# Remarks on Chern-Simons-Higgs Equations inR

Authors: Hyungjin Huh; Guanghui Jin
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

We prove global existence of solutions to Chern-Simons-Higgs equations under the gauge conditionA
1
=
0. We also find stationary solutions.


---

## Body

## 1. Introduction

We are interested in Chern-Simons-Higgs equations inR
1
+
1. The model was proposed in [1] to study BPS domain wall solutions of the Chern-Simons-Higgs system. The Lagrangian density of the (
1
+
1
) dimensional Chern-Simons-Higgs system is given by(1)
L
=
D
0
ϕ
2
-
D
1
ϕ
2
+
1
λ
N
F
01
-
V
N
,
ϕ
,where N, A
0, A
1 are real fields, ϕ is a complex scalar field, and F
01
=
∂
0
A
1
-
∂
1
A
0. The covariant derivative is defined by D
α
ϕ
=
∂
α
ϕ
-
i
A
α
ϕ and the potential V
(
N
,
ϕ
) is given by(2)
V
N
,
ϕ
=
N
2
ϕ
2
+
λ
2
ϕ
2
ϕ
2
-
1
2
,where λ
>
0 is a coupling constant.The Chern-Simons-Higgs system [2, 3] in R
2
+
1 is derived from the following Lagrangian density:(3)
L
=
D
0
ϕ
2
-
D
1
ϕ
2
-
D
2
ϕ
2
+
1
2
λ
ϵ
μ
ν
ρ
A
μ
F
ν
ρ
-
λ
2
ϕ
2
ϕ
2
-
1
2
,where ϵ
μ
ν
ρ is totally skew symmetric tensor with ϵ
012
=
1. The Lagrangian density (1) is obtained by the dimensional reduction of (3). More precisely, we consider the Lagrangian (3) independent of the y coordinate and renaming A
2 as N to get (1).The Euler-Lagrange equations of (1) are(4)
D
0
D
0
ϕ
-
D
1
D
1
ϕ
+
N
2
ϕ
+
λ
2
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
=
0
,
(5)
∂
0
A
1
-
∂
1
A
0
=
2
N
λ
ϕ
2
,
(6)
∂
0
N
=
2
λ
Im
⁡
ϕ
¯
D
1
ϕ
,
(7)
∂
1
N
=
2
λ
Im
⁡
ϕ
¯
D
0
ϕ
.The conservation of the total energy implies that(8)
E
t
≔
∫
R
E
t
,
x
d
x
=
∫
R
E
0
,
x
d
x
,where the energy density corresponding to the Lagrangian density (1) is given by (9)
E
t
,
x
=
D
0
ϕ
2
+
D
1
ϕ
2
+
N
2
ϕ
2
+
λ
2
ϕ
2
ϕ
2
-
1
2
.For the static configuration, this model with self-dual coupling constantλ
=
1 admits a system of first order self-dual equations whose solutions minimize the total static energy (8). The solutions to the self-dual equations and their nonrelativistic limit have been studied in [1].The system of (4)–(7) is invariant under the following gauge transformations:(10)
ϕ
⟶
ϕ
e
i
χ
,
A
μ
⟶
A
μ
+
∂
μ
χ
,
N
⟶
N
,where χ is a real-valued smooth function on R
1
+
1. Here we will fix the gauge by imposing the condition A
1
≡
0. Note that temporal gauge condition A
0
≡
0 is well known. The motivation considering the gauge condition A
1
≡
0 comes from a standing wave solution of (4)–(7). As shown in Section 2 the usual ansatz of standing wave solution leads to A
1
≡
0. To study stability it seems natural to study the initial value problem of (4)–(7) with the condition A
1
≡
0. We have studied Chern-Simons-Schrödinger [4] and Chern-Simons-Dirac [5] equations in the same gauge condition. Our first result is concerned with standing wave solutions of (4)–(7).Theorem 1.
There are standing wave solutions to (4)–(7) of the form(11)
ϕ
t
,
x
=
u
x
e
i
ω
t
,
A
0
t
,
x
=
A
0
x
,
A
1
t
,
x
=
0
,
N
t
,
x
=
N
x
,where ω
∈
R and u is real-valued function. In fact, we find solutions (26) satisfying the boundary condition u
(
±
∞
)
=
0 and solutions (32) satisfying the boundary condition (
u
(
-
∞
)
,
u
(
+
∞
)
)
=
(
0
,
1
) or (
1
,
0
).Low regularity local solutions have been recently constructed in [6] where it is proved that Chern-Simons-Higgs system under the Lorenz gauge is locally well posed with the regularity s
>
3
/
4: (12)
ϕ
∈
C
0
,
T
;
H
s
R
2
∩
C
1
0
,
T
;
H
s
-
1
R
2
,
A
μ
∈
C
0
,
T
;
H
s
-
1
/
2
R
2
,
N
∈
C
0
,
T
;
H
s
-
1
/
2
R
2
.In [6], they observed that the Chern-Simons-Higgs system has null form structures corresponding to the Lorenz gauge and used several null form estimates. Moreover the global energy solution has been constructed when s
=
1. We study the initial value problem of Chern-Simons-Higgs equations under the gauge condition A
1
≡
0 to obtain the following result.Theorem 2.
For the initial dataϕ
(
0
,
x
)
∈
H
1
(
R
) and ∂
t
ϕ
(
0
,
x
)
∈
L
2
(
R
), the initial value problem for (44) has a global solution which belongs to (13)
ϕ
∈
C
0
,
∞
,
H
1
R
∩
C
1
0
,
∞
,
L
2
R
,
A
0
∈
L
∞
0
,
∞
,
L
∞
R
,
∂
t
A
0
∈
L
∞
0
,
∞
,
L
∞
R
,
N
∈
L
∞
0
,
∞
,
L
∞
R
.In Section2 we find standing wave solutions of CSH to prove Theorem 1. Theorem 2 is proved in Section 3. We conclude this section by giving a few notations. We denote by □ the D’Alembertian operator □
=
∂
t
t
-
∂
x
x. We use C to denote various constants. When we are interested in local solutions, we may assume that T
≤
1. Thus we shall replace smooth function of T, C
(
T
) by C. We use A
≲
B to denote an estimate of the form A
≤
C
B.


## 2. Standing Wave Solutions

In this section we consider standing wave solutions of the form(14)
ϕ
t
,
x
=
u
x
e
i
ω
t
,
A
0
t
,
x
=
A
0
x
,
A
1
t
,
x
=
A
1
x
,
N
t
,
x
=
N
x
,where ω is a real constant and u, A
0, A
1, N are real-valued functions. From (6) we derive 0
=
-
2
A
1
u
2 which leads us to A
1
=
0. Then we can rewrite (4)–(7) as follows:(15)
u
′
′
+
ω
2
u
=
2
ω
A
0
u
-
A
0
2
u
+
N
2
u
+
λ
2
3
u
5
-
4
u
3
+
u
,
(16)
N
′
=
2
λ
ω
-
A
0
u
2
,
(17)
A
0
′
=
-
2
λ
N
u
2
,where ′ denotes a derivative d
/
d
x.Using (16) and (17), we may check(18)
d
d
x
2
ω
A
0
-
A
0
2
+
N
2
=
0
,from which we set(19)
2
ω
A
0
-
A
0
2
+
N
2
=
a
2
.Then (15) becomes(20)
u
′
′
=
λ
2
3
u
5
-
4
u
3
+
u
-
ω
2
-
a
2
u
.The energy of (20) becomes(21)
E
=
∫
R
u
′
2
+
λ
2
u
2
u
2
-
1
2
-
ω
2
-
a
2
u
2
d
x
.We consider two cases ω
2
-
a
2
≠
0 and ω
2
-
a
2
=
0.(1) The case ofω
2
-
a
2
≠
0: to make the energy (21) finite, we consider a boundary condition (22)
u
′
±
∞
=
0
,
u
±
∞
=
0
,which is called “symmetric phase.” Multiplying (20) by 2
u
′ and considering symmetry phase boundary condition, we obtain (23)
u
′
2
=
λ
2
u
2
u
2
-
1
2
-
ω
2
-
a
2
u
2
.Foru
′
=
u
λ
2
u
4
-
2
λ
2
u
2
+
λ
2
-
ω
2
+
a
2, changing of variables u
2
=
v and using the following formula (24)
∫
1
v
α
v
2
+
β
v
+
γ
d
v
=
-
1
γ
log
⁡
2
γ
α
v
2
+
β
v
+
γ
+
β
v
+
2
γ
v
f
o
r
γ
>
0
,we obtain (25)
u
=
4
λ
2
-
ω
2
+
a
2
4
λ
2
±
4
λ
2
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
±
e
-
2
λ
2
-
ω
2
+
a
2
x
1
/
2
f
o
r
λ
2
-
ω
2
+
a
2
>
0
.To guarantee a smooth solution, we choose + sign and impose ω
2
-
a
2
>
0 to get(26)
u
=
4
λ
2
-
ω
2
+
a
2
4
λ
2
+
4
λ
2
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
+
e
-
2
λ
2
-
ω
2
+
a
2
x
1
/
2
,where λ
2
-
ω
2
+
a
2
>
0 and ω
2
-
a
2
>
0.Considering2
ω
A
0
-
A
0
2
+
N
2
=
a
2 and (16), we have N
′
=
±
2
λ
N
2
+
ω
2
-
a
2
u
2 which can be written as(27)
N
′
N
2
+
ω
2
-
a
2
=
±
2
λ
4
λ
2
-
ω
2
+
a
2
4
λ
2
+
4
λ
2
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
+
e
-
2
λ
2
-
ω
2
+
a
2
x
.Integrating (27), we obtain (28)
s
i
n
h
-
1
N
ω
2
-
a
2
=
±
log
⁡
2
λ
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
-
λ
2
-
ω
2
+
a
2
+
λ
2
λ
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
+
λ
2
-
ω
2
+
a
2
+
λ
,which can be rewritten as(29)
N
=
±
ω
2
-
a
2
sinh
⁡
log
⁡
2
λ
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
-
λ
2
-
ω
2
+
a
2
+
λ
2
λ
ω
2
-
a
2
e
2
λ
2
-
ω
2
+
a
2
x
+
λ
2
-
ω
2
+
a
2
+
λ
.
A
0 can be obtained by (19). We can treat the case u
′
=
-
u
λ
2
u
4
-
2
λ
2
u
2
+
λ
2
-
ω
2
+
a
2 in the same manner.(2) The case ofω
2
-
a
2
=
0: to make the energy (21) finite, we consider a boundary condition (30)
u
′
±
∞
=
0
,
u
-
∞
,
u
+
∞
=
0,1
o
r
1,0
,which is called “asymmetric phase.” We can check that if u is a solution of (20), then -
u is also a solution of (20).Considering asymmetry phase boundary condition, we obtain(31)
u
′
2
=
λ
2
u
2
u
2
-
1
2
,which implies(32)
u
x
=
b
2
e
±
2
λ
x
1
-
b
2
+
b
2
e
±
2
λ
x
1
/
2
,where 0
<
b
2
<
1 denotes b
=
u
(
0
). Since 2
ω
A
0
-
A
0
2
+
N
2
=
a
2 and ω
2
-
a
2
=
0, we have N
′
=
±
2
λ
N
u
2, which gives us (33)
N
x
=
±
c
1
-
b
2
+
b
2
e
±
2
λ
x
,for some constant c. We also have A
0
=
ω
±
N
=
ω
±
c
(
1
-
b
2
+
b
2
e
±
2
λ
x
).


## 3. Proof of Theorem2

Here we study the initial value problem of Chern-Simons-Higgs system in one space dimension under the gauge conditionA
1
=
0. We are interested in the symmetry phase boundary condition and assume λ
=
1 for a simpler presentation.


### 3.1. Local Existence

The system (4)–(7) with the gauge condition A
1
=
0 can be rewritten as(34)
□
ϕ
=
2
i
A
0
∂
t
ϕ
+
i
∂
t
A
0
ϕ
+
A
0
2
ϕ
-
N
2
ϕ
-
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
,
(35)
∂
x
A
0
=
-
2
N
ϕ
2
,
(36)
∂
x
N
=
2
Im
⁡
ϕ
¯
D
0
ϕ
,
(37)
∂
t
N
=
2
Im
⁡
ϕ
¯
∂
x
ϕ
.We obtain from (35)(38)
∂
x
∂
t
A
0
=
-
∂
t
2
N
ϕ
2
.From (19) and (29), we can check N
(
∞
)
=
0 and A
0
(
∞
)
=
ω
±
ω
2
-
a
2. We will impose a boundary condition N
(
∞
)
=
0, A
0
(
∞
)
=
b, and ∂
t
A
0
(
∞
)
=
0. Let us check compatibility of (37) with other equations (34)–(36). Using (34) and (36), we can check that (39)
∂
x
∂
t
N
=
∂
t
∂
x
N
=
∂
0
2
Im
⁡
ϕ
¯
D
0
ϕ
=
2
Im
⁡
ϕ
¯
D
0
D
0
ϕ
=
2
Im
⁡
ϕ
¯
∂
x
x
ϕ
-
N
2
ϕ
-
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
=
2
Im
⁡
ϕ
¯
∂
x
x
ϕ
=
2
∂
x
Im
⁡
ϕ
¯
∂
x
ϕ
,which implies ∂
t
N
=
2
Im
⁡
(
ϕ
¯
∂
x
ϕ
).Now we obtain, from (35) and (36),(40)
∂
x
A
0
+
N
+
2
A
0
+
N
ϕ
2
=
2
Im
⁡
ϕ
¯
∂
t
ϕ
,
(41)
∂
x
A
0
-
N
-
2
A
0
-
N
ϕ
2
=
-
2
Im
⁡
ϕ
¯
∂
t
ϕ
.Multiplying (40), (41) by e
2
∫
x
∞
‍
|
ϕ
|
2
d
y, e
-
2
∫
x
∞
‍
|
ϕ
|
2
d
y, respectively, and integrating with a boundary condition N
(
∞
)
=
0 and A
0
(
∞
)
=
b, we obtain A
0 and N as follows:(42)
A
0
=
e
-
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
-
e
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
-
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
+
b
,
N
=
-
e
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
-
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
-
e
-
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
.From (37) and (38), we have(43)
∂
t
A
0
t
,
x
=
∫
x
∞
4
Im
⁡
ϕ
¯
∂
x
ϕ
ϕ
2
+
2
N
∂
t
ϕ
2
d
y
,where a boundary condition ∂
t
A
0
(
∞
)
=
0 is used. In the following, we assume b
=
0 for simple presentation. In fact, the boundary condition A
0
(
∞
)
=
b
≠
0 does not affect the proof of Theorem 2.We have shown that the initial value problem of (34)–(37) reduces to the study of the following equation:(44)
□
ϕ
=
2
i
A
0
∂
t
ϕ
+
i
∂
t
A
0
ϕ
+
A
0
2
ϕ
-
N
2
ϕ
-
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
,
ϕ
0
,
x
=
ϕ
0
x
,
∂
t
ϕ
0
,
x
=
ϕ
1
x
,where N, A
0, and ∂
t
A
0 are defined by (42)-(43).We are ready to prove the existence of local solutions to (44). Proof follows by standard arguments from a priori estimates of the following propositions.Proposition 3.
Letϕ be a solution of (44) in a strip R
×
[
0
,
T
] with(45)
ϕ
∈
C
0
,
T
,
H
1
R
∩
C
1
0
,
T
,
L
2
R
.Define (46)
J
T
=
s
u
p
0
≤
t
≤
T
ϕ
H
1
R
+
∂
t
ϕ
L
2
R
.Then there exist constants C
>
0 and T
∗
>
0, depending only on J
(
0
), such that if T
<
T
∗ then J
(
T
)
≤
C
J
(
0
).Proposition 4.
Letϕ and ϕ
′ be two solutions of (44) verifying the hypothesis of Proposition 3 in a strip R
×
[
0
,
T
] and let J
(
T
) as in Proposition 3 and J
′
(
T
) be the corresponding quantity for the primed solution. Also define (47)
Δ
T
=
s
u
p
0
≤
t
≤
T
ϕ
t
,
·
-
ϕ
′
t
,
·
H
1
+
∂
t
ϕ
t
,
·
-
∂
t
ϕ
t
,
·
L
2
.Then there exist constants C
>
0 and T
∗
>
0, depending only on J
(
0
) and J
′
(
0
), such that if T
<
T
∗ then Δ
(
T
)
≤
C
Δ
(
0
).Proposition4 follows by the similar argument to Proposition 3. We will only present the proof of Proposition 3. We define (48)
Y
T
=
∫
0
T
A
0
∂
t
ϕ
s
,
·
L
2
R
+
∂
t
A
0
ϕ
s
,
·
L
2
R
+
A
0
2
+
N
2
ϕ
s
,
·
L
2
R
+
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
s
,
·
L
2
R
d
s
.It is easily shown that J
(
T
)
≲
(
J
(
0
)
+
Y
(
T
)
) by applying energy estimate. We will derive the inequality(49)
Y
T
≲
T
1
+
e
4
J
0
+
Y
T
2
J
0
+
Y
T
5
.Then a bootstrap argument completes the proof of Proposition 3.From the representation (42), we have the following L
∞ bounds: (50)
A
0
t
,
x
L
∞
≲
e
2
J
2
T
ϕ
L
2
∂
t
ϕ
L
2
≲
e
2
J
2
T
J
2
T
,
N
t
,
x
L
∞
≲
e
2
J
2
T
J
2
T
.Considering (43), we have (51)
∂
t
A
0
t
,
x
L
∞
≤
C
∂
x
ϕ
L
2
ϕ
L
6
3
+
N
L
∞
ϕ
L
2
∂
0
ϕ
L
2
≲
1
+
e
2
J
2
T
J
4
T
.Applying the above bounds, the integrals in Y
(
T
) can be treated as follows: (52)
∫
0
T
A
0
∂
t
ϕ
s
,
·
L
2
R
d
s
≲
T
e
2
J
2
T
J
3
T
,
∫
0
T
∂
t
A
0
ϕ
s
,
·
L
2
R
d
s
≲
T
1
+
e
2
J
2
T
J
5
T
,
∫
0
T
A
0
2
ϕ
s
,
·
L
2
R
d
s
≲
T
e
4
J
2
T
J
5
T
,
∫
0
T
N
2
ϕ
s
,
·
L
2
R
d
s
≲
T
e
4
J
2
T
J
5
T
,
∫
0
T
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
s
,
·
L
2
R
d
s
≲
T
1
+
J
5
T
.Combining the above estimates, we get the relation (49).


### 3.2. Global Existence of the Energy Solution

Here we prove the global existence of the solution to (44) for the initial data ϕ
0
∈
H
1
(
R
) and ϕ
1
∈
L
2
(
R
). The existence of global energy solutions is proved by showing that the corresponding norm of the local solution is uniformly bounded on each finite time interval.The energy conservation (8) implies that the quantity (53)
E
t
=
∫
R
D
0
ϕ
2
+
∂
1
ϕ
2
+
N
2
ϕ
2
+
ϕ
2
ϕ
2
-
1
2
d
xis bounded by E
(
0
). First of all, we have(54)
∂
1
ϕ
L
2
≤
E
1
/
2
0
.To control L
2 norm of ϕ, we consider(55)
1
2
∂
t
ϕ
L
2
2
=
R
e
∫
R
ϕ
¯
D
0
ϕ
t
,
·
d
x
≤
ϕ
L
2
D
0
ϕ
t
,
·
L
2
.Taking into account D
0
ϕ
(
t
,
·
)
L
2
≤
E
1
/
2
(
0
), it follows that(56)
ϕ
t
,
·
L
2
≤
ϕ
0
L
2
+
t
E
1
/
2
0
.From (36), we have the following L
∞ bound of N: (57)
N
t
,
x
L
∞
≲
ϕ
L
2
D
0
ϕ
L
2
≲
ϕ
0
L
2
+
t
E
1
/
2
0
E
1
/
2
0
≤
f
ϕ
0
L
2
,
E
1
/
2
0
1
+
t
,where f is a polynomial of ϕ
0
L
2 and E
1
/
2
(
0
). Then we obtain from (35) (58)
A
0
t
,
x
L
∞
≲
N
L
∞
ϕ
L
2
2
≤
f
ϕ
0
L
2
,
E
1
/
2
0
1
+
t
3
.Making use of the above information of A
0 and N, we can control the L
2 norm of ∂
0
ϕ as follows:(59)
∂
0
ϕ
t
,
x
L
2
≤
D
0
ϕ
L
2
+
A
0
L
∞
ϕ
L
2
≤
f
ϕ
0
L
2
,
E
1
/
2
0
1
+
t
4
.Combining (54)–(59), we conclude that (60)
ϕ
H
1
+
∂
0
ϕ
L
2
≤
h
t
,where h
(
t
) is a bounded function on finite time interval 0
≤
t
≤
T.


## 3.1. Local Existence

The system (4)–(7) with the gauge condition A
1
=
0 can be rewritten as(34)
□
ϕ
=
2
i
A
0
∂
t
ϕ
+
i
∂
t
A
0
ϕ
+
A
0
2
ϕ
-
N
2
ϕ
-
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
,
(35)
∂
x
A
0
=
-
2
N
ϕ
2
,
(36)
∂
x
N
=
2
Im
⁡
ϕ
¯
D
0
ϕ
,
(37)
∂
t
N
=
2
Im
⁡
ϕ
¯
∂
x
ϕ
.We obtain from (35)(38)
∂
x
∂
t
A
0
=
-
∂
t
2
N
ϕ
2
.From (19) and (29), we can check N
(
∞
)
=
0 and A
0
(
∞
)
=
ω
±
ω
2
-
a
2. We will impose a boundary condition N
(
∞
)
=
0, A
0
(
∞
)
=
b, and ∂
t
A
0
(
∞
)
=
0. Let us check compatibility of (37) with other equations (34)–(36). Using (34) and (36), we can check that (39)
∂
x
∂
t
N
=
∂
t
∂
x
N
=
∂
0
2
Im
⁡
ϕ
¯
D
0
ϕ
=
2
Im
⁡
ϕ
¯
D
0
D
0
ϕ
=
2
Im
⁡
ϕ
¯
∂
x
x
ϕ
-
N
2
ϕ
-
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
=
2
Im
⁡
ϕ
¯
∂
x
x
ϕ
=
2
∂
x
Im
⁡
ϕ
¯
∂
x
ϕ
,which implies ∂
t
N
=
2
Im
⁡
(
ϕ
¯
∂
x
ϕ
).Now we obtain, from (35) and (36),(40)
∂
x
A
0
+
N
+
2
A
0
+
N
ϕ
2
=
2
Im
⁡
ϕ
¯
∂
t
ϕ
,
(41)
∂
x
A
0
-
N
-
2
A
0
-
N
ϕ
2
=
-
2
Im
⁡
ϕ
¯
∂
t
ϕ
.Multiplying (40), (41) by e
2
∫
x
∞
‍
|
ϕ
|
2
d
y, e
-
2
∫
x
∞
‍
|
ϕ
|
2
d
y, respectively, and integrating with a boundary condition N
(
∞
)
=
0 and A
0
(
∞
)
=
b, we obtain A
0 and N as follows:(42)
A
0
=
e
-
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
-
e
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
-
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
+
b
,
N
=
-
e
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
-
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
-
e
-
2
∫
x
∞
ϕ
2
d
y
∫
x
∞
e
2
∫
y
∞
ϕ
2
d
z
Im
⁡
ϕ
¯
∂
t
ϕ
y
d
y
.From (37) and (38), we have(43)
∂
t
A
0
t
,
x
=
∫
x
∞
4
Im
⁡
ϕ
¯
∂
x
ϕ
ϕ
2
+
2
N
∂
t
ϕ
2
d
y
,where a boundary condition ∂
t
A
0
(
∞
)
=
0 is used. In the following, we assume b
=
0 for simple presentation. In fact, the boundary condition A
0
(
∞
)
=
b
≠
0 does not affect the proof of Theorem 2.We have shown that the initial value problem of (34)–(37) reduces to the study of the following equation:(44)
□
ϕ
=
2
i
A
0
∂
t
ϕ
+
i
∂
t
A
0
ϕ
+
A
0
2
ϕ
-
N
2
ϕ
-
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
,
ϕ
0
,
x
=
ϕ
0
x
,
∂
t
ϕ
0
,
x
=
ϕ
1
x
,where N, A
0, and ∂
t
A
0 are defined by (42)-(43).We are ready to prove the existence of local solutions to (44). Proof follows by standard arguments from a priori estimates of the following propositions.Proposition 3.
Letϕ be a solution of (44) in a strip R
×
[
0
,
T
] with(45)
ϕ
∈
C
0
,
T
,
H
1
R
∩
C
1
0
,
T
,
L
2
R
.Define (46)
J
T
=
s
u
p
0
≤
t
≤
T
ϕ
H
1
R
+
∂
t
ϕ
L
2
R
.Then there exist constants C
>
0 and T
∗
>
0, depending only on J
(
0
), such that if T
<
T
∗ then J
(
T
)
≤
C
J
(
0
).Proposition 4.
Letϕ and ϕ
′ be two solutions of (44) verifying the hypothesis of Proposition 3 in a strip R
×
[
0
,
T
] and let J
(
T
) as in Proposition 3 and J
′
(
T
) be the corresponding quantity for the primed solution. Also define (47)
Δ
T
=
s
u
p
0
≤
t
≤
T
ϕ
t
,
·
-
ϕ
′
t
,
·
H
1
+
∂
t
ϕ
t
,
·
-
∂
t
ϕ
t
,
·
L
2
.Then there exist constants C
>
0 and T
∗
>
0, depending only on J
(
0
) and J
′
(
0
), such that if T
<
T
∗ then Δ
(
T
)
≤
C
Δ
(
0
).Proposition4 follows by the similar argument to Proposition 3. We will only present the proof of Proposition 3. We define (48)
Y
T
=
∫
0
T
A
0
∂
t
ϕ
s
,
·
L
2
R
+
∂
t
A
0
ϕ
s
,
·
L
2
R
+
A
0
2
+
N
2
ϕ
s
,
·
L
2
R
+
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
s
,
·
L
2
R
d
s
.It is easily shown that J
(
T
)
≲
(
J
(
0
)
+
Y
(
T
)
) by applying energy estimate. We will derive the inequality(49)
Y
T
≲
T
1
+
e
4
J
0
+
Y
T
2
J
0
+
Y
T
5
.Then a bootstrap argument completes the proof of Proposition 3.From the representation (42), we have the following L
∞ bounds: (50)
A
0
t
,
x
L
∞
≲
e
2
J
2
T
ϕ
L
2
∂
t
ϕ
L
2
≲
e
2
J
2
T
J
2
T
,
N
t
,
x
L
∞
≲
e
2
J
2
T
J
2
T
.Considering (43), we have (51)
∂
t
A
0
t
,
x
L
∞
≤
C
∂
x
ϕ
L
2
ϕ
L
6
3
+
N
L
∞
ϕ
L
2
∂
0
ϕ
L
2
≲
1
+
e
2
J
2
T
J
4
T
.Applying the above bounds, the integrals in Y
(
T
) can be treated as follows: (52)
∫
0
T
A
0
∂
t
ϕ
s
,
·
L
2
R
d
s
≲
T
e
2
J
2
T
J
3
T
,
∫
0
T
∂
t
A
0
ϕ
s
,
·
L
2
R
d
s
≲
T
1
+
e
2
J
2
T
J
5
T
,
∫
0
T
A
0
2
ϕ
s
,
·
L
2
R
d
s
≲
T
e
4
J
2
T
J
5
T
,
∫
0
T
N
2
ϕ
s
,
·
L
2
R
d
s
≲
T
e
4
J
2
T
J
5
T
,
∫
0
T
ϕ
2
-
1
3
ϕ
2
-
1
ϕ
s
,
·
L
2
R
d
s
≲
T
1
+
J
5
T
.Combining the above estimates, we get the relation (49).


## 3.2. Global Existence of the Energy Solution

Here we prove the global existence of the solution to (44) for the initial data ϕ
0
∈
H
1
(
R
) and ϕ
1
∈
L
2
(
R
). The existence of global energy solutions is proved by showing that the corresponding norm of the local solution is uniformly bounded on each finite time interval.The energy conservation (8) implies that the quantity (53)
E
t
=
∫
R
D
0
ϕ
2
+
∂
1
ϕ
2
+
N
2
ϕ
2
+
ϕ
2
ϕ
2
-
1
2
d
xis bounded by E
(
0
). First of all, we have(54)
∂
1
ϕ
L
2
≤
E
1
/
2
0
.To control L
2 norm of ϕ, we consider(55)
1
2
∂
t
ϕ
L
2
2
=
R
e
∫
R
ϕ
¯
D
0
ϕ
t
,
·
d
x
≤
ϕ
L
2
D
0
ϕ
t
,
·
L
2
.Taking into account D
0
ϕ
(
t
,
·
)
L
2
≤
E
1
/
2
(
0
), it follows that(56)
ϕ
t
,
·
L
2
≤
ϕ
0
L
2
+
t
E
1
/
2
0
.From (36), we have the following L
∞ bound of N: (57)
N
t
,
x
L
∞
≲
ϕ
L
2
D
0
ϕ
L
2
≲
ϕ
0
L
2
+
t
E
1
/
2
0
E
1
/
2
0
≤
f
ϕ
0
L
2
,
E
1
/
2
0
1
+
t
,where f is a polynomial of ϕ
0
L
2 and E
1
/
2
(
0
). Then we obtain from (35) (58)
A
0
t
,
x
L
∞
≲
N
L
∞
ϕ
L
2
2
≤
f
ϕ
0
L
2
,
E
1
/
2
0
1
+
t
3
.Making use of the above information of A
0 and N, we can control the L
2 norm of ∂
0
ϕ as follows:(59)
∂
0
ϕ
t
,
x
L
2
≤
D
0
ϕ
L
2
+
A
0
L
∞
ϕ
L
2
≤
f
ϕ
0
L
2
,
E
1
/
2
0
1
+
t
4
.Combining (54)–(59), we conclude that (60)
ϕ
H
1
+
∂
0
ϕ
L
2
≤
h
t
,where h
(
t
) is a bounded function on finite time interval 0
≤
t
≤
T.



---

*Source: 9830474-2016-09-22.xml*