# The Cauchy Problem for a Fifth-Order Dispersive Equation

Authors: Hongjun Wang; Yongqi Liu; Yongqiang Chen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper is devoted to studying the Cauchy problem for a fifth-order
equation. We prove that it is locally well-posed for the initial data in the Sobolev
spaceH
s
(
R
) with s
≥
1
/
4. We also establish the ill-posedness for the initial data in
H
s
(
R
) with s
<
1
/
4. Thus, the regularity requirement for the fifth-order dispersive equations s
≥
1
/
4 is sharp.


---

## Body

## 1. Introduction

In this paper, we consider the Cauchy problem and existence of solitary waves of the following fifth-order dispersive equation:(1)
∂
t
u
-
∂
t
∂
x
2
u
+
∂
x
3
u
-
∂
x
5
u
+
1
3
∂
x
(
u
3
)
-
1
3
∂
x
3
(
u
3
)
+
∂
x
[
5
3
u
3
+
u
(
∂
x
u
)
2
]
=
0
,
(2)
u
(
x
,
0
)
=
ϕ
.
Obviously, (1) can be seen as the higher modification of the following dispersive equation:
(3)
∂
t
u
-
∂
t
∂
x
2
u
+
1
3
∂
x
(
u
3
)
-
1
3
∂
x
3
(
u
3
)
+
∂
x
[
5
3
u
3
+
u
(
∂
x
u
)
2
]
=
0
.We recall that(4)
u
t
+
(
a
(
u
)
)
x
-
u
x
t
t
-
(
b
′
(
u
)
u
x
2
2
+
b
(
u
)
u
x
x
)
x
=
0
,
where
(5)
b
(
u
)
=
u
p
,
a
(
u
)
=
2
k
u
+
p
+
2
2
u
p
+
1
,
which has been studied in [1], is the generalized Camassa-Holm equation. In fact, Hakkaev and Kirchev [1] studied the local well-posedness and orbital stability and instability of (4) with the aid of the pseudoparabolic regularization and spectral analysis. Obviously, when p
=
2, (4) is the well-known Camassa-Holm equation; when p
=
3 and k
=
0, (4) becomes (3). Some people consider the Cauchy problem for the higher modification of the nonlocal form of Camassa-Holm equation; we refer the readers to [2–5].By acting(
1
-
∂
x
2
)
-
1 on both sides of (1), we obtain the following equivalent form:
(6)
∂
t
u
+
∂
x
3
u
+
1
3
∂
x
(
u
3
)
+
(
1
-
∂
x
2
)
-
1
∂
x
[
5
3
u
3
+
u
(
∂
x
u
)
2
]
=
0
.
Obviously, (6) is a modification of mKdV equation
(7)
∂
t
u
+
∂
x
3
u
+
1
3
∂
x
(
u
3
)
=
0
,
which has been intensively studied; we refer the readers to [6–11]. In [8], the authors proved that (7) is locally well-posed for the initial data in H
s
(
R
) with s
≥
1
/
4. The regularity requirement for the modified KdV equation s
≥
1
/
4 is sharp; see [9]. In [11], by using the I-method which can be seen in [11–14] and Mirua transformation which can be seen in [15], the authors proved that the modified KdV equation is globally well-posed for the initial data in H
s
(
R
) with s
>
1
/
4. In [16], by using the dyadic bilinear estimates and resolution spaces which can be seen in [17, 18], the author proved that the modified KdV equation is globally well-posed for the initial data in H
s
(
R
) with s
=
1
/
4.In this paper, by using the Fourier restriction norm method introduced in [19], we prove that (1) and (2) are locally well-posed for the initial data in H
s
(
R
) with s
≥
1
/
4. When ϕ
∈
H
s
(
R
), s
≥
1, we prove that the Cauchy problem for (1) is globally well-posed in H
s
(
R
). By using the general well-posedness principle proposed by [20], we establish the ill-posedness for the initial data in H
s
(
R
) with s
<
1
/
4. Thus, the requirement for regularity s
≥
1
/
4 is sharp.Before stating the main results, we introduce some notations and definitions. We useX
~
Y when C
1
|
X
|
≤
|
Y
|
≤
C
2
|
X
|, where C
j
>
0
(
j
=
1,2
); we use X
≫
Y when |
X
|
>
C
|
Y
|, where C is some positive number which is larger than 2. 〈
ξ
〉
s
=
(
1
+
ξ
2
)
s
/
2 for any ξ
∈
R, and ℱ
u is the Fourier transform of u with respect to its all variables. ℱ
x
u is the Fourier transform of u with respect to its space variables. Denote D
x
=
(
-
∂
x
2
)
1
/
2. 𝒮
(
R
n
) (n
=
1,2
,
3) is the Schwartz space and 𝒮
′
(
R
n
) is its dual space. H
s
(
R
) is the usual Sobolev space with norm ∥
f
∥
H
s
(
R
)
≜
∥
〈
ξ
〉
s
ℱ
x
f
∥
L
ξ
2
(
R
). For any s
,
α
∈
R, X
s
,
α
(
R
2
) is the Bourgain space with phase function ϕ
(
ξ
)
=
-
ξ
3. That is, a function u
(
x
,
t
) in 𝒮
′
(
R
2
) belongs to X
s
,
α
(
R
2
) if and only if
(8)
∥
u
∥
X
s
,
α
(
R
2
)
≜
∥
〈
ξ
〉
s
〈
τ
-
ξ
3
〉
α
ℱ
u
(
ξ
,
τ
)
∥
L
τ
ξ
2
(
R
2
)
<
∞
.
For any given interval L, X
s
,
α
(
R
×
L
) is the space of the restriction of all functions in X
s
,
α
(
R
2
) on R
×
L, and for u
∈
X
s
,
α
(
R
×
L
) its norm is
(9)
∥
u
∥
X
s
,
α
(
R
×
L
)
=
inf
⁡
{
∥
U
∥
X
s
,
α
(
R
2
)
;
U
∣
R
×
L
=
u
}
.
We write X
s
,
α
T for X
s
,
α
(
R
×
L
) when L
=
[
0
,
T
]. Let σ
=
τ
-
ξ
3
,
σ
j
=
τ
j
-
ξ
j
3
(
j
=
1,2
,
3
).The main results of this paper are as follows.Theorem 1.
The Cauchy problem for (1) is locally well-posed for the initial data ϕ in H
s
(
R
) with s
≥
1
/
4.Theorem 2.
Lets
<
1
/
4. Then (1) and (2) are ill-posed in H
s
(
R
) in the sense that the solution map S
t of the Cauchy problem for (1) is not Lipschitz continuous at zero. More precisely, for any T
>
0, the solution map
(10)
S
t

:
ϕ
∈
H
s
(
R
)
⟶
u
∈
C
(
[
0
,
T
]
;
H
s
(
R
)
)
is not Lipschitz continuous at zero.The rest of this paper is arranged as follows. In Section2, we make some preliminaries. In Section 3, we establish two crucial trilinear estimates. In Section 4, we prove Theorem 1. In Section 5, we prove Theorem 2.


## 2. Preliminaries

Lemma 3.
Letb
>
1
/
2. Then
(11)
∥
u
∥
L
x
t
4
≤
C
∥
u
∥
X
0
,
2
b
/
3
,
(12)
∥
D
x
-
1
/
4
u
∥
L
x
4
L
t
∞
≤
C
∥
u
∥
X
0
,
b
,
(13)
∥
D
x
u
∥
L
x
∞
L
t
2
≤
C
∥
u
∥
X
0
,
b
,
(14)
∥
u
∥
L
x
t
6
≤
C
∥
u
∥
X
0
,
8
b
/
9
.Proof.
From [21], we have
(15)
∥
u
∥
L
x
t
8
≤
C
∥
u
∥
X
0
,
b
.
From Plancherel’s identity, we have
(16)
∥
u
∥
L
x
t
2
=
C
∥
u
∥
X
0
,
b
.
Interpolating (15) with (16) yields (14). Equations (11)–(13) can be seen in [21, 22].Lemma 4.
For0
<
δ
<
1, s
∈
R, and 1
/
2
<
b
≤
1, we have
(17)
∥
ψ
δ
(
t
)
W
(
t
)
u
0
∥
X
s
,
b
≤
C
δ
1
/
2
-
b
∥
u
0
∥
H
s
,
and for -
1
/
2
<
b
′
≤
0
≤
b
≤
b
′
+
1, we have
(18)
∥
ψ
δ
(
t
)
∫
0
t
‍
W
(
t
-
τ
)
f
(
τ
)
d
τ
∥
X
s
,
b
≤
C
δ
1
+
b
′
-
b
∥
f
∥
X
s
,
b
′
.Equations (17) and (18) can be seen in [6, 23].Lemma 5.
Letξ
=
ξ
1
+
ξ
2
+
ξ
3. Then
(19)
ξ
3
-
ξ
1
3
-
ξ
2
3
-
ξ
3
3
=
3
(
ξ
1
+
ξ
2
)
(
ξ
1
+
ξ
3
)
(
ξ
2
+
ξ
3
)
.Lemma5 can be found in [11].Lemma 6.
LetI
s denote the Riesz potential of order -
s and b
>
1
/
2
≥
s
≥
0, b
~
>
1
/
6
+
2
s
/
3. Then the following estimate holds true:
(20)
∥
I
s
I
-
s
(
u
,
v
)
∥
L
x
t
2
≤
C
∥
u
∥
X
0
,
b
∥
v
∥
X
0
,
b
~
,
where
(21)
ℱ
x
I
-
s
(
u
,
v
)
(
ξ
)
=
∫
ξ
=
ξ
1
+
ξ
2
‍
|
ξ
1
-
ξ
2
|
s
ℱ
x
u
(
ξ
1
)
ℱ
x
v
(
ξ
2
)
d
ξ
1
.In particular, whens
=
1
/
2 and b
>
1
/
2, we have
(22)
∥
I
1
/
2
I
-
1
/
2
(
u
,
v
)
∥
L
x
t
2
≤
C
∥
u
∥
X
0
,
(
1
/
2
)
+
∥
v
∥
X
0
,
(
1
/
2
)
+
.Lemma6 can be seen in [6].


## 3. Trilinear Estimates

In this section, we will prove two crucial trilinear estimates.Lemma 7.
Lets
≥
1
/
4, b
=
1
/
2
+
2
ϵ, b
′
=
-
1
/
2
+
3
ϵ, 0
<
ϵ
≪
1. Then
(23)
∥
∂
x
(
1
-
∂
x
2
)
-
1
[
u
1
(
∂
x
u
2
)
(
∂
x
u
3
)
]
∥
X
s
,
b
′
≤
C
∏
j
=
1
3
‍
∥
u
j
∥
X
s
,
b
.Proof.
Let(24)
F
j
(
ξ
j
,
τ
j
)
=
〈
ξ
j
〉
s
〈
σ
j
〉
b
ℱ
u
j
(
ξ
j
,
τ
j
)
(
j
=
1,2
,
3
)
,
F
(
ξ
,
τ
)
=
〈
ξ
〉
-
s
〈
σ
〉
-
b
′
ℱ
u
(
ξ
,
τ
)
.
By duality and Plancherel’s identity, to derive (23), it suffices to prove
(25)
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
|
F
|
∏
j
=
1
3
‍
|
F
j
|
‍
×
K
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
3
,
τ
3
,
ξ
,
τ
)
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
,
where
(26)
K
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
=
|
ξ
|
(
|
ξ
2
|
|
ξ
3
|
+
|
ξ
1
|
|
ξ
2
|
+
|
ξ
1
|
|
ξ
3
|
)
〈
ξ
〉
s
(
1
+
ξ
2
)
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
ξ
j
〉
s
〈
σ
〉
b
.
Without loss of generality, we may assume that F
j
(
ξ
j
,
τ
j
)
(
j
=
1,2
,
3
)
≥
0 and F
(
ξ
,
τ
)
≥
0. By the symmetry among |
ξ
1
|, |
ξ
2
|, |
ξ
3
|, without loss of generality, we may assume that |
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|. Obviously, {
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
}
⊆
⋃
j
=
1
6
‍
Ω
j, where
(27)
Ω
1
=
{
8
≥
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
}
,
Ω
2
=
{
|
ξ
1
|
≥
8
,
|
ξ
1
|
≫
|
ξ
2
|
≥
|
ξ
3
|
}
,
Ω
3
=
{
|
ξ
1
|
≥
8
,
|
ξ
1
|
~
|
ξ
2
|
≫
|
ξ
3
|
,
c
c
c
c
i
x
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
,
|
ξ
3
|
≤
8
}
,
Ω
4
=
{
|
ξ
1
|
≥
8
,
|
ξ
1
|
~
|
ξ
2
|
≫
|
ξ
3
|
,
c
c
c
c
i
x
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
≥
8
}
,
Ω
5
=
{
|
ξ
3
|
3
|
ξ
1
|
≥
8
,
|
ξ
1
|
~
|
ξ
2
|
~
|
ξ
3
|
,
c
c
c
c
c
x
i
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
,
|
ξ
|
≥
|
ξ
3
|
3
}
,
Ω
6
=
{
|
ξ
3
|
3
|
ξ
1
|
≥
8
,
|
ξ
1
|
~
|
ξ
2
|
~
|
ξ
3
|
,
c
c
c
c
c
i
x
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
,
|
ξ
|
≤
|
ξ
3
|
3
}
.
In this lemma, integrals over the subregion Ω
k’s are, respectively, denoted as J
k (1
≤
k
≤
6, k
∈
Z). Consider
(28)
ℱ
f
j
=
F
j
〈
σ
j
〉
b
(
j
=
1,2
,
3
)
,
ℱ
f
=
F
〈
σ
〉
-
b
′
.
(1) Subregion
{
8
≥
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
}. In this subregion, since s
≥
1
/
4 and |
ξ
|
≤
3, we have
(29)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
.
By using Cauchy Schwartz’s inequality and Plancherel’s identity as well as Hölder’s inequality, (11), 2
b
/
3
<
b and 2
b
/
3
<
-
b
′, we derive
(30)
J
1
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
F
∏
j
=
1
3
F
j
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
f
∥
L
x
t
4
∏
j
=
1
3
‍
∥
f
j
∥
L
x
t
4
≤
C
∥
f
∥
X
0,2
b
/
3
∏
j
=
1
3
‍
∥
f
j
∥
X
0
,
2
b
/
3
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
(2) Subregion
{
|
ξ
1
|
≥
8
,
|
ξ
1
|
≫
|
ξ
2
|
≥
|
ξ
3
|
}. In this subregion, |
ξ
|
~
|
ξ
1
| and |
ξ
1
|
≫
|
ξ
2
|, since s
≥
1
/
4, we have
(31)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
1
2
-
ξ
2
2
|
1
/
2
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
.
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (22) of Lemma 6, (11), 2
b
/
3
<
b, and 2
b
/
3
<
-
b
′, since s
≥
1
/
4, we obtain
(32)
J
2
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
|
ξ
1
2
-
ξ
2
2
|
1
/
2
F
∏
j
=
1
3
F
j
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
I
(
f
1
,
2
)
∥
L
x
t
2
∥
f
∥
L
x
t
4
∥
f
3
∥
L
x
t
4
≤
C
∥
f
∥
X
0
,
2
b
/
3
∥
f
3
∥
X
0
,
2
b
/
3
∏
j
=
1
2
‍
∥
f
j
∥
X
0
,
b
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
(3) Subregion
{
|
ξ
1
|
≥
8
,
|
ξ
1
|
~
|
ξ
2
|
≫
|
ξ
3
|
,
|
ξ
3
|
≤
8
}. In this subregion, since s
≥
1
/
4 and |
ξ
|
≤
C
|
ξ
2
|, we have
(33)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
1
-
s
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
2
2
-
ξ
3
2
|
1
/
2
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
.
This case can be treated similarly to J
2.
(4) Subregion
{
|
ξ
1
|
≥
8
,
|
ξ
1
|
~
|
ξ
2
|
≫
|
ξ
3
|
≥
8
}. In this subregion |
ξ
1
|
~
|
ξ
2
| and |
ξ
|
≤
C
|
ξ
1
|.
Whens
≥
1, since |
ξ
2
|
≫
|
ξ
3
| and |
ξ
|
≤
C
|
ξ
2
|, we have
(34)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
|
ξ
1
|
〈
ξ
〉
s
〈
σ
〉
-
b
′
(
1
+
ξ
2
)
∏
j
=
1
3
〈
ξ
j
〉
s
〈
σ
j
〉
b
≤
C
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
.
This case can be treated similarly to J
1.
When1
/
4
≤
s
≤
1, we consider the case |
ξ
1
+
ξ
2
|
≤
|
ξ
3
|
/
3 and the case |
ξ
1
+
ξ
2
|
≥
|
ξ
3
|
/
3.
When|
ξ
1
+
ξ
2
|
≤
|
ξ
3
|
/
3, since 1
/
4
≤
s
≤
1, we have
(35)
K
1
(
ξ
1
,
ξ
2
,
ξ
,
τ
1
,
τ
2
,
τ
)
≤
C
|
ξ
2
|
2
-
2
s
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
2
|
3
/
2
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
.
This case can be treated similarly to J
2.
When|
ξ
1
+
ξ
2
|
≥
|
ξ
3
|
/
3, From Lemma 5, we have
(36)
|
σ
-
(
∑
j
=
1
3
‍
σ
j
)
|
=
|
ξ
3
-
ξ
1
3
-
ξ
2
3
-
ξ
3
3
|
=
3
|
(
ξ
1
+
ξ
2
)
(
ξ
1
+
ξ
3
)
(
ξ
2
+
ξ
3
)
|
≥
C
|
ξ
1
|
2
|
ξ
3
|
,
which yields max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
2
|
ξ
3
|; thus one of the following four cases must occur:
(37)
|
σ
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
2
|
ξ
3
|
,
(38)
|
σ
1
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
2
|
ξ
3
|
,
(39)
|
σ
2
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
2
|
ξ
3
|
,
(40)
|
σ
3
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
2
|
ξ
3
|
.
When (37) is valid, since
(41)
|
ξ
|
〈
ξ
〉
s
1
+
ξ
2
≤
1
,
we have
(42)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
|
ξ
3
|
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
ξ
j
〉
s
〈
σ
j
〉
b
≤
C
|
ξ
2
|
1
-
2
s
|
ξ
3
|
1
-
s
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
2
|
1
-
2
s
+
2
b
′
|
ξ
3
|
1
-
s
+
b
′
∏
j
=
1
3
〈
σ
j
〉
b
;
when 1
-
s
+
b
′
≤
0, since 1
/
4
≤
s
≤
1 and b
′
=
-
1
/
2
+
3
ϵ, we have
(43)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
1
-
2
s
+
2
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
∏
j
=
1
3
〈
σ
j
〉
b
;
when 1
-
s
+
b
′
≥
0, since 1
/
4
≤
s
≤
1 and b
′
=
-
1
/
2
+
3
ϵ, we have
(44)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
2
-
3
s
+
3
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
∏
j
=
1
3
〈
σ
j
〉
b
.
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), 8
b
/
9
<
b, we obtain
(45)
J
4
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
F
∏
j
=
1
3
F
j
∏
j
=
1
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
f
j
∥
L
x
t
6
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
f
j
∥
X
0
,
8
b
/
9
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
When (38) is valid, since
(46)
|
ξ
|
〈
ξ
〉
s
1
+
ξ
2
≤
1
,
we have
(47)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
|
ξ
3
|
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
ξ
j
〉
s
〈
σ
j
〉
b
≤
C
|
ξ
2
|
1
-
2
s
|
ξ
3
|
1
-
s
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
2
|
1
-
2
s
+
2
b
′
|
ξ
3
|
1
-
s
+
b
′
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
;
when 1
-
s
+
b
′
≤
0, we have
(48)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
1
-
2
s
+
2
b
′
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
≤
C
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
;
when 1
-
s
+
b
′
≥
0, since 1
/
4
≤
s
≤
1 and b
′
=
-
1
/
2
+
3
ϵ, we have
(49)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
2
|
2
-
3
s
+
3
b
′
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
≤
C
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
.
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), 8
b
/
9
<
b, and 8
b
/
9
<
-
b
′, we obtain
(50)
J
4
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
F
∏
j
=
1
3
F
j
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
F
1
∥
L
ξ
τ
2
∥
f
∥
L
x
t
6
∏
j
=
2
3
‍
∥
f
j
∥
L
x
t
6
≤
C
∥
F
1
∥
L
ξ
τ
2
∥
f
∥
X
0
,
8
b
/
9
∏
j
=
2
3
‍
∥
f
j
∥
X
0
,
8
b
/
9
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
Cases (39) and (40) can be treated similarly to case (38).
(5) Subregion
{
|
ξ
1
|
≥
1
,
|
ξ
1
|
~
|
ξ
2
|
~
|
ξ
3
|
,
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
,
|
ξ
|
≥
|
ξ
3
|
/
3
}. In this subregion, since s
≥
1
/
4, |
ξ
1
|
~
|
ξ
|
~
|
ξ
2
|
~
|
ξ
3
| which yields
(51)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
1
|
1
-
2
s
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
1
|
1
/
2
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
.
Thus, applying Cauchy-Schwartz’s inequality and Plancherel’s identity, (12) and (13), we have
(52)
J
5
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
|
ξ
1
|
1
/
2
F
∏
j
=
1
3
F
j
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
D
x
f
1
∥
L
x
∞
L
t
2
∥
D
x
-
1
/
4
f
2
∥
L
x
4
L
t
∞
∥
D
x
-
1
/
4
f
3
∥
L
x
4
L
t
∞
∥
f
∥
L
x
t
2
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
f
j
∥
X
0
,
b
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
(6) Subregion
{
|
ξ
1
|
≥
1
,
|
ξ
1
|
~
|
ξ
2
|
~
|
ξ
3
|
,
|
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
,
|
ξ
|
≤
|
ξ
3
|
/
3
}. In this subregion, since |
ξ
1
|
≥
|
ξ
2
|
≥
|
ξ
3
|
≥
3
|
ξ
|, we have (
2
/
3
)
|
ξ
3
|
≤
|
ξ
1
+
ξ
2
|
≤
(
4
/
3
)
|
ξ
3
| and (
2
/
3
)
|
ξ
2
|
≤
|
ξ
1
+
ξ
3
|
≤
(
4
/
3
)
|
ξ
2
| and (
2
/
3
)
|
ξ
1
|
≤
|
ξ
2
+
ξ
3
|
≤
(
4
/
3
)
|
ξ
1
|. From Lemma 5, we have
(53)
|
σ
-
(
∑
j
=
1
3
‍
σ
j
)
|
=
|
ξ
3
-
ξ
1
3
-
ξ
2
3
-
ξ
3
3
|
=
3
|
(
ξ
1
+
ξ
2
)
(
ξ
1
+
ξ
3
)
(
ξ
2
+
ξ
3
)
|
~
|
ξ
1
|
3
~
|
ξ
2
|
3
~
|
ξ
3
|
3
which yields max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
3; thus one of the following four cases must occur:
(54)
|
σ
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
3
,
(55)
|
σ
1
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
3
,
(56)
|
σ
2
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
3
,
(57)
|
σ
3
|
=
max
⁡
{
|
σ
|
,
|
σ
1
|
,
|
σ
2
|
,
|
σ
3
|
}
≥
C
|
ξ
1
|
3
.
When s
≥
1, we have
(58)
K
1
(
ξ
1
,
τ
1
,
τ
2
,
ξ
2
,
ξ
,
τ
)
≤
C
|
ξ
1
|
2
-
2
s
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
;
this case can be handled similarly to J
1.
When1
/
4
≤
s
≤
1, we consider (54), (55), (56), and (57), respectively.
When (37) is valid, since |
ξ
1
|
~
|
ξ
2
|
~
|
ξ
3
|, |
ξ
|
≤
|
ξ
3
| and b
′
=
-
1
/
2
+
3
ϵ as well as 1
/
4
≤
s
≤
1, we have
(59)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
|
〈
ξ
〉
s
|
ξ
1
|
2
-
3
s
(
1
+
ξ
2
)
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
1
|
2
-
3
s
+
3
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
|
ξ
1
|
9
ϵ
-
(
1
/
4
)
∏
j
=
1
3
〈
σ
j
〉
b
≤
C
∏
j
=
1
3
〈
σ
j
〉
b
.
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), 8
b
/
9
<
b, we obtain
(60)
J
6
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
F
∏
j
=
1
3
F
j
∏
j
=
1
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
f
j
∥
L
x
t
6
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
f
j
∥
X
0
,
8
b
/
9
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
When (55) holds, since |
ξ
1
|
~
|
ξ
2
|
~
|
ξ
3
|, |
ξ
|
≤
|
ξ
3
| and b
′
=
-
1
/
2
+
3
ϵ as well as 1
/
4
≤
s
≤
1, we have
(61)
K
1
(
ξ
1
,
τ
1
,
ξ
2
,
τ
2
,
ξ
,
τ
)
≤
C
|
ξ
|
〈
ξ
〉
s
|
ξ
1
|
2
-
3
s
(
1
+
ξ
2
)
〈
σ
〉
-
b
′
∏
j
=
1
3
〈
σ
j
〉
b
≤
|
ξ
1
|
2
-
3
s
+
3
b
′
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
≤
C
|
ξ
1
|
9
ϵ
-
(
1
/
4
)
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
≤
C
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
.
By using Cauchy-Schwartz’s inequality and Plancherel’s identity, (11), 8
b
/
9
<
b, and 8
b
/
9
<
-
b
′, we obtain
(62)
J
6
≤
C
∫
R
2
‍
∫
ξ
=
ξ
1
+
ξ
2
+
ξ
3
τ
=
τ
1
+
τ
2
+
τ
3
F
∏
j
=
1
3
F
j
〈
σ
〉
-
b
′
∏
j
=
2
3
〈
σ
j
〉
b
d
ξ
1
d
τ
1
d
ξ
2
d
τ
2
d
ξ
d
τ
‍
≤
C
∥
F
1
∥
L
ξ
τ
2
∥
f
∥
L
x
t
6
∏
j
=
2
3
‍
∥
f
j
∥
L
x
t
6
≤
C
∥
F
1
∥
L
ξ
τ
2
∥
f
∥
X
0
,
8
b
/
9
∏
j
=
2
3
‍
∥
f
j
∥
X
0
,
8
b
/
9
≤
C
∥
F
∥
L
ξ
τ
2
∏
j
=
1
3
‍
∥
F
j
∥
L
ξ
τ
2
.
Cases (56) and (57) can be treated similarly to case (55).
Putting the estimates ofJ
k (1
≤
k
≤
6, k
∈
Z) together, we have (25).
Thus, we complete the proof of Lemma7.Lemma 8.
Lets
≥
1
/
4, b
=
1
/
2
+
2
ϵ, b
′
=
-
1
/
2
+
3
ϵ, 0
<
ϵ
≪
1. Then
(63)
∥
∂
x
(
u
1
u
2
u
3
)
∥
X
s
,
b
′
≤
C
∏
j
=
1
3
‍
∥
u
j
∥
X
s
,
b
.Lemma8 can be proved similarly to Lemma 7.


## 4. The Proof of Theorem1

In order to prove Theorem1, firstly, for ϕ
∈
H
s
(
R
) and δ
∈
(
0,1
], u
∈
X
s
,
b
(
R
2
), we define G
ϕ
(
u
) by
(64)
G
ϕ
(
v
)
=
ψ
(
t
)
W
(
t
)
ϕ
-
ψ
δ
∫
0
t
‍
W
(
t
-
t
′
)
g
(
u
,
∂
x
u
)
d
t
′
,
where
(65)
g
(
u
,
∂
x
u
)
=
1
3
∂
x
(
u
3
)
+
(
1
-
∂
x
2
)
-
1
∂
x
[
5
3
u
3
+
u
(
∂
x
u
)
2
]
.
By using Lemmas 4, 7, and 8, we have
(66)
∥
G
ϕ
(
u
)
∥
X
s
,
b
(
R
2
)
≤
C
∥
ϕ
∥
H
s
(
R
)
+
C
δ
b
′
+
1
-
b
∥
u
∥
X
s
,
b
(
R
2
)
3
,
where s, b, b
′ of (66) concord with s, b, and b
′ of Lemmas 7 and 8. Let
(67)
δ
=
(
1
16
C
3
(
∥
ϕ
∥
H
s
(
R
)
2
+
2
)
)
1
/
(
b
′
+
1
-
b
)
,
r
=
2
C
∥
ϕ
∥
H
s
(
R
)
,
where 0
<
δ
<
1. From (66) and (67), we know that G
ϕ is a mapping from the closed ball B
(
0
,
r
)
=
{
u
∈
X
s
,
b
(
R
2
)
,
∥
u
∥
X
s
,
b
(
R
2
)
≤
r
} into itself. Similarly, we have
(68)
∥
G
ϕ
(
v
)
-
G
ϕ
(
u
)
∥
X
s
,
b
(
R
2
)
≤
2
C
δ
b
′
+
1
-
b
∥
v
-
u
∥
X
s
,
b
(
R
2
)
(
∥
u
∥
X
s
,
b
(
R
2
)
2
+
∥
v
∥
X
s
,
b
(
R
2
)
2
)
≤
1
2
∥
v
-
u
∥
X
s
,
b
(
R
2
)
;
thus, G
ϕ is a contraction mapping from the closed ball B
(
0
,
r
)
=
{
u
∈
X
s
,
b
(
R
2
)
,
∥
u
∥
X
s
,
b
(
R
2
)
≤
r
} into itself; by using Banach fixed point Theorem, we have G
ϕ
(
u
)
=
u.The rest of local well-posedness of Theorem1 follows from a standard proof.


## 5. The Proof of Theorem2

In this section, we prove Theorem2.By contradiction, we assume that the solution map of (1) and (2)
(69)
S
t

:
u
0
∈
H
s
(
R
)
⟶
u
∈
C
(
[
0
,
T
]
;
H
s
(
R
)
)
is Lipschitz continuous at zero with s
<
1
/
4. From the general well-posedness principle of [20], we must have
(70)
sup
⁡
t
∈
[
0
,
T
]
∥
B
3
(
u
0
)
∥
H
s
≤
C
∥
u
0
∥
H
s
3
,
∀
u
0
∈
H
s
(
R
)
,
where
(71)
B
3
(
u
0
)
(
x
,
t
)
=
∫
0
t
‍
W
(
t
-
τ
)
∂
x
(
B
1
(
u
0
)
B
1
(
u
0
)
B
1
(
u
0
)
)
(
τ
)
d
τ
,
B
1
(
u
0
)
(
x
,
t
)
=
W
(
t
)
u
0
.
We consider the initial data
(72)
u
0
(
x
)
=
r
-
1
/
2
N
-
s
{
e
-
i
N
x
(
∫
0
r
‍
e
i
x
ξ
d
ξ
)
+
e
i
N
x
(
∫
r
2
r
‍
e
i
x
ξ
d
ξ
)
}
,
where r
2
N
=
O
(
1
) and N
≫
1. Thus, we have
(73)
ℱ
x
u
0
(
ξ
)
=
C
r
-
1
/
2
N
-
s
{
χ
[
-
N
,
-
N
+
r
]
(
ξ
)
+
χ
[
N
+
r
,
N
+
2
r
]
(
ξ
)
}
,
which can be seen in [24], where χ
I denotes the characteristic function of a set I
⊂
R. It is easy to check that ∥
u
0
∥
H
s
~
1. Let I
1
=
[
-
N
,
-
N
+
r
] and I
2
=
[
N
+
r
,
N
+
2
r
] and I
1
∪
I
2
=
Ω. We have
(74)
ℱ
x
B
1
(
u
0
)
(
ξ
)
=
C
e
i
t
ϕ
(
ξ
)
ℱ
x
u
0
(
ξ
)
and thus
(75)
B
1
(
u
0
)
(
x
,
t
)
=
C
r
-
1
/
2
N
-
s
∫
ξ
∈
Ω
‍
e
i
x
ξ
e
i
t
ϕ
(
ξ
)
d
ξ
,
B
3
(
u
0
)
(
x
,
t
)
=
C
f
,
where
(76)
f
=
r
-
3
/
2
N
-
3
s
∫
ξ
1
∈
Ω
‍
∫
ξ
2
∈
Ω
‍
∫
ξ
3
∈
Ω
‍
ξ
3
(
1
+
ξ
2
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
ξ
2
+
6
-
3
ξ
1
ξ
2
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
e
i
x
(
ξ
1
+
ξ
2
+
ξ
3
)
Q
d
ξ
1
d
ξ
2
d
ξ
3
,
where
(77)
Q
=
e
i
t
(
ϕ
(
ξ
1
)
+
ϕ
(
ξ
2
)
+
ϕ
(
ξ
3
)
)
-
e
i
t
ϕ
(
ξ
1
+
ξ
2
+
ξ
3
)
ϕ
(
ξ
1
)
+
ϕ
(
ξ
2
)
+
ϕ
(
ξ
3
)
-
ϕ
(
ξ
1
+
ξ
2
+
ξ
3
)
.
Since ϕ
(
ξ
)
=
-
ξ
3, we define
(78)
θ
≔
ϕ
(
ξ
1
)
+
ϕ
(
ξ
2
)
+
ϕ
(
ξ
3
)
-
ϕ
(
ξ
1
+
ξ
2
+
ξ
3
)
=
3
(
ξ
1
+
ξ
2
)
(
ξ
1
+
ξ
3
)
(
ξ
2
+
ξ
3
)
resulting from Lemma 5 and ξ
=
ξ
1
+
ξ
2
+
ξ
3. To estimate ∥
f
∥
H
s, we consider the following three cases. Case 1:ξ
j
(
j
=
1,2
,
3
)
∈
I
1.
Case 2:ξ
j
(
j
=
1,2
,
3
)
∈
I
2.
Case 3:ξ
j
(
j
=
1,2
)
∈
I
1, ξ
3
∈
I
2, or ξ
1
∈
I
1, ξ
j
(
j
=
2,3
)
∈
I
2, or ξ
j
(
j
=
1,2
)
∈
I
2, ξ
3
∈
I
1, or ξ
1
∈
I
2, ξ
j
(
j
=
2,3
)
∈
I
1.The integrals in (76) corresponding to Cases 1, 2, and 3 are denoted as f
1, f
2, f
3, respectively.Case 1.
In this case, from Lemma6, we have |
θ
|
~
N
3 and |
ξ
2
+
6
-
3
ξ
1
ξ
2
|
~
N
2. Since r
2
N
=
O
(
1
), we have
(79)
∥
f
1
∥
H
s
≤
C
r
-
3
/
2
N
-
3
s
N
s
r
5
/
2
N
-
2
~
N
-
2
s
-
5
/
2
.Case 2.
In this case, from Lemma6, we have |
θ
|
~
N
3 and |
ξ
2
+
6
-
3
ξ
1
ξ
2
|
~
N
2. Since r
2
N
=
O
(
1
), we have
(80)
∥
f
2
∥
H
s
≤
C
r
-
3
/
2
N
-
3
s
N
s
r
5
/
2
N
-
2
~
N
-
2
s
-
5
/
2
.Case 3.
In this case|
θ
|
~
r
2
N
=
O
(
1
), |
Q
|
≥
Constant due to Lemma 6, and |
ξ
2
+
6
-
3
ξ
1
ξ
2
|
~
N
2. Since r
2
N
=
O
(
1
), we have
(81)
∥
f
3
∥
H
s
≥
C
r
-
3
/
2
N
-
3
s
N
s
r
5
/
2
N
~
N
-
2
s
+
1
/
2
.
From (70), we have
(82)
C
~
C
∥
u
0
∥
H
s
3
≥
∥
f
1
+
f
2
+
f
3
∥
H
s
≥
∥
f
3
+
f
2
∥
H
s
-
∥
f
1
∥
H
s
≥
∥
f
3
∥
H
s
-
∥
f
2
∥
H
s
-
∥
f
1
∥
H
s
≥
C
(
N
-
2
s
+
1
/
2
-
N
-
2
s
-
5
/
2
)
which yields
(83)
N
-
2
s
+
1
/
2
≤
C
(
1
+
N
-
2
s
-
5
/
2
)
.
When s
<
-
5
/
4, from (83), we have N
-
2
s
+
1
/
2
≤
C
N
-
2
s
-
5
/
2 which yields N
3
≤
C; we obtain a contradiction since N
≫
1. When -
5
/
4
≤
s
<
1
/
4, from (83), we have N
-
2
s
+
1
/
2
≤
C; we obtain a contradiction since N
≫
1.The proof of Theorem2 is completed.



---

*Source: 404781-2014-03-27.xml*