# Asymptotic Study of the 2D-DQGE Solutions

Authors: Jamel Benameur; Mongi Blel
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We study the regularity of the solutions of the surface quasi-geostrophic equation with subcritical exponent1
/
2
<
α
≤
1. We prove that if the initial data is small enough in the critical space H
˙
2
-
2
α
(
R
2
), then the regularity of the solution is of exponential growth type with respect to time and its H
˙
2
-
2
α
(
R
2
) norm decays exponentially fast. It becomes then infinitely differentiable with respect to time and has value in all homogeneous Sobolev spaces H
˙
s
(
R
2
) for s
≥
2
-
2
α. Moreover, we give some general properties of the global solutions.


---

## Body

## 1. Introduction

We consider the2
D dissipative quasi-geostrophic equation with subcritical exponent 1
/
2
<
α
≤
1:
(
S
α
)
∂
t
θ
+
(
-
Δ
)
α
θ
+
(
u
·
∇
)
θ
=
0
in
R
+
×
R
2
,
θ
(
0
,
x
)
=
θ
0
(
x
)
in
R
2
,
where x
∈
R
2, t
>
0, θ
=
θ
(
x
,
t
) is the unknown potential temperature, and u
=
(
u
1
,
u
2
) is the divergence free velocity which is determined by the Riesz transformation of θ in the following sense:
(1)
u
1
=
-
R
2
θ
=
-
∂
2
(
-
Δ
)
-
1
/
2
θ
,
u
2
=
R
1
θ
=
∂
1
(
-
Δ
)
-
1
/
2
θ
.
The critical homogeneous Sobolev space is H
˙
2
-
2
α
(
R
2
) and we have
(2)
∥
λ
2
α
-
1
f
(
λ
.
)
∥
H
˙
2
-
2
α
=
∥
f
∥
H
˙
2
-
2
α
,
∀
λ
>
0
.In [1], we studied the existence of global solutions of (
S
α
) if the initial data θ
0 is small in the critical space H
˙
2
-
2
α
(
R
2
) and the subcritical exponent α
∈
(
1
/
2
,
1
]. In use of Theorem 4.2 in [2] with p
=
q
=
2, we proved the following Theorem.Theorem 1 (see [1]).
Forα
∈
(
1
/
2
,
1
] and θ
0
∈
H
˙
2
-
2
α
(
R
2
), there exists a constant c
α
>
0 such that if
(3)
∥
θ
0
∥
H
˙
2
-
2
α
<
c
α
,
the initial value problem (
S
α
) has a unique solution in C
b
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
∩
L
2
(
R
+
,
H
˙
2
-
α
(
R
2
)
). Moreover,
(4)
∥
θ
(
t
)
∥
H
˙
2
-
2
α
2
+
∫
0
t
‍
∥
θ
(
τ
)
∥
H
˙
2
-
α
2
d
τ
≤
∥
θ
0
∥
H
˙
2
-
2
α
2
,
∀
t
≥
0
.We proved also the following result.Theorem 2 (see [1]).
Let2
/
3
≤
α
≤
1.(i)
Ifθ
∈
C
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
) is a global solution of (
S
α
), then
(5)
lim
⁡
t
→
∞
∥
θ
(
t
)
∥
H
˙
2
-
2
α
=
0
.
(ii)
Ifθ
∈
C
(
R
+
,
H
2
-
2
α
(
R
2
)
) is a global solution of (
S
α
), then
(6)
lim
⁡
t
→
∞
∥
θ
(
t
)
∥
H
2
-
2
α
=
0
.In this paper, we describe the long time behavior of these solutions with respect to the homogeneous Sobolev norm∥
·
∥
H
˙
s, for s
≥
2
-
2
α. We prove the following.Theorem 3.
There existsc
α
>
0 such that, for all θ
0
∈
H
˙
2
-
2
α
(
R
2
), ∥
θ
0
∥
H
˙
2
-
2
α
<
c
α, and there exists a global solution θ
∈
C
b
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
∩
L
2
(
R
+
,
H
˙
2
-
α
(
R
2
)
) such that, for all s
>
2
-
2
α, θ
(
t
)
∈
H
˙
s
(
R
2
) for all t
>
0, and
(7)
∥
θ
(
t
)
∥
H
˙
s
=
O
(
t
-
(
s
-
(
2
-
2
α
)
)
/
2
α
)
,
t
⟶
∞
.When the initial data is inH
˙
2
-
2
α
(
R
2
)
∩
L
2
(
R
2
) and small enough in the homogeneous space H
˙
2
-
2
α
(
R
2
), we prove that the Leray solution is also in all Sobolev spaces H
˙
s
(
R
2
). Moreover, we describe the long time behavior of its homogeneous Sobolev norm ∥
·
∥
H
˙
s, for s
≥
0. We state also the following.Theorem 4.
There existsc
α
>
0 such that, for all θ
0
∈
H
2
-
2
α
(
R
2
), ∥
θ
0
∥
H
˙
2
-
2
α
<
c
α, and there exists a global solution θ
∈
C
b
(
R
+
,
H
2
-
2
α
(
R
2
)
)
∩
L
2
(
R
+
,
H
˙
2
-
α
(
R
2
)
) such that, for all s
≥
0,
(8)
∥
θ
(
t
)
∥
H
˙
s
=
O
(
t
-
s
/
2
α
)
,
t
⟶
∞
.The paper is organized as follows. We start by recalling some preliminary background and stating useful preliminary results on Sobolev spaces. Sections3 and 4 are devoted to the proof of the main results, Theorems 3 and 4. In Section 5, we give some general properties for any global solutions of the system (
S
α
).


## 2. Notations and Preliminary Results

### 2.1. Notations and Technical Lemmas

In this short section, we collect some notations and definitions that will be used later and we give some technical lemmas.(i)
The Fourier transformation inR
2 is normalized as
(9)
F
(
f
)
(
ξ
)
=
f
^
(
ξ
)
=
∫
R
2
‍
exp
⁡
⁡
(
-
i
x
·
ξ
)
f
(
x
)
d
x
,
kkkkkkkkkkkkkk
ξ
=
(
ξ
1
,
ξ
2
)
∈
R
2
.
(ii)
The inverse Fourier formula is(10)
F
-
1
(
g
)
(
x
)
=
(
2
π
)
-
2
∫
‍
exp
⁡
(
i
ξ
·
x
)
f
(
ξ
)
d
ξ
,
k
k
k
k
k
k
k
k
k
k
x
=
(
x
1
,
x
2
)
∈
R
2
.
(iii)
Fors
∈
R, H
s
(
R
2
) denotes the usual nonhomogeneous Sobolev space on R
2 and 〈
·
,
·
〉
H
s its scalar product.
(iv)
Fors
∈
R, H
˙
s
(
R
2
) denotes the usual homogeneous Sobolev space on R
2 and 〈
·
,
·
〉
H
˙
s its scalar product.
(v)
The convolution product of a suitable pair of functionsf and g on R
2 is given by
(11)
(
f
*
g
)
(
x
)
∶
=
∫
R
2
‍
f
(
y
)
g
(
x
-
y
)
d
y
.
(vi)
For any Banach space(
B
,
∥
·
∥
), any real number 1
≤
p
≤
∞, and any time T
>
0, we will denote by L
T
p
(
B
) the space of measurable functions t
∈
[
0
,
T
]
↦
f
(
t
)
∈
B such that (
t
↦
∥
f
(
t
)
∥
)
∈
L
p
(
[
0
,
T
]
).
(vii)
Iff
=
(
f
1
,
f
2
) and g
=
(
g
1
,
g
2
) are two vector fields, we set
(12)
f
⊗
g
∶
=
(
g
1
f
,
g
2
f
)
,
div
⁡
(
f
⊗
g
)
:
=
(
div
⁡
(
g
1
f
)
,
div
⁡
(
g
2
f
)
)
.
(viii)
For any subsetX of a set E, 1
X denotes the characteristic function of X.We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.Lemma 5 (see [3]).
Lets
1, s
2 be two real numbers such that
(13)
s
1
<
1
,
s
1
+
s
2
>
0
.
There exists a constant C
∶
=
C
(
s
1
,
s
2
), such that, for all f
,
g
∈
H
˙
s
1
(
R
2
)
∩
H
˙
s
2
(
R
2
),
(14)
∥
f
g
∥
H
˙
s
1
+
s
2
-
1
≤
C
(
∥
f
∥
H
˙
s
1
∥
g
∥
H
˙
s
2
+
∥
f
∥
H
˙
s
2
∥
g
∥
H
˙
s
1
)
.
If s
1
,
s
2
<
1 and s
1
+
s
2
>
0, there exists a constant c
=
c
(
s
1
,
s
2
), such that, for all f
∈
H
˙
s
1
(
R
2
) and g
∈
H
˙
s
2
(
R
2
),
(15)
∥
f
g
∥
H
˙
s
1
+
s
2
-
1
≤
c
∥
f
∥
H
˙
s
1
∥
g
∥
H
˙
s
2
.For the proof of the main results, we need the following lemma.Lemma 6.
Under the same conditions as in Theorem3, for all σ
≥
0 and ε
≥
0,
(16)
∫
R
2
‍
|
ξ
|
2
σ
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
F
(
w
)
|
d
ξ
≤
C
∥
θ
ε
∥
H
˙
2
-
2
α
∥
θ
ε
∥
H
˙
σ
+
α
∥
w
ε
∥
H
˙
σ
+
α
,
where F
(
θ
ε
)
=
e
2
ε
t
|
ξ
|
F
(
θ
) and F
(
w
ε
)
=
e
2
ε
t
|
ξ
|
F
(
w
)
(
t
,
ξ
) and w
ε
∈
H
˙
σ
+
α
(
R
2
).Remarks 7.
(i) Ifσ
=
0, formula (16) gives
(17)
∫
R
2
‍
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
C
∥
θ
ε
∥
H
˙
2
-
2
α
∥
θ
ε
∥
H
˙
α
∥
w
ε
∥
H
˙
α
.
(ii)
Ifε
=
0 and σ
=
0, formula (16) gives
(18)
∫
R
2
‍
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
C
∥
θ
∥
H
˙
2
-
2
α
∥
θ
∥
H
˙
α
∥
w
∥
H
˙
α
.
(iii)
Ifε
=
0 and σ
=
2
-
2
α, formula (16) gives
(19)
∫
R
2
‍
|
ξ
|
2
(
2
-
2
α
)
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
C
∥
θ
∥
H
˙
2
-
2
α
∥
θ
∥
H
˙
2
-
α
∥
w
∥
H
˙
2
-
α
.Proof of Lemma6.
By the Cauchy-Schwarz inequality, we get(20)
∫
‍
|
ξ
|
2
σ
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
∫
‍
|
ξ
|
σ
-
α
e
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
|
e
ε
t
|
ξ
|
|
ξ
|
σ
+
α
|
w
^
|
d
ξ
≤
(
∫
‍
|
ξ
|
2
(
σ
-
α
)
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
|
2
d
ξ
)
1
/
2
∥
w
ε
∥
H
˙
σ
+
α
.
Using the weak derivatives properties, the elementary inequality e
a
|
ξ
|
≤
e
a
|
ξ
-
η
|
e
a
|
ξ
|, with a
≥
0 and ξ
,
η
∈
R
2, and the product laws (Lemma 5), with s
1
+
s
2
=
σ
-
α
+
2
>
0, s
1
=
2
-
2
α
<
1, and s
2
=
σ
+
α, we can dominate the nonlinear part of (20) as follows:
(21)
∫
‍
|
ξ
|
2
(
σ
-
α
)
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
|
2
d
ξ
≤
∫
‍
|
ξ
|
2
(
σ
-
α
+
1
)
e
2
ε
t
|
ξ
|
(
|
θ
^
|
*
|
θ
^
|
)
2
d
ξ
≤
∫
‍
|
ξ
|
2
(
σ
-
α
+
1
)
·
(
e
2
ε
t
|
·
|
|
θ
^
|
*
e
2
ε
t
|
·
|
|
θ
^
|
)
2
d
ξ
≤
C
∥
θ
ε
∥
H
˙
2
-
2
α
2
∥
θ
ε
∥
H
˙
σ
+
α
2
.


## 2.1. Notations and Technical Lemmas

In this short section, we collect some notations and definitions that will be used later and we give some technical lemmas.(i)
The Fourier transformation inR
2 is normalized as
(9)
F
(
f
)
(
ξ
)
=
f
^
(
ξ
)
=
∫
R
2
‍
exp
⁡
⁡
(
-
i
x
·
ξ
)
f
(
x
)
d
x
,
kkkkkkkkkkkkkk
ξ
=
(
ξ
1
,
ξ
2
)
∈
R
2
.
(ii)
The inverse Fourier formula is(10)
F
-
1
(
g
)
(
x
)
=
(
2
π
)
-
2
∫
‍
exp
⁡
(
i
ξ
·
x
)
f
(
ξ
)
d
ξ
,
k
k
k
k
k
k
k
k
k
k
x
=
(
x
1
,
x
2
)
∈
R
2
.
(iii)
Fors
∈
R, H
s
(
R
2
) denotes the usual nonhomogeneous Sobolev space on R
2 and 〈
·
,
·
〉
H
s its scalar product.
(iv)
Fors
∈
R, H
˙
s
(
R
2
) denotes the usual homogeneous Sobolev space on R
2 and 〈
·
,
·
〉
H
˙
s its scalar product.
(v)
The convolution product of a suitable pair of functionsf and g on R
2 is given by
(11)
(
f
*
g
)
(
x
)
∶
=
∫
R
2
‍
f
(
y
)
g
(
x
-
y
)
d
y
.
(vi)
For any Banach space(
B
,
∥
·
∥
), any real number 1
≤
p
≤
∞, and any time T
>
0, we will denote by L
T
p
(
B
) the space of measurable functions t
∈
[
0
,
T
]
↦
f
(
t
)
∈
B such that (
t
↦
∥
f
(
t
)
∥
)
∈
L
p
(
[
0
,
T
]
).
(vii)
Iff
=
(
f
1
,
f
2
) and g
=
(
g
1
,
g
2
) are two vector fields, we set
(12)
f
⊗
g
∶
=
(
g
1
f
,
g
2
f
)
,
div
⁡
(
f
⊗
g
)
:
=
(
div
⁡
(
g
1
f
)
,
div
⁡
(
g
2
f
)
)
.
(viii)
For any subsetX of a set E, 1
X denotes the characteristic function of X.We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.Lemma 5 (see [3]).
Lets
1, s
2 be two real numbers such that
(13)
s
1
<
1
,
s
1
+
s
2
>
0
.
There exists a constant C
∶
=
C
(
s
1
,
s
2
), such that, for all f
,
g
∈
H
˙
s
1
(
R
2
)
∩
H
˙
s
2
(
R
2
),
(14)
∥
f
g
∥
H
˙
s
1
+
s
2
-
1
≤
C
(
∥
f
∥
H
˙
s
1
∥
g
∥
H
˙
s
2
+
∥
f
∥
H
˙
s
2
∥
g
∥
H
˙
s
1
)
.
If s
1
,
s
2
<
1 and s
1
+
s
2
>
0, there exists a constant c
=
c
(
s
1
,
s
2
), such that, for all f
∈
H
˙
s
1
(
R
2
) and g
∈
H
˙
s
2
(
R
2
),
(15)
∥
f
g
∥
H
˙
s
1
+
s
2
-
1
≤
c
∥
f
∥
H
˙
s
1
∥
g
∥
H
˙
s
2
.For the proof of the main results, we need the following lemma.Lemma 6.
Under the same conditions as in Theorem3, for all σ
≥
0 and ε
≥
0,
(16)
∫
R
2
‍
|
ξ
|
2
σ
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
F
(
w
)
|
d
ξ
≤
C
∥
θ
ε
∥
H
˙
2
-
2
α
∥
θ
ε
∥
H
˙
σ
+
α
∥
w
ε
∥
H
˙
σ
+
α
,
where F
(
θ
ε
)
=
e
2
ε
t
|
ξ
|
F
(
θ
) and F
(
w
ε
)
=
e
2
ε
t
|
ξ
|
F
(
w
)
(
t
,
ξ
) and w
ε
∈
H
˙
σ
+
α
(
R
2
).Remarks 7.
(i) Ifσ
=
0, formula (16) gives
(17)
∫
R
2
‍
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
C
∥
θ
ε
∥
H
˙
2
-
2
α
∥
θ
ε
∥
H
˙
α
∥
w
ε
∥
H
˙
α
.
(ii)
Ifε
=
0 and σ
=
0, formula (16) gives
(18)
∫
R
2
‍
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
C
∥
θ
∥
H
˙
2
-
2
α
∥
θ
∥
H
˙
α
∥
w
∥
H
˙
α
.
(iii)
Ifε
=
0 and σ
=
2
-
2
α, formula (16) gives
(19)
∫
R
2
‍
|
ξ
|
2
(
2
-
2
α
)
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
C
∥
θ
∥
H
˙
2
-
2
α
∥
θ
∥
H
˙
2
-
α
∥
w
∥
H
˙
2
-
α
.Proof of Lemma6.
By the Cauchy-Schwarz inequality, we get(20)
∫
‍
|
ξ
|
2
σ
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
w
^
|
d
ξ
≤
∫
‍
|
ξ
|
σ
-
α
e
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
|
e
ε
t
|
ξ
|
|
ξ
|
σ
+
α
|
w
^
|
d
ξ
≤
(
∫
‍
|
ξ
|
2
(
σ
-
α
)
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
|
2
d
ξ
)
1
/
2
∥
w
ε
∥
H
˙
σ
+
α
.
Using the weak derivatives properties, the elementary inequality e
a
|
ξ
|
≤
e
a
|
ξ
-
η
|
e
a
|
ξ
|, with a
≥
0 and ξ
,
η
∈
R
2, and the product laws (Lemma 5), with s
1
+
s
2
=
σ
-
α
+
2
>
0, s
1
=
2
-
2
α
<
1, and s
2
=
σ
+
α, we can dominate the nonlinear part of (20) as follows:
(21)
∫
‍
|
ξ
|
2
(
σ
-
α
)
e
2
ε
t
|
ξ
|
|
F
(
(
u
·
∇
)
θ
)
|
2
d
ξ
≤
∫
‍
|
ξ
|
2
(
σ
-
α
+
1
)
e
2
ε
t
|
ξ
|
(
|
θ
^
|
*
|
θ
^
|
)
2
d
ξ
≤
∫
‍
|
ξ
|
2
(
σ
-
α
+
1
)
·
(
e
2
ε
t
|
·
|
|
θ
^
|
*
e
2
ε
t
|
·
|
|
θ
^
|
)
2
d
ξ
≤
C
∥
θ
ε
∥
H
˙
2
-
2
α
2
∥
θ
ε
∥
H
˙
σ
+
α
2
.


## 3. Proof of Theorem3

To prove Theorem3, we need the following result.Proposition 8.
There existsc
α
>
0 such that, for all θ
0
∈
H
˙
2
-
2
α
(
R
2
), ∥
θ
0
∥
H
˙
2
-
2
α
(
R
2
)
<
c
α, and there exists a global solution θ
∈
C
b
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
∩
L
2
(
R
+
,
H
˙
2
-
α
(
R
2
)
) such that
(22)
∫
‍
e
t
1
/
2
α
|
ξ
|
|
ξ
|
2
(
2
-
2
α
)
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
+
∫
0
t
‍
∫
‍
e
τ
t
-
1
+
1
/
2
α
|
ξ
|
|
ξ
|
2
(
2
-
α
)
|
θ
^
(
τ
,
ξ
)
|
2
d
ξ
d
τ
≤
2
∥
θ
0
∥
H
˙
2
-
2
α
2
.Proof.
The proof is done in two steps.
First Step. For a nonnegative integer n, Friedrich’s operator J
n is defined by
(23)
J
n
(
f
)
∶
=
F
-
1
(
1
{
1
/
n
<
|
ξ
|
<
n
}
f
^
)
.
Consider the following approximate system (
S
α
n
) on R
+
×
R
2:
(24)
∂
t
θ
+
(
-
Δ
)
α
J
n
θ
+
J
n
(
J
n
u
·
∇
J
n
θ
)
=
0
,
u
=
(
-
Δ
)
-
1
/
2
(
-
∂
2
θ
,
∂
1
θ
)
,
θ
∣
t
=
0
=
J
n
θ
0
.
Then, by the ordinary differential equations theory, the system (
S
α
n
) has a unique maximal solution θ
n in the space C
1
(
[
0
,
T
n
*
)
,
L
2
(
R
2
)
), T
n
*
>
0. Using the uniqueness and the fact that J
n
2
=
J
n, we obtain J
n
θ
n
=
θ
n and
(25)
∂
t
θ
n
+
(
-
Δ
)
α
θ
n
+
J
n
(
u
n
·
∇
θ
n
)
=
0
,
u
n
=
(
-
Δ
)
-
1
/
2
(
-
∂
2
θ
n
,
∂
1
θ
n
)
,
θ
n
∣
t
=
0
=
J
n
θ
0
.
Taking the scalar product in L
2
(
R
2
), we obtain, for t
∈
[
0
,
T
n
*
),
(26)
∂
t
∥
θ
n
∥
L
2
2
+
2
∥
θ
n
∥
H
˙
α
2
≤
0
.
It follows that, for all t
∈
[
0
,
T
n
*
), ∥
θ
n
(
t
)
∥
L
2
2
≤
∥
J
n
θ
0
∥
L
2
2, which implies that T
n
*
=
+
∞.
Now, taking scalar product inH
˙
2
-
2
α
(
R
2
), we obtain
(27)
∂
t
∥
θ
n
∥
H
˙
2
-
2
α
2
+
2
∥
θ
n
∥
H
˙
2
-
α
2
≤
2
|
〈
J
n
(
u
n
·
∇
θ
n
)
,
θ
n
〉
H
˙
2
-
2
α
|
,
|
〈
J
n
(
u
n
·
∇
θ
n
)
,
θ
n
〉
H
˙
2
-
2
α
|
=
|
〈
u
n
·
∇
θ
n
,
J
n
θ
n
〉
H
˙
2
-
2
α
|
=
|
〈
div
⁡
(
θ
n
u
n
)
,
θ
n
〉
H
˙
2
-
2
α
|
≤
∥
div
⁡
(
θ
n
u
n
)
∥
H
˙
2
-
α
∥
θ
n
∥
H
˙
2
-
α
≤
∥
θ
n
u
n
∥
H
˙
2
-
3
α
+
1
∥
θ
n
∥
H
˙
2
-
α
.
Using product law (15) with s
1
=
2
-
2
α
<
1 and s
2
=
2
-
α, we obtain
(28)
|
〈
J
n
(
u
n
·
∇
θ
n
)
,
θ
n
〉
H
˙
2
-
2
α
|
≤
c
(
α
)
(
∥
θ
n
∥
H
˙
2
-
2
α
∥
u
n
∥
H
˙
2
-
α
+
∥
θ
n
∥
H
˙
2
-
α
∥
u
n
∥
H
˙
2
-
2
α
)
∥
θ
n
∥
H
˙
2
-
α
.
But
(29)
∥
u
n
∥
H
˙
s
2
=
∫
‍
|
ξ
|
2
s
|
θ
n
^
(
t
,
ξ
)
|
2
|
(
i
ξ
2
|
ξ
|
,
-
i
ξ
1
|
ξ
|
)
|
2
d
ξ
=
∥
θ
n
∥
H
˙
s
2
,
|
〈
J
n
(
u
n
·
∇
θ
n
)
,
θ
n
〉
H
˙
2
-
2
α
|
≤
c
(
α
)
∥
θ
n
∥
H
˙
2
-
2
α
∥
θ
n
∥
H
˙
2
-
α
2
.
Then,
(30)
∂
t
∥
θ
n
∥
H
˙
2
-
2
α
2
+
2
∥
θ
n
∥
H
˙
2
-
α
2
≤
c
(
α
)
∥
θ
n
∥
H
˙
2
-
2
α
∥
θ
n
∥
H
˙
2
-
α
2
.
Let(31)
T
n
∶
=
sup
⁡
⁡
{
t
≥
0
,
∥
θ
n
∥
L
t
∞
(
H
˙
2
-
2
α
)
<
2
c
α
}
.
For 0
≤
t
<
T
n, by (30), we have
(32)
∥
θ
n
(
t
)
∥
H
˙
2
-
2
α
2
+
2
∫
0
t
‍
∥
θ
n
∥
H
˙
2
-
α
2
≤
∥
θ
0
∥
H
˙
2
-
2
α
2
+
∫
0
t
‍
∥
θ
n
∥
H
˙
2
-
α
2
d
τ
;
∥
θ
n
(
t
)
∥
H
˙
2
-
2
α
2
+
∫
0
t
‍
∥
θ
n
∥
H
˙
2
-
α
2
≤
∥
θ
0
∥
H
˙
2
-
2
α
2
<
c
α
2
,
then T
n
=
∞, and, for all t
≥
0, we have
(33)
∥
θ
n
(
t
)
∥
H
˙
2
-
2
α
2
+
∫
0
t
‍
∥
θ
n
∥
H
˙
2
-
α
2
≤
∥
θ
0
∥
H
˙
2
-
2
α
2
.
If we take the limit when n goes to the infinity, we find a solution θ
∈
C
b
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
∩
L
2
(
R
+
,
H
˙
2
-
α
(
R
2
)
) which satisfies
(34)
∥
θ
(
t
)
∥
H
˙
2
-
2
α
2
+
∫
0
t
‍
∥
θ
∥
H
˙
2
-
α
2
≤
∥
θ
0
∥
H
˙
2
-
2
α
2
,
which proves the first result of Theorem 3.
Second Step. Back to the approximate system,
(35)
∂
t
θ
n
^
+
|
ξ
|
2
α
θ
n
^
+
1
1
/
n
<
|
ξ
|
<
n
F
(
u
n
·
∇
θ
n
)
=
0
,
∂
t
|
θ
n
^
|
2
+
2
|
ξ
|
2
α
|
θ
n
^
|
2
+
2
Re
(
F
(
u
n
·
∇
θ
n
)
(
ξ
)
·
θ
n
^
(
-
ξ
)
)
=
0
.
For ε
>
0, we define
(36)
f
n
=
f
n
,
ε
∶
=
F
-
1
(
e
ε
t
|
ξ
|
θ
n
^
)
,
f
n
^
=
e
ε
t
|
ξ
|
θ
n
^
.
Then,
(37)
∂
t
|
f
n
^
|
2
+
2
|
ξ
|
2
α
|
f
n
^
|
2
=
2
ε
|
f
n
^
|
2
-
2
Re
e
2
ε
t
|
ξ
|
(
F
(
u
n
·
∇
θ
n
)
(
ξ
)
·
θ
n
^
(
-
ξ
)
)
=
2
ε
|
ξ
|
·
|
f
n
^
|
2
-
2
Re
(
e
ε
t
|
ξ
|
F
(
u
n
·
∇
θ
n
)
(
ξ
)
·
e
ε
t
|
ξ
|
θ
n
^
(
-
ξ
)
)
=
2
ε
|
ξ
|
·
|
f
n
^
|
2
-
2
Re
(
e
ε
t
|
ξ
|
F
(
u
n
·
∇
θ
n
)
(
ξ
)
·
f
n
^
(
-
ξ
)
)
=
2
ε
|
ξ
|
·
|
f
n
^
|
2
-
2
Re
(
e
ε
t
|
ξ
|
F
(
div
⁡
(
θ
n
u
n
)
)
(
ξ
)
·
f
n
^
(
-
ξ
)
)
≤
2
ε
|
ξ
|
·
|
f
n
^
|
2
+
2
e
ε
t
|
ξ
|
|
ξ
|
·
|
F
(
θ
n
u
n
)
|
·
|
f
n
^
(
ξ
)
|
≤
2
ε
|
ξ
|
·
|
f
n
^
|
2
+
2
e
ε
t
|
ξ
|
|
ξ
|
·
(
|
θ
n
^
|
*
|
u
n
^
|
)
·
|
f
n
^
(
ξ
)
|
≤
2
ε
|
ξ
|
·
|
f
n
^
|
2
+
2
e
ε
t
|
ξ
|
|
ξ
|
·
(
|
θ
n
^
|
*
|
θ
n
^
|
)
·
|
f
n
^
(
ξ
)
|
.
Using the classical inequality(38)
e
λ
|
ξ
|
≤
e
λ
|
ξ
-
η
|
e
λ
|
η
|
,
∀
λ
≥
0
,
we let
(39)
∂
t
|
f
n
^
|
2
+
2
|
ξ
|
2
α
|
f
n
^
|
2
≤
2
ε
|
ξ
|
·
|
f
n
^
|
2
+
2
|
ξ
|
(
|
f
n
^
|
*
|
f
n
^
|
)
·
|
f
n
^
|
.
Taking the norm in H
˙
2
-
2
α, we obtain
(40)
∂
t
∥
f
n
∥
H
˙
2
-
2
α
2
+
2
∥
f
n
∥
H
˙
2
-
α
2
≤
2
ε
∥
f
n
∥
H
˙
2
-
2
α
+
1
/
2
2
+
2
∫
‍
|
ξ
|
2
(
2
-
2
α
)
+
1
(
|
f
n
^
|
*
|
f
n
^
|
)
·
|
f
n
^
|
d
ξ
≤
2
ε
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
+
2
∫
‍
|
ξ
|
(
2
-
2
α
)
+
1
-
α
(
|
f
n
^
|
*
|
f
n
^
|
)
|
ξ
|
2
-
α
|
f
n
^
|
d
ξ
.
By Cauchy-Schwarz inequality, we have
(41)
∂
t
∥
f
n
∥
H
˙
2
-
2
α
2
+
2
∥
f
n
∥
H
˙
2
-
α
2
≤
2
ε
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
+
2
[
∫
‍
|
ξ
|
2
(
(
2
-
2
α
)
+
1
-
α
)
(
|
f
n
^
|
*
|
f
n
^
|
)
d
ξ
]
2
∥
f
n
∥
H
˙
s
c
+
α
≤
2
ε
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
+
2
∥
g
n
·
g
n
∥
H
˙
(
2
-
2
α
)
+
1
-
α
∥
f
n
∥
H
˙
2
-
α
,
where
(42)
g
n
∶
=
F
-
1
(
|
f
n
^
|
)
.
Using product law (15) in the homogeneous Sobolev space with s
1
=
2
-
2
α
<
1, s
2
=
2
-
α, and s
1
+
s
2
=
(
2
-
2
α
)
+
(
2
-
α
)
>
0, we obtain
(43)
∂
t
∥
f
n
∥
H
˙
2
-
2
α
2
+
2
∥
f
n
∥
H
˙
2
-
α
2
≤
2
ε
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
+
C
α
∥
g
n
∥
H
˙
2
-
2
α
∥
g
n
∥
H
˙
2
-
α
∥
f
n
∥
H
˙
2
-
α
.
Then,
(44)
∂
t
∥
f
n
∥
H
˙
2
-
2
α
2
+
2
∥
f
n
∥
H
˙
2
-
α
2
≤
2
ε
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
+
C
α
∥
f
n
∥
H
˙
2
-
2
α
∥
f
n
∥
H
˙
2
-
α
2
.
To estimate the term ∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2, we use the Hölder inequality and we get
(45)
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
≤
∥
f
n
∥
H
˙
2
-
2
α
2
(
1
-
1
/
2
α
)
∥
f
n
∥
H
˙
2
-
α
1
/
α
.
The convex inequality a
b
≤
a
p
/
p
+
b
q
/
q, with p
=
2
α
/
(
2
α
-
1
) and q
=
2
α, gives
(46)
ε
∥
f
n
∥
H
˙
(
2
-
2
α
)
+
1
/
2
2
≤
C
α
ε
2
α
/
(
2
α
-
1
)
∥
f
n
∥
H
˙
2
-
2
α
2
+
∥
f
n
∥
H
˙
2
-
α
2
.
Thus,
(47)
∂
t
∥
f
n
∥
H
˙
2
-
2
α
2
+
3
2
∥
f
n
∥
H
˙
2
-
α
2
≤
C
α
ε
2
α
/
(
2
α
-
1
)
∥
f
n
∥
H
˙
2
-
2
α
2
+
C
α
∥
f
n
∥
H
˙
2
-
2
α
∥
f
n
∥
H
˙
2
-
α
2
,
where f
n is in C
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
) and ∥
f
n
(
0
)
∥
H
˙
2
-
2
α
=
∥
θ
n
(
0
)
∥
H
˙
2
-
2
α
≤
∥
θ
0
∥
H
˙
2
-
2
α
<
c
α.
LetT
>
0 and ε
=
(
ln
⁡
⁡
(
2
)
/
C
α
T
)
(
2
α
-
1
)
/
2
α; we set
(48)
T
n
∶
=
sup
⁡
⁡
{
t
≥
0
;
∥
f
n
∥
L
t
∞
(
H
˙
2
-
2
α
)
<
2
c
α
}
.
For 0
≤
t
<
inf
⁡
⁡
(
T
,
T
n
), we have
(49)
∂
t
∥
f
n
∥
H
˙
2
-
2
α
2
+
∥
f
n
∥
H
˙
2
-
α
2
≤
C
α
ε
2
α
/
(
2
α
-
1
)
∥
f
n
∥
H
˙
2
-
2
α
2
.
By Gronwall lemma, we get that, for all t
∈
[
0
,
inf
⁡
⁡
(
T
,
T
n
)
),
(50)
∥
f
n
(
t
)
∥
H
˙
2
-
2
α
2
≤
∥
f
n
(
0
)
∥
H
˙
2
-
2
α
2
e
C
α
t
ε
2
α
/
(
2
α
-
1
)
≤
∥
J
n
θ
0
∥
H
˙
2
-
2
α
2
e
C
α
ε
2
α
/
(
2
α
-
1
)
inf
⁡
⁡
(
T
,
T
n
)
≤
∥
J
n
θ
0
∥
H
˙
2
-
2
α
2
e
C
α
ε
2
α
/
(
2
α
-
1
)
T
.
For the given value of ε, we have that, for all t
∈
[
0
,
inf
⁡
⁡
(
T
,
T
n
)
),
(51)
∥
f
n
(
t
)
∥
H
˙
2
-
2
α
2
≤
2
∥
θ
0
∥
H
˙
2
-
2
α
2
;
thus,
(52)
sup
⁡
t
∈
[
0
,
inf
⁡
⁡
(
T
,
T
n
)
)
∥
f
n
(
t
)
∥
H
˙
2
-
2
α
≤
2
∥
θ
0
∥
H
˙
2
-
2
α
<
2
c
α
.
It follows that T
n
>
T and, for all t
∈
[
0
,
T
],
(53)
∥
f
n
(
t
)
∥
H
˙
2
-
2
α
2
+
∫
0
t
‍
∥
f
n
(
τ
)
∥
H
˙
2
-
α
2
d
τ
≤
2
∥
θ
0
∥
H
˙
2
-
2
α
2
,
which proves formula (22), and the proof of Proposition 8 is finished.
Now we intend to study the behavior of the solution at infinity. We claim to prove that, for alls
≥
2
-
2
α,
(54)
∥
θ
(
t
)
∥
H
˙
s
=
O
(
1
t
(
s
-
(
2
-
2
α
)
)
/
2
α
)
,
t
⟶
∞
.
We can suppose that s
>
2
-
2
α. We have
(55)
∥
θ
(
t
)
∥
H
˙
s
2
=
∫
‍
|
ξ
|
2
s
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
=
∫
‍
|
ξ
|
2
s
-
2
(
2
-
2
α
)
e
-
t
1
/
2
α
|
ξ
|
|
ξ
|
2
(
2
-
2
α
)
e
t
1
/
2
α
|
ξ
|
|
θ
^
|
2
d
ξ
≤
t
-
(
2
s
-
2
(
2
-
2
α
)
)
/
2
α
×
∫
‍
(
|
ξ
|
t
1
/
2
α
e
-
1
/
(
2
s
-
2
(
2
-
2
α
)
)
t
1
/
2
α
|
ξ
|
)
2
s
-
2
(
2
-
2
α
)
k
k
k
k
k
k
×
|
ξ
|
2
(
2
-
2
α
)
e
t
1
/
2
α
|
ξ
|
|
θ
^
|
2
d
ξ
≤
M
t
-
(
2
s
-
2
(
2
-
2
α
)
)
/
2
α
∫
‍
|
ξ
|
2
(
2
-
2
α
)
e
t
1
/
2
α
|
ξ
|
|
θ
^
|
2
d
ξ
,
where M
:
=
sup
⁡
x
>
0
(
x
e
-
(
1
/
(
2
s
-
2
(
2
-
2
α
)
)
)
x
)
2
s
-
2
(
2
-
2
α
).
Using (22), we get
(56)
∥
θ
(
t
)
∥
H
˙
s
2
≤
M
∥
θ
0
∥
H
˙
2
-
2
α
2
t
-
(
s
-
(
2
-
2
α
)
)
/
α
and the proof of Theorem 3 is finished.Remark 9.
(i) Combining Theorems2 and 3, we can obtain, for 2
/
3
≤
α
≤
1 and s
≥
2
-
2
α,
(57)
∥
θ
(
t
)
∥
H
˙
s
2
=
o
(
t
-
(
s
-
(
2
-
2
α
)
)
/
2
α
)
,
t
⟶
∞
.
Indeed, from (34), ∥
θ
(
t
)
∥
H
˙
2
-
2
α
<
c
α, for all t
≥
0. For T
>
0, we consider the following system:
(58)
∂
t
v
+
(
-
Δ
)
α
v
+
(
V
·
∇
)
v
=
0
in
R
+
×
R
2
,
V
∶
=
(
-
Δ
)
-
1
/
2
(
-
∂
2
v
,
∂
1
v
)
,
v
(
0
,
·
)
=
θ
(
T
2
,
·
)
in
R
2
.
This system has a Leray unique solution v
T that satisfies, for all t
>
0,
(59)
∥
v
T
(
t
,
·
)
∥
H
˙
s
≤
M
∥
v
T
(
0
,
·
)
∥
H
˙
2
-
2
α
t
(
s
-
(
2
-
2
α
)
)
/
2
α
.
From the uniqueness of the solution, we have v
T
(
t
,
·
)
=
θ
(
t
+
T
/
2
,
·
); then
(60)
∥
θ
(
t
+
T
2
,
·
)
∥
H
˙
s
≤
M
∥
θ
(
T
/
2
,
·
)
∥
H
˙
2
-
2
α
t
(
s
-
(
2
-
2
α
)
)
/
2
α
.
For t
=
T
/
2, we have
(61)
∥
θ
(
T
,
·
)
∥
H
˙
s
≤
2
(
s
-
(
2
-
2
α
)
)
/
2
α
M
∥
θ
(
T
/
2
,
·
)
∥
H
˙
2
-
2
α
T
(
s
-
(
2
-
2
α
)
)
/
2
α
.
Combining this inequality with the result of Theorem 2, we obtain the desired result.
(ii) Ifα
∈
(
1
/
2
,
2
/
3
), we do not know if limsup
⁡
t
→
∞
∥
θ
(
t
)
∥
H
˙
(
2
-
2
α
)
=
0 holds. But this result depends on the lower frequencies. Indeed, for δ
>
0 and ε
>
0, we have
(62)
∫
|
ξ
|
>
δ
‍
|
ξ
|
2
(
2
-
2
α
)
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
=
∫
|
ξ
|
>
δ
‍
|
ξ
|
-
2
ε
|
ξ
|
2
(
2
-
2
α
)
+
2
ε
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
≤
δ
-
2
ε
∫
R
2
‍
|
ξ
|
2
(
2
-
2
α
)
+
2
ε
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
≤
δ
-
2
ε
∥
θ
(
t
)
∥
H
˙
(
2
-
2
α
)
+
ε
2
.
By Theorem 3, we obtain
(63)
∫
|
ξ
|
>
δ
‍
|
ξ
|
2
(
2
-
2
α
)
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
≤
M
∥
θ
0
∥
H
˙
(
2
-
2
α
)
2
δ
-
2
ε
t
-
ε
/
α
≤
M
∥
θ
0
∥
H
˙
(
2
-
2
α
)
2
(
δ
2
α
t
)
-
ε
/
α
.Then, forδ
=
t
-
1
/
2
α
+
a, a
>
0, we have
(64)
∫
|
ξ
|
>
δ
‍
|
ξ
|
2
(
2
-
2
α
)
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
≤
M
∥
θ
0
∥
H
˙
(
2
-
2
α
)
2
t
-
a
ε
/
α
,
limsup
⁡
t
→
∞
∫
|
ξ
|
>
t
-
1
/
2
α
+
a
‍
|
ξ
|
2
(
2
-
2
α
)
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
=
0
.
Then, to prove the result,
(65)
limsup
⁡
t
→
∞
∥
θ
(
t
)
∥
H
˙
(
2
-
2
α
)
=
0
.
It suffices to prove that
(66)
limsup
⁡
t
→
∞
∫
|
ξ
|
<
t
-
1
/
2
α
+
a
‍
|
ξ
|
2
(
2
-
2
α
)
|
θ
^
(
t
,
ξ
)
|
2
d
ξ
=
0
.


## 4. Proof of Theorem4

First Step. Using the approximate system (24) and inequality (17),
(67)
∂
t
∥
f
n
∥
L
2
2
+
2
∥
f
n
∥
H
˙
α
2
≤
2
ε
∥
f
n
∥
H
˙
1
/
2
2
+
C
α
∥
f
n
∥
H
˙
2
-
2
α
∥
f
n
∥
H
˙
α
2
.
Then,
(68)
∫
ξ
‍
e
t
1
/
2
α
|
ξ
|
|
θ
∧
(
t
,
ξ
)
|
2
d
ξ
+
∫
0
t
‍
∫
ξ
‍
e
τ
t
-
1
+
1
/
2
α
|
ξ
|
|
ξ
|
2
α
|
θ
∧
(
τ
,
ξ
)
|
2
d
ξ
d
τ
≤
2
∥
θ
0
∥
L
2
2
.Second Step. From relation (22), we deduce that
(69)
∥
θ
∥
H
˙
s
2
≤
∫
ξ
‍
|
ξ
|
2
s
e
-
t
1
/
2
α
|
ξ
|
e
t
1
/
2
α
|
ξ
|
|
F
(
θ
)
|
2
d
ξ
≤
t
-
s
/
α
sup
⁡
x
≥
0
x
2
s
e
-
x
·
2
∥
θ
0
∥
L
2
2
≤
C
t
-
s
/
α
∥
θ
0
∥
L
2
2
.
Then, the proof is achieved.Remark 10.
If2
/
3
≤
α
≤
1 and s
≥
0, we have
(70)
∥
θ
(
t
)
∥
H
˙
s
2
=
o
(
t
-
s
/
2
α
)
,
t
⟶
∞
.


## 5. General Properties of Global Solutions

Theorem 11.
Letθ be a global solution of (
S
α
) such that
(71)
θ
∈
C
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
.
Then, θ
∈
C
b
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
∩
L
2
(
[
T
,
∞
)
,
H
˙
2
-
α
(
R
2
)
) for some T
>
0. Moreover, for all s
≥
2
-
2
α,
(72)
∥
θ
(
t
)
∥
H
˙
s
=
o
(
t
-
(
s
-
(
2
-
2
α
)
)
/
2
α
)
,
t
⟶
∞
.Combining the energy estimate(73)
∥
θ
∥
L
2
2
+
2
∫
0
t
‍
∥
θ
∥
H
˙
α
2
d
τ
≤
∥
θ
0
∥
L
2
2
and the conclusion of Theorems 4 and 11, we get the following.Theorem 12.
Letθ be a global solution of (
S
α
) such that
(74)
θ
∈
C
(
R
+
,
H
2
-
2
α
(
R
2
)
)
.
Then, θ
∈
C
b
(
R
+
,
H
2
-
2
α
(
R
2
)
)
∩
L
2
(
[
T
,
∞
)
,
H
˙
2
-
α
(
R
2
)
) for some T
>
0.
Moreover, for alls
≥
0,
(75)
∥
θ
(
t
)
∥
H
˙
s
=
o
(
t
-
s
/
2
α
)
,
t
⟶
∞
.Remarks 13.
(a) Letα
∈
(
1
/
2
,
1
] and let θ be a global solution of (
S
α
) such that
(76)
θ
∈
C
(
R
+
,
H
˙
2
-
2
α
(
R
2
)
)
.
Using the Sobolev injection,
(77)
L
p
(
R
2
)
↪
H
˙
s
(
R
2
)
,
with
1
p
+
s
2
=
1
2
;
0
<
s
<
1
.
We conclude that, for all p
∈
[
2
/
(
1
-
(
2
-
2
α
)
)
,
∞
),
(78)
∥
θ
(
t
)
∥
L
p
=
o
(
t
-
(
1
-
2
/
p
-
(
2
-
2
α
)
)
/
2
α
)
,
t
⟶
∞
.
(b) Letθ be a global solution of (
S
α
) such that
(79)
θ
∈
C
(
R
+
,
H
2
-
2
α
(
R
2
)
)
;
then, for all p
∈
(
2
,
∞
),
(80)
∥
θ
(
t
)
∥
L
p
=
o
(
t
-
(
p
-
2
)
/
2
p
α
)
,
t
⟶
∞
.
Using the classical interpolation inequality
(81)
∥
f
∥
L
∞
(
R
2
)
≤
C
(
s
)
∥
f
∥
L
2
(
R
2
)
1
-
1
/
s
∥
f
∥
H
˙
s
(
R
2
)
1
/
s
,
s
∈
(
1
,
∞
)
,
and Theorem 12, we get
(82)
∥
θ
(
t
)
∥
L
∞
=
o
(
t
-
1
/
2
α
)
,
t
⟶
∞
.



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*Source: 538374-2014-07-07.xml*