# On the Weak Solution to a Fractional Nonlinear Schrödinger Equation

Authors: Zujin Zhang; Xiaofeng Wang; Zheng-an Yao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We obtain the existence of a global weak
solution to a fractional nonlinear Schrödinger equation
by the Galerkin method.  Its uniqueness is also discussed.
In our proof, we use harmonic analysis techniques and compactness arguments.


---

## Body

## 1. Introduction

This paper is concerned with the following fractional partial differential equations in then dimensional torus 𝕋
n:
(1)
∂
u
∂
t
-
i
Λ
2
α
u
+
|
u
|
ρ
u
=
f
,
where α
∈
(
0,1
) and ρ
∈
(
0
,
∞
) are real numbers and i
=
-
1. f
(
x
,
t
) is given and u
(
x
,
t
) is a complex-valued function. Here, Λ
2
α
=
(
-
Δ
)
α, the fractional Laplacian with respect to x.The fractional Laplacian operator appears in a wide class of physical systems and engineering problems, including Lévy flights, viscoelasticity, electrochemistry, control, porous media, electromagnetic, stochastic interfaces, and anomalous diffusion problems, and attracts the interests of many mathematicians; see [1–5], for example. The quasigeostrophic equation with fractional dissipation has been also extensively studied; see Constantin et al. [6–10], for example. In mathematical physics, the fractional Laplacian is often applied to describe many complicated phenomena via partial differential equations.The Schrödinger equation is a fundamental equation in physics, which describes nonrelativistic quantum mechanical behavior. It is well known that Feynman and Hibbs [11] used path integrals over Brownian paths to derive the standard Schrödinger equation (α
=
1 in (1)). Recently, Laskin [12, 13] showed that the path integral over the Lévy-like quantum mechanical paths allows us to generalize the classical quantum mechanics. Namely, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation. The fractional Schrödinger equation includes the space derivative of order 2
α instead of second-order space derivative in the standard Schrödinger equation. Laskin [14] showed the hermiticity of the fractional Hamilton operator and established the parity conservation law. Guo et al. [15] obtained the existence of a unique global smooth solution to the periodical boundary value problem for the fractional nonlinear Schrödinger equation.Interestingly enough, there are also some other models involving the damping term|
u
|
ρ
u; see [16, 17].In studying (1), there exist some essential difficulties. First, since the fractional differential operator is defined by Fourier series and is nonlocal except when α
∈
ℕ, which means that Λ
2
α
u
(
x
) depends not only on u
(
y
) for y near x, but also on u
(
y
) for all y. Moreover, integration by part for nonlinear term is not valid. These bring new difficulties when doing energy estimate. And thus new harmonic analysis methods must be introduced to overcome these difficulties. Second, there are some difficulties in the convergence of the approximate solutions because of the nonlinear term. A compactness device should be given to treat this case.We now collect the notations in this paper. The fractional LaplacianΛ
2
α
f, for α
∈
ℝ, can be defined as
(2)
Λ
2
α
f
^
k
=
|
k
|
2
α
f
^
k
.
,
k
∈
ℤ
n
,
where f
^
k is the Fourier coefficients of f:
(3)
f
^
k
=
1
(
2
π
)
n
∫
𝕋
n
f
(
x
)
e
-
i
k
·
x
d
x
,
k
∈
ℤ
n
.We shall also invoke the notion of inhomogeneous Sobolev spaceH
˙
s
(
𝕋
n
)
(
s
∈
ℝ
), which comprises all tempered distributions f on ℝ
n such that
(4)
∥
f
∥
H
s
(
𝕋
n
)
=
(
∑
k
∈
ℤ
n
∖
{
0
}
‍
(
1
+
|
k
|
2
)
s
|
u
^
k
|
2
)
1
/
2
.We now end this introduction by outlining the rest of this paper. In Section2, we prove the existence of a weak solution to (1); see Theorem 8. The uniqueness of such weak solutions is discussed in Section 3; see Theorem 9.


## 2. Existence of a Weak Solution

In this section, we prove the existence of a weak solution to the following system:(5)
∂
u
∂
t
-
i
Λ
2
α
u
+
|
u
|
ρ
u
=
f
,
(
x
,
t
)
∈
𝕋
n
×
(
0
,
T
)
(6)
u
(
0
)
=
u
0
,
x
∈
𝕋
n
.Let us first recall and prove some fundamental Lemmas.Lemma 1 (see [18]).
LetX be a Banach space; consider
(7)
f
∈
L
p
(
0
,
T
;
X
)
,
∂
f
∂
t
∈
L
p
(
0
,
T
;
X
)
,
for some 1
≤
p
≤
∞; then f is continuous map from [
0
,
T
] to X.Lemma 2 (see [18]).
Let𝒟 be a bounded domain in ℝ
n
×
ℝ
+, and g
k, g are in L
q
(
𝒟
)
(
1
<
q
<
∞
) with
(8)
∥
g
k
∥
L
q
(
𝒟
)
≤
C
,
g
k
⟶
g
,
a
.
e
.
𝒟
;
then g
k
→
g weakly in L
q
(
𝒟
).Lemma 3 (see [18]).
LetT
>
0, 1
<
p
1, p
2
<
∞, and B
0, B, B
1 be three Banach spaces satisfying B
0
↪
B
↪
B
1, where B
0, B
1 are reflexive and the embedding B
0
↪
B is compact. Endow the space
(9)
W
=
{
w
;
w
∈
L
p
0
(
0
,
T
;
B
0
)
,
∂
w
∂
t
∈
L
p
1
(
0
,
T
;
B
1
)
}
with the norm
(10)
∥
w
∥
W
=
∥
w
∥
L
p
0
(
0
,
T
;
B
0
)
+
∥
∂
w
∂
t
∥
L
p
1
(
0
,
T
;
B
1
)
,
∀
w
∈
W
.
Then the embedding W
↪
L
p
0
(
0
,
T
;
B
) is compact.In the following developments, we modify the methods in [19].Proposition 4.
Let0
<
α
<
1, x
∈
𝕋
n
=
ℝ
n
/
ℤ
n, and u
∈
𝒮
(
ℝ
n
). Then
(11)
Λ
2
α
u
(
x
)
=
C
α
∑
ν
∈
ℤ
n
P
V
∫
𝕋
n
u
(
x
)
-
u
(
y
)
|
x
-
y
-
ν
|
n
+
2
α
d
y
,
with C
α
>
0.Proof.
First, we have(12)
Λ
2
α
u
(
x
)
=
∑
|
ν
|
>
0
|
ν
|
2
α
u
^
(
ν
)
e
i
ν
·
x
=
-
∑
|
ν
|
>
0
|
ν
|
2
α
-
2
Δ
u
^
(
ν
)
e
i
ν
·
x
.
Let Φ
ɛ
(
x
)
=
(
|
x
|
2
α
-
2
)
ɛ
*
φ
ɛ
(
x
), where (
|
x
|
2
α
-
2
)
ɛ
=
|
x
|
2
α
-
2
·
χ
(
|
x
|
/
ɛ
) with χ
∈
C
∞
(
[
0
,
∞
)
),
(13)
χ
(
x
)
=
{
0
,
if
|
x
|
≤
1
,
1
,
if
|
x
|
≥
2
,
and φ
ɛ
(
x
)
=
ɛ
-
n
φ
(
x
/
ɛ
) is a standard approximation of identity, 0
≤
φ
∈
C
∞
(
ℝ
n
), supp φ
⊂
B
1, and ∫
‍
φ
=
1. Now we can write
(14)
Λ
2
α
u
(
x
)
=
-
lim
⁡
ɛ
→
0
∑
ν
∈
ℤ
n
Φ
ɛ
(
ν
)
Δ
u
^
(
ν
)
e
i
ν
·
x
=
-
lim
⁡
ɛ
→
0
(
∑
ν
∈
ℤ
n
Φ
ɛ
(
ν
)
e
i
ν
·
x
)
*
(
∑
ν
∈
ℤ
n
Δ
u
^
(
ν
)
e
i
ν
·
x
)
.
Poisson’s summation then yields
(15)
Λ
2
α
u
(
x
)
=
-
lim
⁡
ɛ
→
0
(
∑
ν
∈
ℤ
n
Φ
ɛ
^
(
x
-
ν
)
)
*
Δ
u
(
x
)
=
lim
⁡
ɛ
→
0
∑
ν
∈
ℤ
n
‍
∫
𝕋
n
Φ
ɛ
^
(
x
-
y
-
ν
)
Δ
[
u
(
x
)
-
u
(
y
)
]
d
y
=
lim
⁡
ɛ
→
0
∑
ν
∈
ℤ
n
‍
∫
𝕋
n
Δ
(
Φ
ɛ
^
)
(
x
-
y
-
ν
)
[
u
(
x
)
-
u
(
y
)
]
d
y
.
Due to the fact that
(16)
Φ
ɛ
^
(
η
)
=
(
|
x
|
2
α
-
2
)
ɛ
^
·
φ
ɛ
^
(
η
)
=
(
|
x
|
2
α
-
2
)
ɛ
^
·
φ
^
(
ɛ
η
)
,
Δ
Φ
ɛ
^
(
η
)
=
Δ
(
(
|
x
|
2
α
-
2
)
ɛ
^
)
(
η
)
·
φ
^
(
ɛ
η
)
+
O
(
ɛ
)
,
(
|
x
|
2
α
-
2
)
ɛ
^
(
y
)
=
c
α
|
y
|
n
+
2
α
-
2
-
∫
𝕋
n
e
-
i
y
·
x
|
x
|
2
α
-
2
(
1
-
χ
(
|
x
|
ɛ
)
)
d
x
,
Δ
(
|
x
|
2
α
-
2
)
ɛ
^
(
y
)
=
c
~
α
|
y
|
n
+
2
α
-
∫
𝕋
n
e
-
i
y
·
x
|
x
|
2
α
(
1
-
χ
(
|
x
|
ɛ
)
)
d
x
,
we obtain
(17)
∑
ν
∈
ℤ
n
‍
Δ
(
Φ
ɛ
^
)
(
y
-
ν
)
=
c
~
α
∑
ν
∈
ℤ
n
1
|
y
-
ν
|
n
+
2
α
+
O
(
∑
ν
∈
ℤ
n
1
|
y
-
ν
|
n
+
δ
O
(
ɛ
δ
)
)
,
for some δ
>
0.
Therefore,(18)
Λ
2
α
u
(
x
)
=
lim
⁡
ɛ
→
0
∑
ν
∈
ℤ
n
‍
∫
𝕋
n
Δ
(
Φ
ɛ
^
)
(
x
-
y
-
ν
)
[
u
(
x
)
-
u
(
y
)
]
d
y
=
C
α
∑
ν
∈
ℤ
n
P
V
∫
𝕋
n
u
(
x
)
-
u
(
y
)
|
x
-
y
-
ν
|
n
+
2
α
d
y
.Lemma 5.
Let𝕋
n
=
ℝ
n
/
ℤ
n, and let u be a complex-valued function satisfying u
,
u
2
α
∈
L
p
(
ℝ
n
) with 0
<
α
<
1 and 1
≤
p
<
∞. Then
(19)
Re
(
∫
𝕋
n
|
u
|
p
-
2
u
Λ
2
α
u
¯
d
x
)
≥
0
.Proof.
For0
<
α
<
1, by invoking Proposition 4 and changing of variables, we have
(20)
Re
(
∫
𝕋
n
|
u
|
p
-
2
u
Λ
2
α
u
¯
d
x
)
=
C
α
lim
⁡
ɛ
→
0
Re
(
∬
|
y
-
x
|
≥
ɛ
|
u
(
x
)
|
p
-
2
u
(
x
)
u
(
x
)
-
u
(
y
)
¯
|
x
-
y
|
n
+
2
α
d
y
d
x
)
=
-
C
α
lim
⁡
ɛ
→
0
Re
(
∫
|
y
-
x
|
≥
ɛ
‍
|
u
(
y
)
|
p
-
2
u
(
y
)
u
(
x
)
-
u
(
y
)
¯
|
x
-
y
|
n
+
2
α
d
y
d
x
)
=
C
α
2
lim
⁡
ɛ
→
0
Re
(
∬
|
y
-
x
|
≥
ɛ
‍
[
|
u
(
x
)
|
p
-
2
u
(
x
)
-
|
u
(
y
)
|
p
-
2
u
(
y
)
]
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
u
(
x
)
-
u
(
y
)
¯
|
x
-
y
|
n
+
2
α
d
y
d
x
)
≥
0
.In the following Lemma, we give a characterization of the eigenvalues and eigenvectors of the pseudodifferential operatorsΛ
2
α.Lemma 6.
Let𝕋
n
=
ℝ
n
/
ℤ
n. Then there exist a sequence of real numbers {
λ
j
}
j
=
1
∞ and a sequence of periodical functions {
ω
j
}
j
=
1
∞
⊂
H
2
α
(
𝕋
n
), such that
(21)
Λ
2
α
ω
j
=
λ
j
ω
j
,
j
=
1,2
,
3
,
…
.
Moreover, {
ω
j
}
j
=
1
∞ is a basis of H
2
α
(
𝕋
n
) satisfying
(22)
(
ω
j
,
ω
k
)
≡
∫
𝕋
n
ω
j
ω
k
¯
d
x
=
δ
j
k
,
(23)
(
Λ
2
α
ω
j
,
ω
k
)
=
λ
j
δ
j
k
,
for j, k
=
1,2
,
3
,
….Notice that the lemma is a direct consequence of elliptic regularity and functional analysis; see [20], for example.Let us now give the weak formulation of (5)-(6).Definition 7.
LetT
>
0, f
∈
L
2
(
0
,
T
;
H
α
(
𝕋
n
)
), ∂
f
/
∂
t
∈
L
2
(
0
,
T
;
L
2
(
𝕋
n
)
), and u
0
∈
H
α
(
𝕋
n
)
∩
L
2
(
ρ
+
1
)
(
𝕋
n
). A measurable complex-valued function u
(
x
,
t
) is said to be a weak solution to (5)-(6) on [
0
,
T
], provided the following:(1)
u
∈
L
∞
(
0
,
T
;
H
α
(
𝕋
n
)
)
∩
L
p
(
0
,
T
;
L
p
(
𝕋
n
)
) and ∂
u
/
∂
t
∈
L
∞
(
0
,
T
;
L
2
(
𝕋
n
)
);
(2)
u satisfies (5) in the sense of distributions;
(3)
u
(
x
,
0
)
=
u
0
(
x
) a.e. 𝕋
n.
Here and hereafter,p
=
ρ
+
2.Now, we state our existence results in the following theorem.Theorem 8.
LetT
>
0 be a given time, and
(24)
u
0
∈
H
α
(
𝕋
n
)
∩
L
2
(
ρ
+
1
)
(
𝕋
n
)
,
f
∈
L
2
(
0
,
T
;
H
α
(
𝕋
n
)
)
,
∂
f
∂
t
∈
L
2
(
0
,
T
;
L
2
(
𝕋
n
)
)
.
Then there exists at least one weak solution u to (5)-(6) on [
0
,
T
], taking u
0 as initial data.Proof.
We use Galerkin method.
Step  1. Construction of approximate solution.
Let{
ω
j
}
j
=
1
∞ be given as in Lemma 6. We consider the approximate solution which has the form
(25)
u
m
(
t
)
=
∑
j
=
1
m
c
j
m
(
t
)
ω
j
,
where c
j
m satisfy the following ordinary differential system:
(26)
(
∂
u
m
∂
t
(
t
)
,
ω
j
)
-
i
(
Λ
2
α
u
m
(
t
)
,
ω
j
)
+
(
|
u
m
(
t
)
|
ρ
u
m
(
t
)
,
ω
j
)
=
(
f
(
t
)
,
ω
j
)
,
1
≤
j
≤
m
,
(27)
u
m
(
0
)
=
u
0
m
=
∑
j
=
1
m
α
j
m
ω
j
⟶
u
0
in
H
α
(
𝕋
n
)
∩
L
2
(
ρ
+
1
)
(
𝕋
n
)
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhh
as
m
⟶
∞
.
Here and hereafter, (
u
,
v
)
=
∫
𝕋
n
‍
u
v
¯
d
x is the inner product in L
2
(
𝕋
n
).
The system (26)-(27) is nonsingular because {
ω
j
} are linear independent. Thus we may apply standard theory of ordinary differential equations to obtain the existence of a local solution to equations (26)-(27) on [
0
,
t
m
], for some t
m
>
0. We shall then, in the next step, establish some a priori estimates of the obtained solutions. This will ensure that t
m
=
T.
Step  2. A priori estimates.
By multiplying (26) by c
j
m
¯
(
t
) and summing with 1
≤
j
≤
m, we have
(28)
(
∂
u
m
∂
t
(
t
)
,
u
m
(
t
)
)
+
i
(
Λ
α
u
m
(
t
)
,
Λ
α
u
m
(
t
)
)
+
(
|
u
m
(
t
)
|
ρ
u
m
(
t
)
,
u
m
(
t
)
)
=
(
f
(
t
)
,
u
m
(
t
)
)
.
Taking real part of (28) and invoking Hölder inequality then yields
(29)
1
2
d
d
t
∥
u
m
(
t
)
∥
L
2
(
𝕋
n
)
2
+
∫
𝕋
n
|
u
m
(
t
)
|
p
d
x
=
Re
(
f
(
t
)
,
u
m
(
t
)
)
≤
∥
u
m
(
t
)
∥
L
2
(
𝕋
n
)
2
+
∥
f
(
t
)
∥
L
2
(
𝕋
n
)
2
.
By Gronwall’s inequality, we have
(30)
u
m
is
bounded
in
L
∞
(
0
,
T
;
L
2
(
𝕋
n
)
)
,
L
p
(
0
,
T
;
L
p
(
𝕋
n
)
)
.
By multiplying (26) by λ
j
c
j
m
¯
(
t
), summing with 1
≤
j
≤
m, and noticing (23), we get
(31)
(
∂
Λ
α
u
m
∂
t
(
t
)
,
Λ
α
u
m
)
-
i
∥
Λ
2
α
u
m
(
t
)
∥
L
2
(
𝕋
n
)
2
+
(
|
u
m
(
t
)
|
ρ
u
m
(
t
)
,
Λ
2
α
u
m
(
t
)
)
=
(
Λ
α
f
(
t
)
,
Λ
α
u
m
(
t
)
)
.
Thanks to Lemma 5, we may consider the real part of (31) to obtain
(32)
1
2
d
d
t
∥
Λ
α
u
m
∥
L
2
(
𝕋
n
)
≤
Re
(
Λ
α
f
(
t
)
,
Λ
α
u
m
)
.
By Gronwall’s inequality again, we have
(33)
u
m
is
bounded
in
L
∞
(
0
,
T
;
H
α
(
𝕋
n
)
)
.
By multiplying (26) by c
j
m
′
¯
(
t
), summing with 1
≤
j
≤
m, and taking t
=
0, we have
(34)
∥
∂
u
m
∂
t
(
0
)
∥
L
2
(
𝕋
n
)
2
-
i
2
d
d
t
|
t
=
0
∥
Λ
α
u
m
(
t
)
∥
L
2
(
𝕋
n
)
+
(
|
u
m
(
0
)
|
ρ
u
m
(
0
)
,
∂
u
m
∂
t
(
0
)
)
=
(
f
(
0
)
,
∂
u
m
∂
t
(
0
)
)
.
Looking into the real part of (34), we see
(35)
∥
∂
u
m
∂
t
(
0
)
∥
L
2
(
𝕋
n
)
2
≤
(
∥
u
m
(
0
)
∥
L
2
(
ρ
+
1
)
(
𝕋
n
)
+
∥
f
(
0
)
∥
L
2
(
𝕋
n
)
)
∥
∂
u
m
∂
t
(
0
)
∥
L
2
(
𝕋
n
)
.
Thus,
(36)
∥
∂
u
m
∂
t
(
0
)
∥
L
2
(
𝕋
n
)
≤
C
.
Now we can obtain the estimate foru
m
′
(
t
). By differentiating (26) with respect to t, we get
(37)
(
∂
2
u
m
∂
t
2
(
t
)
,
ω
j
)
+
i
(
∂
Λ
2
α
u
m
∂
t
(
t
)
,
ω
j
)
+
(
∂
∂
t
(
|
u
m
(
t
)
|
ρ
u
m
(
t
)
)
,
ω
j
)
=
(
∂
f
∂
t
(
t
)
,
ω
j
)
,
1
≤
j
≤
m
.
By multiplying (37) by c
j
m
′
¯
(
t
) and summing with 1
≤
j
≤
m, we have
(38)
1
2
d
d
t
∥
∂
u
m
∂
t
(
t
)
∥
L
2
(
𝕋
n
)
2
-
i
2
∥
∂
Λ
α
u
m
∂
t
(
t
)
∥
L
2
(
𝕋
n
)
+
(
∂
∂
t
(
|
u
m
(
t
)
|
ρ
u
m
(
t
)
)
,
∂
u
m
∂
t
(
t
)
)
=
(
∂
f
∂
t
(
t
)
,
∂
u
m
∂
t
(
t
)
)
.
Notice that
(39)
(
∂
∂
t
(
|
u
m
(
t
)
|
ρ
u
m
(
t
)
)
,
∂
u
m
∂
t
(
t
)
)
=
∫
𝕋
n
|
u
m
(
t
)
|
ρ
|
∂
u
m
∂
t
(
t
)
|
2
d
x
+
ρ
2
∫
𝕋
n
|
u
m
(
t
)
|
ρ
-
2
h
h
h
h
h
h
h
h
h
h
×
(
|
u
m
(
t
)
∂
u
m
¯
∂
t
(
t
)
|
2
+
|
u
m
(
t
)
|
2
|
∂
u
m
∂
t
|
2
)
d
x
≥
0
.
Taking the real part of (38) then yields
(40)
d
d
t
∥
∂
u
m
∂
t
(
t
)
∥
L
2
(
𝕋
n
)
2
≤
2
Re
(
∂
f
∂
t
(
t
)
,
∂
u
m
∂
t
(
t
)
)
≤
∥
∂
f
∂
t
(
t
)
∥
L
2
(
𝕋
n
)
2
+
∥
∂
u
m
∂
t
(
t
)
∥
L
2
(
𝕋
n
)
2
.
By Gronwall’s inequality, we deduce(41)
∂
u
m
∂
t
is
bounded
in
L
∞
(
0
,
T
;
L
2
(
𝕋
n
)
)
.
Step  3. Convergence process.
By (30), (33), and (41), we have, up to a subsequence, still denoted by u
m, that
(42)
u
m
⇀
u
weakly
*
in
L
∞
(
0
,
T
;
H
α
(
𝕋
n
)
)
,
as
m
⟶
∞
,
(43)
u
m
⇀
u
weakly
in
L
p
(
0
,
T
;
L
p
(
𝕋
n
)
)
,
as
m
⟶
∞
,
(44)
∂
u
m
∂
t
⇀
∂
u
∂
t
weakly
in
L
∞
(
0
,
T
;
L
2
(
𝕋
n
)
)
,
as
m
⟶
∞
.
Thus by Lemma 3, we find
(45)
u
m
⟶
u
strongly
in
L
∞
(
0
,
T
;
L
2
(
𝕋
n
)
)
,
a
.
e
.
𝕋
n
,
0068
hhhhhhhhhhhhhhhhhhhhh
as
m
⟶
∞
.
By (44), there exists a function w
=
w
(
x
,
t
) such that
(46)
|
u
m
|
ρ
u
m
⇀
w
(
x
,
t
)
weakly
in
L
q
(
0
,
T
;
L
q
(
𝕋
n
)
)
,
hhhhhhhhhhhhhhhhhhhhh
as
m
⟶
∞
,
with (
1
/
p
)
+
(
1
/
q
)
=
1. By Lemma 2 and the fact that
(47)
|
u
m
|
ρ
u
m
⟶
|
u
|
ρ
u
,
a
.
e
.
𝕋
n
,
we see
(48)
w
=
|
u
|
ρ
u
.
Fix j; we now pass to limit m
→
∞ in (26) to deduce
(49)
(
∂
u
∂
t
(
t
)
,
ω
j
)
-
i
(
Λ
α
u
(
t
)
,
Λ
α
ω
j
)
+
(
|
u
(
t
)
|
ρ
u
(
t
)
,
ω
j
)
=
(
f
(
t
)
,
ω
j
)
.
A simple density argument then shows
(50)
(
∂
u
∂
t
(
t
)
,
v
)
+
i
(
Λ
α
u
(
t
)
,
Λ
α
v
)
+
(
|
u
(
t
)
|
ρ
u
(
t
)
,
v
)
=
(
f
(
t
)
,
v
)
,
for all v
∈
H
α
(
𝕋
n
)
∩
L
p
(
𝕋
n
).
The proof of Theorem8 is completed.


## 3. Uniqueness of Weak Solutions

In this section, we will discuss the uniqueness of weak solutions of (5)-(6). More precisely, we have the following theorem.Theorem 9.
Assume as in Theorem8. Then there exists a unique weak solution of (5)-(6).Proof.
Letu and v be two weak solutions of (5)-(6), given in Theorem 8, with the same datum. Then w
=
u
-
v satisfies
(51)
∂
w
∂
t
+
i
Λ
2
α
w
=
|
v
|
ρ
v
-
|
u
|
ρ
u
,
(52)
w
(
0
)
=
0
.
Also, we have
(53)
w
∈
L
∞
(
0
,
T
;
H
α
(
𝕋
n
)
)
∩
L
p
(
0
,
T
;
L
p
(
𝕋
n
)
)
,
∂
w
∂
t
∈
L
∞
(
0
,
T
;
L
2
(
𝕋
n
)
)
.
By taking the inner product of (51) with w in L
2
(
𝕋
n
), we obtain
(54)
1
2
d
d
t
∥
w
∥
L
2
(
𝕋
n
)
2
-
i
∥
w
∥
H
α
(
𝕋
n
)
2
=
∫
𝕋
n
(
|
v
|
ρ
v
-
|
u
|
ρ
u
)
(
u
-
v
)
¯
d
x
.
By taking the real part of (54) and noticing
(55)
Re
(
∫
𝕋
n
(
|
v
|
ρ
v
-
|
u
|
ρ
u
)
(
u
-
v
)
¯
d
x
)
≤
0
,
we have
(56)
∥
w
∥
L
2
(
𝕋
n
)
2
≤
∥
w
(
0
)
∥
L
2
(
𝕋
n
)
2
=
0
.
Thus w
=
0 and u
≡
v a.e. in 𝕋
n.



---

*Source: 569693-2014-04-02.xml*