t
[
ρ2(0, t)
]+ c(t)ρ2(0, t) − b(t)
σ
[
b˙(t) + 3b(t)c(t)] (28)
+
{− 1σ ∂∂t [b˙(t) + 3b(t)c(t)] − c(t)σ [b˙(t) + 3b(t)c(t)]
− b(t)σ [c˙(t) + 3c2(t)] − 2c(t)σ [b˙(t) + 3b(t)c(t)]
}
x (29)
+
{− 12σ ∂∂t [c˙(t) + 3c2(t)] − 1σ [c˙(t) + 3c2(t)]c(t)
− c(t)[c˙(t)+3c2(t)]σ
}
x2. (30)
By comparing the coeﬃcients of the polynomial, we require the functional differential equations involving (c(t),b(t),
ρ2(0, t)):⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
d
dt
[
ρ2(0, t)
]+ 2c(t)ρ2(0, t) − 2
σ
b(t)
[
b˙(t) + 3b(t)c(t)]= 0,
d
dt
[
b˙(t) + 3b(t)c(t)]+ 3c(t)2[b˙(t) + 3b(t)c(t)]+ b(t)[c˙(t) + 3c2(t)]= 0,
d
dt
[
c˙(t) + 3c2(t)]+ 4[c˙(t) + 3c2(t)]c(t) = 0.
(31)
For details (existence, uniqueness and continuous dependence) about general functional differential equations, the interested
reader may refer to the classical references [11] and [17].
For solving the above ordinary differential system (31), we initially solve Eq. (31)3 about the function c(t). Here we let
the function c(t) be expressed with Hubble’s transformation:
c(t) = a˙(3t)
a(3t)
(32)
where a˙(3t) := da(3t)dt and the function a(3t) could be determined later.
It is transformed to be
d
dt
[
3a¨(3t)
a(3t)
− 3a˙
2(3t)
a2(3t)
+ 3a˙
2(3t)
a2(3t)
]
+ 4
[
3a¨(3t)
a(3t)
− 3a˙
2(3t)
a2(3t)
+ 3a˙
2(3t)
a2(3t)
]
a˙(3t)
a(3t)
= 0, (33)⎧⎨
⎩
d
dt
(
a¨(3t)
a(3t)
)
+ 4a¨(3t)
a(3t)
a˙(3t)
a(3t)
= 0,
a(0) = a0 > 0, a˙(0) = a1, a¨(0) = a2,
(34)
3
...
a(3t)
a(3t)
− 3a˙(3t)a¨(3t)
a2(3t)
+ 4a˙(3t)a¨(3t)
a2(3t)
= 0, (35)
...
a(3t)
a(3t)
+ a˙(3t)a¨(3t)
3a2(3t)
= 0. (36)
Then, we multiply a2(3t) on both sides to have:
a(3t)
...
a(3t) + a˙(3t)a¨(3t)
3
= 0. (37)
It can be reduced to the second-order Emden equation:⎧⎨
⎩
d2
dt2
a(3t) = ξ
a
1
3 (3t)
,
a(0) = a0 > 0, a˙(0) = a1
(38)
where ξ := a
1
3 a2 is an arbitrary constant by choosing a2.0
600 M.W. Yuen / J. Math. Anal. Appl. 390 (2012) 596–602We remark that the well-known Emden equation is well studied in astrophysics and mathematics.
Next, for the second equation (31)2 about b(t) of the functional differential system, we could further simplify it in terms
of the known function a(3t):
d
dt
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
+ 3 a˙(3t)
a(3t)
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
+ 3a¨(3t)
a(3t)
b(t) = 0, (39)
b¨(t) + 6 a˙(3t)
a(3t)
b˙(t) +
[
9
a¨(3t)
a(3t)
− 9 a˙
2(3t)
a2(3t)
+ 9a˙
2(3t)
a2(3t)
+ 3a¨(3t)
a(3t)
]
b(t) = 0, (40)⎧⎪⎨
⎪⎩
b¨(t) + 6 a˙(3t)
a(3t)
b˙(t) + 12ξ
a
4
3 (3t)
b(t) = 0,
b(0) = b0, b˙(0) = b1
(41)
with the Emden equation (38).
We denote f1(t) = 6a˙(3t)a(3t) and f2(t) = 12ξ
a
4
3 (3t)
to have
{
b¨(t) + f1(t)b˙(t) + f2(t)b(t) = 0,
b(0) = b0, b˙(0) = b1.
(42)
Therefore, when the functions f1(t) and f2(t) are bounded, that is∣∣ f1(t)∣∣ F1 and ∣∣ f2(t)∣∣ F2 (43)
with some constants F1 and F2, provided that the functions 1a(3t) and a˙(3t) exist, the functions b(t) and b˙(t) can be
guaranteed for existing by the comparison theorem of ordinary differential equations [18].
Lastly, for the ﬁrst equation (31)1, we denote H(t) = 2a˙(3t)a(3t) and G(t) = 2b(t)σ [b˙(t) + 3b(t) a˙(3t)a(3t) ] in terms of functions 1a(3t) ,
a(3t), b(t) and b˙(t) provided that they exist, to solve⎧⎨
⎩
d
dt
[
ρ2(0, t)
]+ ρ2(0, t)H(t) = G(t),
ρ2(0,0) = α2.
(44)
The formula of the ﬁrst-order ordinary differential equation (44) is
ρ2(0, t) =
∫ t
0 μ(s)G(s)ds + k
μ(t)
(45)
where
μ(t) = e
∫ t
0 H(s)ds. (46)
Therefore, we have the density function from Eq. (22):
ρ2(x, t) = ρ2(0, t) − 2
σ
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
x− 3ξ
σa
4
3 (3t)
x2. (47)
For ρ(x, t) 0, we may set
ρ2(x, t) =max
{
ρ2(0, t) − 2
σ
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
x− 3ξ
σa
4
3 (3t)
x2,0
}
. (48)
In conclusion, we have the corresponding functional differential equations (13) to be the solutions of Camassa–Holm equa-
tions.
The proof is completed. 
We notice that the above solutions are not radially symmetric for the density function ρ with b(t) = 0. Thus, the above
solutions cannot be obtained by the separation method of the self-similar functional [23], as
ρ(x, t) = f
(
x
a(3t)
)
g
(
a(3t)
)
and u(x, t) = a˙(3t)
a(3t)
x+ b(t). (49)
M.W. Yuen / J. Math. Anal. Appl. 390 (2012) 596–602 601On the other hand, for the 2-component Camassa–Holm equations in radial symmetry with linear velocity u(r, t):{
ρt + Vρr + ρVr = 0,
Vt + 3V Vr + σρρr = 0, (50)
we may replace Eq. (20) to have the corresponding step by taking the integration from [0, r]
σ
2
r∫
0
∂
∂s
ρ2 ds = −[b˙(t) + b(t)c(t)]
r∫
0
ds − 3[c˙(t) + c2(t)]
r∫
0
s ds. (51)
It is clear for that the rest of the proof is similar to have the corresponding result for the solutions in radial symmetry:
Theorem 2. For the 2-component Camassa–Holm equations in radial symmetry (2), there exists a family of solutions:⎧⎪⎪⎨
⎪⎪⎩
ρ2(r, t) = max
{
ρ2(0, t) − 2
σ
[
b˙(t) + 3b(t) a˙(3t)
a(3t)
]
r − 3ξ
σa
4
3 (3t)
r2,0
}
,
V (r, t) = a˙(3t)
a(3t)
r + b(t)
(52)
where a(3t), b(t) and ρ2(0, t) are the solutions of Eqs. (13)3–5.
3. Blowup or global solutions
To determine that the solutions are global or local only, we can use the corresponding lemma about the Emden equation:
Lemma 3. For the Emden equation (13)3,⎧⎨
⎩ a¨(3t) =
ξ
a
1
3 (3t)
,
a(0) = a0 > 0, a˙(0) = a1,
(53)
(1) if ξ < 0, there exists a ﬁnite time T , such that
lim
t→T−
a(3t) = 0, (54)
(2) if ξ = 0, with a1 < 0, the solution a(t) blows up in the ﬁnite time:
T = −a0
a1
, (55)
(3) otherwise, the solution a(t) exists globally.
We observe that it is the same lemma for the function a(s) with s = 3t by the separation method in [23]. Therefore, the
proof can be found in Lemma 3 of [23].
The gradient of the velocity in solutions (13) and (52) is
∂
∂x
u(x, t) = ∂
∂r
u(r, t) = a˙(3t)
a(3t)
. (56)
When the function a(t) blows up with a ﬁnite time T , ∂
∂x u(x, T ) also blows up at every point x. And based on the above
lemma about the Emden equation for a(t), it is clear to have the corollary below:
Corollary 4. (1a) For ξ < 0, solutions (13) and (52) blow up in a ﬁnite time T .
(1b) For ξ = 0, with a1 < 0, solutions (13) and (52) blow up in the ﬁnite time:
T = −a0
a1
. (57)
(2) Otherwise, solutions (13) and (52) exist globally.
For the graphical illustration of the classical blowup solution (13) with the inﬁnitive mass by choosing the parameters
α = 1, b1 = σ2 , a0 = 1, a1 = 0 and ξ = −σ3 , we can see the initial shape of the non-radially symmetric solution in Fig. 1.
For the global breaking solutions (13), we choose the parameters α = 1, b1 = σ2 , a0 = 1, a1 = 0 and ξ = σ3 to have the
graph as in Fig. 2.
602 M.W. Yuen / J. Math. Anal. Appl. 390 (2012) 596–602Fig. 1. ρ0(x) =
√
1− x+ x2. Fig. 2. ρ0(x) = max(
√
1− x− x2,0).
Remark 5. We also apply this perturbational method to obtain new solutions with drift phenomena for the 1-dimensional
compressible Euler equations in [25].
References
[1] R. Camassa, D.D. Holm, Integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664.
[2] M. Chen, S.-Q. Liu, Y. Zhang, A 2-component generalization of the Camassa–Holm equation and its solutions, Lett. Math. Phys. 75 (2006) 1–15.
[3] A. Constantin, On the blow-up solutions of a periodic shallow water equation, J. Nonlinear Sci. 10 (2000) 391–399.
[4] A. Constantin, R. Ivanov, On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A 372 (2008) 7129–7132.
[5] Y.B. Deng, J.L. Xiang, T. Yang, Blowup phenomena of solutions to Euler–Poisson equations, J. Math. Anal. Appl. 286 (2003) 295–306.
[6] J. Escher, O. Lechtenfeld, Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation, Discrete Contin. Dyn. Syst.
Ser. A 19 (2007) 493–513.
[7] P. Goldreich, S. Weber, Homologously collapsing stellar cores, Astrophys. J. 238 (1980) 991–997.
[8] C.X. Guan, Z.Y. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system, J. Differential
Equations 248 (2010) 2003–2014.
[9] Z.G. Guo, Blow-up and global solutions to a new integrable model with two components, J. Math. Anal. Appl. 372 (2010) 316–327.
[10] Z.G. Guo, Y. Zhou, On solutions to a two-component generalized Camassa–Holm system, Stud. Appl. Math. 124 (2010) 307–322.
[11] J. Hale, Theory of Functional Differential Equations, 2nd edition, Appl. Math. Sci., vol. 3, Springer-Verlag, New York–Heidelberg, 1977, x+365 pp.
[12] T.H. Li, Some special solutions of the multidimensional Euler equations in RN , Commun. Pure Appl. Anal. 4 (2005) 757–762.
[13] T.H. Li, D.H. Wang, Blowup phenomena of solutions to the Euler equations for compressible ﬂuid ﬂow, J. Differential Equations 221 (2006) 91–101.
[14] Z.L. Liang, Blowup phenomena of the compressible Euler equations, J. Math. Anal. Appl. 379 (2010) 506–510.
[15] T. Makino, Blowing up solutions of the Euler–Poisson equation for the evolution of the gaseous stars, Transport Theory Statist. Phys. 21 (1992) 615–624.
[16] T. Makino, Exact solutions for the compressible Euler equation, J. Osaka Sangyo Univ. Natur. Sci. 95 (1993) 21–35.
[17] W. Walter, Ordinary Differential Equations, Grad. Texts Math. Read. Math., vol. 182, Springer-Verlag, New York, 1998, xii+380 pp., translated from the
6th German (1996) edition by Russell Thompson.
[18] G.B. Whitham, Linear and Nonlinear Waves, Wiley–Interscience, New York–London–Sydney, 1974.
[19] M.W. Yuen, Blowup solutions for a class of ﬂuid dynamical equations in RN , J. Math. Anal. Appl. 329 (2007) 1064–1079.
[20] M.W. Yuen, Analytical blowup solutions to the 2-dimensional isothermal Euler–Poisson equations of gaseous stars, J. Math. Anal. Appl. 341 (2008)
445–456.
[21] M.W. Yuen, Analytical solutions to the Navier–Stokes equations, J. Math. Phys. 49 (2008) 113102, 10 pp.
[22] M.W. Yuen, Analytical blowup solutions to the pressureless Navier–Stokes–Poisson equations with density-dependent viscosity in RN , Nonlinearity 22
(2009) 2261–2268.
[23] M.W. Yuen, Self-similar blowup solutions to the 2-component Camassa–Holm equations, J. Math. Phys. 51 (2010) 093524, 14 pp.
[24] M.W. Yuen, Self-similar blowup solutions to the 2-component Degasperis–Procesi shallow water system, Commun. Nonlinear Sci. Numer. Simul. 16
(2011) 2993–2998.
[25] M.W. Yuen, Perturbational blowup solutions to the 1-dimensional compressible Euler equations, preprint, arXiv:1012.2033.


Perturbational blowup solutions to the 2-component Camassa–Holm equations 

Crossref

Journal of Mathematical Analysis and Applications

Perturbational blowup solutions to the 2-component Camassa–Holm equations

file:///data/core-remote/dit/data/elsevier/pdf/be2/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAyMjI0N3gxMTAwNDYxNg==.pdf

Perturbational Blowup Solutions to the 2-Component Camassa-Holm
  Equations

Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector

Crossref