Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 172956, 15 pages
doi:10.1155/2012/172956
Research Article
Global Attractor of Atmospheric Circulation
Equations with Humidity Effect
Hong Luo
College of Mathematics and Software Science, Sichuan Normal University, Sichuan,
Chengdu 610066, China
Correspondence should be addressed to Hong Luo, lhscnu@hotmail.com
Received 1 June 2012; Accepted 15 July 2012
Academic Editor: Jinhu Lü
Copyright q 2012 Hong Luo. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Global attractor of atmospheric circulation equations is considered in this paper. Firstly, it is proved
that this system possesses a unique global weak solution in L2 Ω, R4 . Secondly, by using Ccondition, it is obtained that atmospheric circulation equations have a global attractor in L2 Ω, R4 .
1. Introduction
This paper is concerned with global attractor of the following initial-boundary problem of
atmospheric circulation equations involving unknown functions u, T, q, p at x, t x1 , x2 ,
t ∈ Ω × 0, ∞ Ω 0, 2π × 0, 1 is a period of C∞ field −∞, ∞ × 0, 1:
∂u
κ
Pr Δu − ∇p − σu Pr RT − Rq
− u · ∇u,
∂t
∂T
ΔT u2 − u · ∇T Q,
∂t
∂q
Le Δq u2 − u · ∇q G,
∂t
div u 0,
1.1
1.2
1.3
1.4
and Le are constants, u u1 , u2 , T , q, and p denote velocity field, temwhere Pr , R, R,
perature, humidity, and pressure, respectively; Q, G are known functions, and σ is constant
matrix:
σ0 ω
.
1.5
σ
ω σ1
2
Abstract and Applied Analysis
The problems 1.1–1.4 are supplemented with the following Dirichlet boundary
condition at x2 0, 1 and periodic condition for x1 :
u, T, q 0,
x2 0, 1,
u, T, q 0, x2 u, T, q 2π, x2 ,
1.6
and initial value conditions
u, T, q u0 , T0 , q0 ,
t 0.
1.7
The partial differential equations 1.1–1.7 were firstly presented in atmospheric
circulation with humidity effect 1
. Atmospheric circulation is one of the main factors
affecting the global climate, so it is very necessary to understand and master its mysteries
and laws. Atmospheric circulation is an important mechanism to complete the transports
and balance of atmospheric heat and moisture and the conversion between various energies.
On the contrary, it is also the important result of these physical transports, balance and
conversion. Thus, it is of necessity to study the characteristics, formation, preservation,
change and effects of the atmospheric circulation and master its evolution law, which is not
only the essential part of human’s understanding of nature, but also the helpful method of
changing and improving the accuracy of weather forecasts, exploring global climate change,
and making effective use of climate resources.
The atmosphere and ocean around the earth are rotating geophysical fluids, which
are also two important components of the climate system. The phenomena of the atmosphere
and ocean are extremely rich in their organization and complexity, and a lot of them cannot be
produced by laboratory experiments. The atmosphere or the ocean or the couple atmosphere
and ocean can be viewed as an initial and boundary value problem 2–5
, or an infinite
dimensional dynamical system 6–8
. We deduce the atmospheric circulation model 1.1–
1.7 which is able to show features of atmospheric circulation and is easy to be studied
from the very complex atmospheric circulation model based on the actual background and
meteorological data, and we present global solutions of atmospheric circulation equations
with the use of the T -weakly continuous operator 1
. In fact, there are numerous papers
on this topic 9–13
. Compared with some similar papers, we add humidity function in this
paper. We propose firstly the atmospheric circulation equation with humidity function which
does not appear in the previous literature.
As far as the theory of infinite-dimensional dynamical system is concerned, we refer to
9–11, 14–18
. In the study of infinite dimensional dynamical system, the long-time behavior
of the solution to equations is an important issue. The long-time behavior of the solution to
equations can be shown by the global attractor with the finite-dimensional characteristics.
Some authors have already studied the existence of the global attractor for some evolution
equations 2, 3, 13, 19–21
. The global attractor strictly defined as ω-limit set of ball,
which under additional assumptions is nonempty, compact, and invariant 13, 17
. Attractor
theory has been intensively investigated within the science, mathematics, and engineering
communities. Lü et al. 22–25
apply the current theoretical results or approaches to
investigate the global attractor of complex multiscroll chaotic systems. We obtain existence of
global attractor for the atmospheric circulation equations from the mathematical perspective
in this paper.
Abstract and Applied Analysis
3
The paper is organized as follows. In Section 2, we recall preliminary results. In
Section 3, we present uniqueness of the solution to the atmospheric circulation equations.
In Section 4, we obtain global attractor of the equations.
· X denote norm of the space X; C and Ci are variable constants. Let H {φ u, T, q ∈ L2 Ω, R4 | φ satisfy 1.4, 1.6}, and H1 {φ u, T, q ∈ H 1 Ω, R4 | φ satisfy
1.4, 1.6}.
2. Preliminaries
Let X and X1 be two Banach spaces, X1 ⊂ X a compact and dense inclusion. Consider the
abstract nonlinear evolution equation defined on X, given by
du
Lu Gu,
dt
2.1
ux, 0 u0 ,
where ut is an unknown function, L : X1 → X a linear operator, and G : X1 → X a
nonlinear operator.
A family of operators St : X → X t ≥ 0 is called a semigroup generated by 2.1 if
it satisfies the following properties:
1 St : X → X is a continuous map for any t ≥ 0;
2 S0 id : X → X is the identity;
3 St s St · Ss, for all t, s ≥ 0. Then, the solution of 2.1 can be expressed as
ut, u0 Stu0 .
2.2
Next, we introduce the concepts and definitions of invariant sets, global attractors, and ωlimit sets for the semigroup St.
Definition 2.1. Let St be a semigroup defined on X. A set Σ ⊂ X is called an invariant set of
St if StΣ Σ, for all t ≥ 0. An invariant set Σ is an attractor of St if Σ is compact, and
there exists a neighborhood U ⊂ X of Σ such that for any u0 ∈ U,
inf Stu0 − vX −→ 0,
v∈Σ
as t −→ ∞.
2.3
In this case, we say that Σ attracts U. Particularly, if Σ attracts any bounded set of X, Σ
is called a global attractor of St in X.
For a set D ⊂ X, we define the ω-limit set of D as follows:
ωD StD,
2.4
s≥0 t≥s
where the closure is taken in the X-norm. Lemma 2.2 is the classical existence theorem of
global attractor by Temam 13
.
4
Abstract and Applied Analysis
Lemma 2.2. Let St : X → X be the semigroup generated by 2.1. Assume that the following
conditions hold:
1 St has a bounded absorbing set B ⊂ X, that is, for any bounded set A ⊂ X there exists a
time tA ≥ 0 such that Stu0 ∈ B, for all u0 ∈ A and t > tA ;
2 St is uniformly compact, that is, for any bounded set U ⊂ X and some T > 0 sufficiently
large, the set t≥T StU is compact in X.
Then the ω-limit set A ωB of B is a global attractor of 2.1, and A is connected providing
B is connected.
Definition 2.3 see 19
. We say that St : X → X satisfies C-condition, if for any bounded
set B ⊂ X and ε > 0, there exist tB > 0 and a finite dimensional subspace X1 ⊂ X such that
{P StB} is bounded, and
∀t ≥ tB , u ∈ B,
I − P StuX < ε,
2.5
where P : X → X1 is a projection.
Lemma 2.4 see 19
. Let St : X → X (t ≥ 0) be a dynamical systems. If the following conditions
are satisfied:
1 there exists a bounded absorbing set B ⊂ X;
2 St satisfies C-condition,
then St has a global attractor in X.
From Linear elliptic equation theory, one has the following.
Lemma 2.5. The eigenvalue equation:
−ΔT x1 , x2 βT x1 , x2 ,
x1 , x2 ∈ 0, 2π × 0, 1,
2.6
x2 0, 1,
T 0,
T 0, x2 T 2π, x2 has eigenvalue {βk }∞
k1 , and
0 < β1 ≤ β2 ≤ · · · ,
βk −→ ∞,
as k −→ ∞.
2.7
3. Uniqueness of Global Solution
2
− 1 /Le }, and β1 is the first eigenvalue of elliptic
Theorem 3.1. If σ β1 ≥ max{R 12 , R
equation 2.6, then the weak solution to 1.1–1.7 is unique.
Abstract and Applied Analysis
5
Proof. From 1
, u, T, q ∈ L∞ 0, T , H ∩ L2 0, T , H1 , 0 < T < ∞ is the weak solution to
1.1–1.7. Then for all v, S, z ∈ H1 , 0 ≤ t ≤ T , we have
1
Pr
Ω
uvdx Ω
T Sdx Ω
t v2
−∇u∇v − σuv RT − Rq
qzdx 0
Ω
1
u · ∇uv − ∇T ∇S u2 S − u · ∇T S
Pr
QS − Le ∇q∇z u2 z − u · ∇qz Gz dx dt
1
u0 vdx T0 Sdx q0 zdx.
Pr Ω
Ω
Ω
3.1
−
Set u1 , T 1 , q1 and u2 , T 2 , q2 are two weak solutions to 1.1–1.7, which satisfy 3.1.
Let u, T, q u1 , T 1 , q1 − u2 , T 2 , q2 . Then,
1
Pr
Ω
uvdx T Sdx Ω
Ω
qzdx t 0
Ω
v2
−∇u∇v − σuv RT − Rq
1 2
1 1
u · ∇ u2 v −
u · ∇ u1 v − ∇T ∇S
Pr
Pr
u2 S u2 · ∇ T 2 S − u1 · ∇ T 1 S
− Le ∇q∇z u2 z u2 · ∇ q2 z
− u1 · ∇ q1 z dx dt.
3.2
Let v, S, z u, T, q. We obtain from 3.2 the following:
1
Pr
2
Ω
|u| dx 2
Ω
|T | dx Ω
2
q dx
t u2 1 u2 · ∇ u2 u − 1 u1 · ∇ u1 u
−|∇u|2 − σu · u RT − Rq
Pr
Pr
0 Ω
2
− |∇T |2 u2 T u2 · ∇ T 2 T − u1 · ∇ T 1 T − Le ∇q u2 q
2
2
1
u · ∇ q q − u · ∇ q1 q dx dt
≤
t 2 −|∇u|2 − |∇T |2 − Le ∇q dx dt
0
Ω
t − 1 qu2 dx dt
−
σ |u|2 R 1T u2 − R
0
Ω
6
Abstract and Applied Analysis
t 1
u · ∇u2 u u · ∇T 2 T u · ∇q2 q dx dt
0 Ω Pr
t 2 ≤
−|∇u|2 − |∇T |2 − Le ∇q dx dt
0
Ω
t 0
⎡
⎤
2
−1 R
R 1
⎢
q2 ⎥
σ |u|2 σ |u2 |2 |T |2 ⎣−
⎦dx dt
2
σ
2
σ
Ω
2
t √
√
2
1/2 2
uL2 ∇u2 2 ∇uL2 2u1/2
∇u
∇T 1/2
∇T
2 T 1/2
2
2
L
L
L2
L2
L
L
0 Pr
√
1/2
1/2
1/2 2 1/2 2u 2 ∇u 2 ∇q q 2 ∇q 2 dt
L
L
L
L
t 1
L e 2
≤
−|∇u|2 − |∇T |2 − ∇q dx dt
2
2
0 Ω
t √
2
uL2 ∇u2 2 ∇uL2
L
0 Pr
√
√ 2uL2 ∇u∇T 2 2 2∇T 2 2 T L2 ∇T L2
L
2uL2 ∇uL2 ∇q2 √
L2
L
√ 2qL2 ∇qL2 ∇q2 L2
dt
t 1
Le 2
2
2
≤
∇q dx dt
−|∇u| − |∇T | −
2
2
0 Ω
t
2
2
2
3
∇u2L2 2 u2L2 ∇u2 2 3u2L2 ∇T 2 2 3u2L2 ∇q2 2
L
L
L
P
0
r
2
2
1
Le 2
2 2 ∇T 2L2 ∇T 2 2 T 2L2 ∇qL2 qL2 ∇q2 2 dt
L
L
2
2
Le
t
2
2
2
3
2 2 2 2
2
2
3u
3u
≤
u
∇u
∇T
∇q
2
2
2
2
L
L
L
2
L2
L2
L
0 Pr
2
2 2 2
∇T 2 2 T 2L2 qL2 ∇q2 2 dt.
L
L
Le
3.3
Then,
2
u2L2 T 2L2 qL2
t 2 2 2 2 ≤C
u2L2 T 2L2 qL2 ∇u2 2 ∇T 2 2 ∇q2 2 dt
L
0
≤C
t
0
2 u2L2 T 2L2 qL2 dt.
L
L
3.4
Abstract and Applied Analysis
7
By using the Gronwall inequality, it follows that
2
u2L2 T 2L2 qL2 ≤ 0,
3.5
which imply u, T, q ≡ 0. Thus, the weak solution to 1.1–1.7 is unique.
4. Existence of Global Attractor
− 12 /Le }, and β1 is the first eigenvalue of elliptic equation
Theorem 4.1. If σ β1 ≥ max{R 12 , R
2.6, then 1.1–1.7 have a global attractor in L2 Ω, R4 .
Proof. According to Lemma 2.4, we prove Theorem 4.1 in the following two steps.
Step 1. Equations 1.1–1.7 have an absorbing set in H.
Multiply 1.1 by u and integrate the product in Ω:
1
Pr
Ω
du
udx dt
1
κ
Δu − ∇p − σu RT − Rq
− u · ∇u udx.
Pr
Ω
4.1
Then,
1 d
2Pr dt
u2 dx Ω
u2 dx.
−|∇u|2 − σu · u RT − Rq
Ω
4.2
Multiply 1.2 by T and integrate the product in Ω:
Ω
dT
T dx dt
Ω
ΔT u2 − u · ∇T Q
T dx.
4.3
Then,
1 d
2 dt
Ω
T 2 dx Ω
−|∇T |2 u2 T dx QT dx.
4.4
Multiply 1.3 by q and integrate the product in Ω:
Ω
dq
qdx dt
Ω
Le Δq u2 − u · ∇q Q qdx.
4.5
Then,
1 d
2 dt
q dx 2
Ω
Ω
2
−Le ∇q u2 q dx Gq dx.
4.6
8
Abstract and Applied Analysis
We deduce from 4.2–4.6 the following:
1 d
2 dt
Ω
2
1 2
−|∇u|2 − |∇T |2 − Le ∇q − σu · u
u T 2 q2 dx Pr
Ω
− 1 qu2 QT Gq dx
R 1T u2 − R
≤
Ω
2
− |∇u|2 − |∇T |2 − Le ∇q − σ |u|2
4.7
2
−1 R
R 1
q2
σ |u2 |2 |T |2 2
σ
2
σ
2 1 2
2
2
ε|T | ε q |Q| |G| dx.
ε
2
Let ε > 0 be appropriate small such that
d
dt
Ω
u2 T 2 q2 dx
4.8
2 2
2
≤ C1
−|∇u| − |∇T | − ∇q dx C2
|Q|2 |G|2 dx.
Ω
Ω
Then,
d
dt
Ω
u T q
2
2
2
dx ≤ −C3
Ω
2 |u|2 |T |2 q dx C4 .
4.9
Applying the Gronwall inequality, it follows that
u, T, q t2 2 ≤ u, T, q 02 2 e−C3 t C4 1 − e−C3 t .
L
L
C3
4.10
Then, when M2 > C4 /C3 , for any u0 , T0 , q0 ∈ B, here B is a bounded in H, there exists
t∗ > 0 such that
St u0 , T0 , q0 ut, T t, qt ∈ BM ,
t > t∗ ,
4.11
where BM is a ball in H, at 0 of radius M. Thus, 1.1–1.7 have an absorbing BM in H.
Abstract and Applied Analysis
9
Step 2. C-condition is satisfied.
The eigenvalue equation:
Δu λu,
ux1 , 0 ux2 , 0 0,
4.12
u0, x2 u2π, x2 ,
div u 0
has eigenvalues λ1 , λ2 , . . . , λk , . . . and eigenvector {ek | k 1, 2, 3, . . .}, and λ1 ≥ λ2 ≥ · · · ≥ λk ≥
· · · . If k → ∞, then λk → −∞. {ek | k 1, 2, 3, . . .} constitutes an orthogonal base of L2 Ω.
For all u, T, q ∈ H, we have
∞
u
T
q
k1
∞
k1
∞
uk e k ,
T k ek ,
qk ek ,
k1
u2L2 T 2L2 2
q 2 L
∞
k1
∞
k1
∞
u2k ,
Tk2 ,
4.13
qk .
k1
When k → ∞, λk → −∞. Let δ be small positive constant, and N 1/δ. There exists
positive integer k such that
−N ≥ λj ,
j ≥ k 1.
4.14
Introduce subspace E1 span{e1 , e2 , . . . , ek } ⊂ L2 Ω. Let E2 be an orthogonal subspace of E1 in L2 Ω.
For all u, T, q ∈ H, we find that
u v1 v2 ,
v1 k
T T1 T2 ,
xi ei ∈ E1 × E1 ,
v2 i1
T1 xj ej ∈ E2 × E2 ,
jk1
k
Ti ei ∈ E1 ,
T2 i1
q1 q q1 q2 ,
∞
k
∞
Tj ej ∈ E2 ,
4.15
jk1
qi ei ∈ E1 ,
i1
q2 ∞
qj ej ∈ E2 .
jk1
Let Pi : L2 Ω → Ei be the orthogonal projection. Thanks to Definition 2.3, we will
prove that for any bounded set B ⊂ H and ε > 0, there exists t0 > 0 such that
P1 StBH ≤ M,
P2 StBH ≤ ε,
∀t > t0 , M is a constant,
∀t > t0 ,
u0 , T0 , q0 ∈ B.
4.16
4.17
10
Abstract and Applied Analysis
From Step 1, St has an absorbing set BM . Then for any bounded set B ⊂ H, there
exists t∗ > 0 such that StB ⊂ BM , for all t > t∗ , which imply 4.16.
Multiply 1.1 by u and integrate over Ω. We obtain
du
,u
dt
Pr Δu, u − Pr σu, u Pr
κ
RT Rq
, u − u · ∇u, u.
4.18
Then,
u2L2 Pr
t
Δu, udt − Pr
0
t
ε1 Pr
t
0
Δu, udt 1 − ε1 Pr
t
0
Pr
t 0
t σu, udt Pr
κ
RT Rq
, u dt u0 2L2
0
Δu, udt − Pr
t
0
σu, udt
4.19
0
κ
RT Rq
, u dt u0 2L2 ,
where ε1 is a constant which needs to be determined.
From 4.14, we find that
Δu, u ∞
λi u2i i1
≤λ
k
k
λi u2i i1
u2i − N
i1
∞
λj u2j
jk1
∞
4.20
u2j
ik1
≤ λu2L2 − Nν2 2L2 ,
where λ max{λ1 , λ2 , . . . , λk }.
Thanks to Δu, u − Ω |∇u|2 dx −∇u2L2 and uL2 ≤ C∇uL2 , it follows that
Δu, u −∇u2L2 ≤ −
1
u2L2 .
C2
4.21
We deduce from 4.11 the following:
2
u2L2 T 2L2 qL2 ≤ M2 ,
t ≥ t∗ .
Using 4.19–4.22, we find that
ν2 2L2 ≤ u2L2
t
t
t
ε1 Pr Δu, udt 1 − ε1 Pr Δu, udt − Pr σu, udt
0
Pr
t 0
0
κ
RT Rq
, u dt u0 2L2
0
4.22
Abstract and Applied Analysis
≤ ε1 λPr
t
0
Pr R
2
11
u2L2 dt − ε1 NPr
t
0
u2L2 dt t
0
t
0
1 − ε1 Pr t
u2L2 dt
C2
0
t
t
2
Pr R
u2L2 dt qL2 dt
2
0
0
ν2 2L2 dt −
T 2L2 dt
u0 2L2 .
4.23
Let ε1 satisfy ε1 λ ≤ 1 − ε1 /C2 and K1 Pr RM Pr RM.
Then,
ν2 2L2
≤ −ε1 NPr
t
0
ν2 2L2 dt K1 t u0 2L2 ,
4.24
t > t∗ .
By the Gronwall inequality, we find that
ν2 2L2 ≤ e−ε1 NPr t u0 2L2 K1 1 − e−ε1 NPr t ,
ε1 NPr
t > t∗ .
4.25
Then, there exists t1 > t∗ satisfying
e−ε1 NPr t1 u0 2L2 ≤
K1
K1 1 − e−ε1 NPr t1 ≤
.
ε1 NPr
2ε1 NPr
K1
,
2ε1 NPr
4.26
Since δ 1/N, for t > t1 it follows that
ν2 2L2 ≤
K1
K1
δ.
ε1 NPr ε1 Pr
4.27
Multiply 1.2 by T and integrate over Ω. We obtain
dT
,T
dt
4.28
ΔT, T u2 , T − u · ∇T, T Q, T .
Then,
T 2L2
t
0
ε2
ΔT, T dt t
0
t
0
u2 , T dt T0 2L2
ΔT, T dt 1 − ε2 t
ΔT, T dt 0
where ε2 is a constant which needs to be determined.
t
0
4.29
u2 , T dt T0 2L2 ,
12
Abstract and Applied Analysis
From 4.14, we find that
ΔT, T ∞
λi Ti2 i1
≤λ
k
λi Ti2 i1
k
Ti2 − N
i1
∞
λj Tj2
jk1
∞
4.30
Tj2
jk1
≤ λT 2L2 − NT2 2L2 ,
where λ max{λ1 , λ2 , . . ., λk }.
Since ΔT, T − Ω |∇T |2 dx −∇T 2L2 and T L2 ≤ C∇T L2 , it follows that
ΔT, T −∇T 2L2 ≤ −
1
T 2L2 .
C2
4.31
Using 4.22 and 4.29–4.31, we find that
T2 2L2 ≤ T 2L2
t
t
t
ε2 ΔT, T dt 1 − ε2 ΔT, T dt u2 , T dt T0 2L2
0
≤ ε2 λ
t
0
1
2
T 2L2 dt − ε2 N
t
0
u2L2 dt t
0
t
0
0
0
T2 2L2 dt −
T 2L2 dt
1 − ε2 C2
t
0
T 2L2 dt
4.32
T0 2L2 .
Let ε2 satisfy ε2 λ ≤ 1 − ε2 /C2 . Then
T2 2L2 ≤ −ε2 N
t
0
T2 2L2 dt Mt T0 2L2 ,
t > t∗ .
4.33
t > t∗ .
4.34
By the Gronwall inequality, we find that
T2 2L2 ≤ e−ε2 Nt T0 2L2 M 1 − e−ε2 Nt ,
ε2 N
Then, there exists t2 > t∗ satisfying
e−ε2 Nt2 T0 2L2 ≤
M
,
2ε2 N
M M
1 − e−ε2 Nt2 ≤
.
ε2 N
2ε2 N
4.35
Abstract and Applied Analysis
13
Since δ 1/N, for t > t2 , it follows that
T2 2L2 ≤
M
M
δ.
ε2 N
ε2
4.36
Multiply 1.3 by q and integrate over Ω. We obtain
dq
,q
dt
Le Δq, q u2 , q − u · ∇q, q G, q .
4.37
Then,
2
q 2 Le
L
t
Δq, q dt t
0
Le ε3
t
2
u2 , q dt q0 L2
0
Δq, q dt 1 − ε3 t
0
Δq, q dt t
0
0
2
u2 , q dt q0 L2 ,
4.38
where ε3 is a constant which needs to be determined.
From 4.14, we find that
∞
k
∞
λi qi2 λi qi2 λj qj2
Δq, q i1
≤λ
i1
k
qi2 − N
i1
jk1
∞
4.39
qj2
jk1
2
2
≤ λqL2 − N q2 L2 ,
where λ max{λ1 , λ2 , . .. , λk }.
Since Δq, q − Ω |∇q|2 dx −∇q2L2 and qL2 ≤ C∇qL2 , we see that
2
1 2
Δq, q −∇qL2 ≤ − 2 qL2 .
C
4.40
Using 4.22 and 4.38–4.40, we obtain
2
q2 2 ≤ q2 2
L
L
t
t
t
2
ε3 Le
Δq, q dt 1 − ε3 Le
Δq, q dt u2 , q dt q0 L2
0
≤ ε3 λLe
t
0
1
2
t
0
2
q 2 dt − ε3 NLe
L
u2L2 dt t
0
t
0
2
q2 2 dt − 1 − ε3 Le
L
C2
0
2
q 2 dt
L
0
2
q0 L2 .
t
0
2
q 2 dt
L
4.41
14
Abstract and Applied Analysis
Let ε3 satisfy ε3 λ ≤ 1 − ε3 /C2 . Then,
2
q2 2 ≤ −ε3 NLe
L
t
0
2
q2 2 dt Mt q0 2 2 ,
L
L
t > t∗ .
4.42
By the Gronwall inequality, we find that
2
q2 2 ≤ e−ε3 Le Nt q0 2 2 L
L
M 1 − e−ε3 Le Nt ,
ε3 Le N
t > t∗ .
4.43
Then, there exists t3 > t∗ satisfying
2
e−ε3 Le Nt3 q0 L2 ≤
M
,
2ε3 Le N
M M
1 − e−ε3 Le Nt3 ≤
.
ε3 Le N
2ε3 Le N
4.44
Since δ 1/N, for t > t3 , it follows that
2
q2 2 ≤
L
M
M
δ.
ε3 Le N ε3 Le
4.45
From 4.27; 4.36 and 4.45 for all δ > 0 there exists t0 max{t1 , t2 , t3 } such that
when t > t0 , it follows that
P2 St u0 , T0 , q0 2 v2 2 2 T2 2 2 q2 2 2 ≤
L
L
L
H
K1
M
M
δ,
ε1 Pr
ε2 ε3 Le
4.46
which imply 4.17. From Lemma 2.4, 1.1–1.7 have a global attractor in L2 Ω, R4 .
Acknowledgments
This work was supported by national natural science foundation of China no. 11271271 and
the NSF of Sichuan Education Department of China no. 11ZA102.
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