Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 758730, 16 pages
http://dx.doi.org/10.1155/2013/758730
Research Article
Global Attractors for the Three-Dimensional
Viscous Primitive Equations of Large-Scale Atmosphere
in Log-Pressure Coordinate
Bo You and Shan Ma
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
Correspondence should be addressed to Bo You; youb03@126.com
Received 4 September 2012; Revised 14 December 2012; Accepted 21 December 2012
Academic Editor: Grzegorz Lukaszewicz
Copyright © 2013 B. You and S. Ma. 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.
We will first prove the existence of (𝑉, 𝑉)- and (𝑉, 𝐻2 )-absorbing sets for the three-dimensional primitive equations of large-scale
atmosphere in log-pressure coordinate and then prove the existence of (𝑉, 𝐻3 )-absorbing set by the use of the elliptic regularity
theory. Finally, we obtain the existence of (𝑉, 𝑉)- and (𝑉, 𝐻2 )-global attractors for the three-dimensional viscous primitive equations
of large-scale atmosphere in log-pressure coordinate by using the Sobolev compactness embedding theory.
1. Introduction
The fundamental equations governing the motion of the
atmosphere consist of the Navier-Stokes equations with
Coriolis force and thermodynamics, which account for the
buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, due to the shallowness of
the atmosphere, that is, the depth of the fluid layer is very
small in comparison with the radius of the earth, the vertical
large-scale motion in the atmosphere is much smaller than
the horizontal one, which in turn leads to modeling the
vertical motion by hydrostatic balance [1–3]. But for largescale atmospheric flows, it is often necessary to take account
of compressibility. This may be done most easily by a change
of vertical coordinates to log-pressure coordinate rather than
a function of geometric height. As a result, one can obtain the
system (1)–(4), which is known as the primitive equations for
large-scale atmosphere in log-pressure coordinate.
In recent years, the primitive equations of the atmosphere,
the ocean, and the coupled atmosphere-ocean have been
extensively studied from the mathematical point of view [1, 4–
13]. The mathematical framework of the primitive equations
of the ocean was formulated, and the existence of weak
solutions was proved by Lions et al. in [11]. In [1], the
authors have studied the primitive equations of large-scale
atmosphere in pressure coordinate, where the continuity
equation is changed into the incompressible equation, but the
thermodynamical equation is very complicated. For the sake
of simplicity, the authors have made some approximation of
the thermodynamical equation, and in this case, the authors
have proved the existence and the uniqueness of strong, localin-time solutions. By assuming that the initial data is small
enough, the authors have studied the existence of strong,
global-in-time solutions to the three-dimensional primitive
equations of large-scale ocean and obtained the existence of
strong, local-in-time solutions to the equations for all initial
data in [14]. In [5], the authors have proved the maximum
principles of the temperature for the primitive equations of
the atmosphere in pressure coordinate. The existence and the
uniqueness of strong, global-in-time solutions to the primitive equations in thin domains for a large set of initial
datum whose sizes depend inversely on the thickness were
established in [9]. In [15], Temam and Ziane have considered
the existence of strong, local-in-time solutions for the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean. Asymptotic analysis of the primitive
equations under the small depth assumption was established
in [10]. In [8], the authors have proved the existence of weak
solutions and a trajectory attractors for the moist atmospheric
equations in geophysics. The existence and the uniqueness
2
Abstract and Applied Analysis
of strong, global-in-time solutions for the three-dimensional
viscous primitive equations of large-scale ocean and atmosphere dynamics were established by the authors in [4, 16].
In [6], the authors have considered the long-time dynamics
of the primitive equations of large-scale atmosphere and
obtained a weakly compact global attractor A which captures
all the trajectories with respect to the weak topology of 𝑉.
The existence and the uniqueness of strong solutions, the
existence of smooth solutions, and the existence of a compact
global attractor in 𝑉 for the primitive equations with more
regular initial data than that in [6] were established in [7]. In
[12], the authors have proved the existence and the uniqueness of 𝑧-weak, global-in-time solutions when the initial
conditions satisfy some regularity. The existence of global
attractor in 𝑉 for the primitive equations with initial datum
in 𝑉 and the heat source 𝑄 ∈ 𝐿2 (Ω) was proved by use
of the Aubin-Lions compactness lemma in [17]. In [18], the
authors have proved the existence of the global attractor in
(𝐻2 (Ω))3 for the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics. In this
paper, we will consider the primitive equations of large-scale
atmosphere in log-pressure coordinate, where the continuity
equation is transformed into the incompressible equation
with the weighted function 𝑒−𝑧/𝐻𝑠 , which is more complicated
than that in pressure coordinate, but we do not need to make
any approximation of the thermodynamical equation such
that the accuracy of the temperature is higher than that in
pressure coordinate which will reveal the real world better.
Moreover, the appearance of the weighted function 𝑒−𝑧/𝐻𝑠
makes the calculations more difficult. In order to make sure
of the existence of the absorbing set, we need to add some
additional conditions (10) on ℎ, which means that the thickness of Ω is very small. In [18], there are not any restrictions
on the thickness of Ω to ensure the existence of the absorbing
sets, and the 𝐻1 (Ω) estimate of 𝑇 can be obtained directly
by taking the inner product (15) with −Δ 3 𝑇. But in this
paper, thanks to the appearance of the weighted function
𝑒−𝑧/𝐻𝑠 , we have to first estimate 𝑇𝑧 in 𝐿2 and then ∇2 𝑇 in 𝐿2 to
obtain a priori estimates on 𝑇 in 𝐻1 . In this paper, we
will prove the existence of (𝑉, 𝐻2 )-global attractor for the
three-dimensional primitive equations of large-scale atmosphere in log-pressure coordinate under the assumptions (10)
on ℎ.
In this paper, we will consider the following threedimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate [3]:
𝜕𝑣
𝜕𝑣
+ (𝑣 ⋅ ∇) 𝑣 + 𝑤 + 𝑓0 𝑘⃗ × 𝑣 + ∇Φ + 𝐿 1 𝑣 = 0,
𝜕𝑡
𝜕𝑧
𝜕Φ 𝑅𝑇
,
=
𝜕𝑧
𝐻𝑠
(1)
(2)
(3)
𝜕𝑇
𝜕𝑇
+ 𝑣 ⋅ ∇𝑇 + 𝑤
+ 𝐿 2 𝑇 = 𝑄,
𝜕𝑡
𝜕𝑧
(4)
∇ ⋅ (𝑒
Ω = 𝑀 × (−ℎ, 0) ⊂ R3 ,
(5)
where 𝑀 is a bounded domain in R2 with smooth boundary.
Here, 𝑣 = (𝑣1 , 𝑣2 ), (𝑣1 , 𝑣2 , 𝑤) is the velocity field, 𝑇 is the
temperature, Φ is the geopotential, 𝑓0 = 2Ω1 sin 𝜈0 is the
Coriolis parameter, 𝑘⃗ is the vertical unit vector, 𝑅 is a positive
constant, and 𝑄 is the heat source. The operators 𝐿 1 and 𝐿 2
are given by
𝐿1 = −
1
1 𝜕2
Δ−
,
Re1
Re2 𝜕𝑧2
1
1 𝜕2
𝐿2 = −
Δ−
,
𝑅𝑡1
𝑅𝑡2 𝜕𝑧2
(6)
where Re1 and Re2 are positive constants representing the
horizontal and the vertical Reynolds numbers, respectively,
and 𝑅𝑡1 and 𝑅𝑡2 are positive constants which stand for the
horizontal and the vertical heat diffusivities, respectively. And
𝑧 = −𝐻𝑠 log(𝑝/𝑝∗ ) is used as a vertical coordinate, where
𝐻𝑠 is a constant “scale height” and 𝑝∗ is a constant reference
pressure. For the sake of simplicity, let ∇ = (𝜕𝑥 , 𝜕𝑦 ) be the
horizontal gradient operator and Δ = 𝜕𝑥2 + 𝜕𝑦2 the horizontal
Laplacian.
We denote the different parts of the boundary 𝜕Ω by
Γ𝑢 = {(𝑥, 𝑦, 𝑧) ∈ Ω : 𝑧 = 0} ,
Γ𝑏 = {(𝑥, 𝑦, 𝑧) ∈ Ω : 𝑧 = −ℎ} ,
(7)
Γ𝑙 = {(𝑥, 𝑦, 𝑧) ∈ Ω : (𝑥, 𝑦) ∈ 𝜕𝑀, −ℎ ≤ 𝑧 ≤ 0} .
We equip (1)–(4) with the following boundary conditions,
with nonslip and nonflux on the side walls and bottom [4]:
󵄨󵄨
𝜕𝑣 󵄨󵄨󵄨󵄨
1 𝜕𝑇
󵄨
𝑤|Γ𝑢 = 0,
(
+ 𝛼𝑇)󵄨󵄨󵄨 = 0,
󵄨󵄨 = 0,
𝜕𝑧 󵄨󵄨Γ𝑢
𝑅𝑡2 𝜕𝑧
󵄨󵄨Γ𝑢
𝜕𝑣 󵄨󵄨󵄨󵄨
𝜕𝑇 󵄨󵄨󵄨󵄨
𝑤|Γ𝑏 = 0,
󵄨󵄨 = 0,
󵄨 = 0,
𝜕𝑧 󵄨󵄨Γ𝑏
𝜕𝑧 󵄨󵄨󵄨Γ𝑏
󵄨󵄨
𝜕𝑣
𝜕𝑇 󵄨󵄨󵄨󵄨
󵄨
𝑣 ⋅ 𝑛|⃗ Γ𝑙 = 0,
× 𝑛󵄨󵄨󵄨⃗ = 0,
󵄨 = 0,
𝜕𝑛 ⃗
𝜕𝑛⃗ 󵄨󵄨󵄨Γ𝑙
󵄨󵄨Γ𝑙
(8)
where 𝛼 is a positive constant.
In addition, (1)–(8) are supplemented with the following
initial conditions:
𝑣 (𝑥, 𝑦, 𝑧, 0) = 𝑣0 (𝑥, 𝑦, 𝑧) ,
𝑇 (𝑥, 𝑦, 𝑧, 0) = 𝑇0 (𝑥, 𝑦, 𝑧) .
(9)
Additionally, assume that ℎ satisfy
𝜕 −𝑧/𝐻𝑠
𝑣) +
𝑤) = 0,
(𝑒
𝜕𝑧
−𝑧/𝐻𝑠
in the domain
4ℎ
9ℎ2
+
< 1,
4𝐻𝑠2 𝛼𝐻𝑠2 𝑅𝑡2
𝐻𝑠2 Re2 𝜆 > 1.
(10)
Abstract and Applied Analysis
3
This paper is organized as follows. Section 2 is devoted
to introduce the mathematical framework and the definition
of solutions for (1)–(10) and give some auxiliary lemmas
used in the sequel. In Section 3, we will give the existence
and the uniqueness of solutions for (1)–(10). Next, some a
priori estimates of solutions corresponding to (1)–(10) will be
established, which imply the existence of (𝑉, 𝑉)- and (𝑉, 𝐻2 )absorbing sets for (1)–(10). In Section 4, we first obtain the
existence of the (𝑉, 𝑉)-global attractor for (1)–(10) by 𝐻2 ⊂ 𝑉
compactly, and then get the existence of (𝑉, 𝐻3 )-absorbing
set of (1)–(10) by the elliptic regularity theory. Finally, we
prove the existence of the (𝑉, 𝐻2 )-global attractor for the
three-dimensional viscous primitive equations of large-scale
atmosphere in log-pressure coordinate by the Sobolev compactness embedding theory.
Throughout this paper, let 𝐶 be positive constants independent of 𝑡 which may be different from line to line.
with the following boundary conditions:
𝜕𝑣 󵄨󵄨󵄨󵄨
𝜕𝑣 󵄨󵄨󵄨󵄨
󵄨󵄨 = 0,
󵄨 = 0,
𝜕𝑧 󵄨󵄨Γ𝑏
𝜕𝑧 󵄨󵄨󵄨Γ𝑢
(16)
󵄨󵄨
𝜕𝑣
󵄨
󵄨
𝑣 ⋅ 𝑛|⃗ Γ𝑙 = 0,
× 𝑛󵄨󵄨⃗ = 0,
󵄨󵄨Γ𝑙
𝜕𝑛 ⃗
󵄨󵄨󵄨
𝜕𝑇 󵄨󵄨󵄨󵄨
1 𝜕𝑇
𝜕𝑇 󵄨󵄨󵄨󵄨
(
+ 𝛼𝑇)󵄨󵄨󵄨 = 0,
󵄨󵄨 = 0,
󵄨 = 0,
󵄨󵄨Γ
𝜕𝑧 󵄨󵄨Γ𝑏
𝑅𝑡2 𝜕𝑧
𝜕𝑛⃗ 󵄨󵄨󵄨Γ𝑙
𝑢
(17)
and the initial data
𝑣 (𝑥, 𝑦, 𝑧, 0) = 𝑣0 (𝑥, 𝑦, 𝑧) ,
At last, we will divide (14) into two systems with respect to 𝑣
and 𝑣̃ [4], where 𝑣 and 𝑣̃ are defined by
2. Mathematical Setting of (1)–(10)
𝑣=
2.1. Reformulation of (1)–(9). Integrating the continuity
equation (3) and taking the boundary conditions for 𝑤 into
account, we obtain
𝑧
(𝑧−𝜁)/𝐻𝑠
𝑤 (𝑥, 𝑦, 𝑧, 𝑡) = − ∫ 𝑒
−ℎ
0
And then, we consider the vertical momentum equation,
that is, the second equation (2). First, we define an unknown
function on 𝑧 = 0 (i.e., 𝑝 = 𝑝∗ ), where 𝑝∗ is a constant
reference pressure, say
(12)
which is the geopotential of the atmosphere on 𝑧 = 0.
Then,
𝑅 𝑧
Φ (𝑥, 𝑦, 𝑧, 𝑡) = Φ𝑠 (𝑥, 𝑦, 𝑡) +
∫ 𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁.
𝐻𝑠 0
+ ∇Φ𝑠 (𝑥, 𝑦, 𝑡) + 𝑓0 𝑘⃗ × 𝑣
(13)
𝑣 ⋅ 𝑛|⃗ Γ𝑙 = 0,
󵄨󵄨
𝜕𝑣
󵄨
× 𝑛󵄨󵄨󵄨⃗ = 0,
󵄨󵄨Γ𝑙
𝜕𝑛 ⃗
(22)
𝑅 𝑧
𝑣 ⋅∇) 𝑣 + (𝑣 ⋅∇) 𝑣̃
∫ ∇𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁 + 𝑓0 𝑘⃗ × 𝑣̃ + 𝐿 1 𝑣̃ + (̃
𝐻𝑠 0
(14)
𝑣) 𝑣̃ + (̃
𝑣 ⋅∇) 𝑣̃ +
− (∇ ⋅̃
𝜕𝑇
+ 𝑣 ⋅ ∇𝑇
𝜕𝑡
𝜕𝑇
+ 𝐿 2 𝑇 = 𝑄,
𝜕𝑧
(21)
and 𝑣̃ satisfies the following systems
+
𝑅
∫ ∇𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁 + 𝐿 1 𝑣 = 0,
𝐻𝑠 0
−ℎ
𝑅 𝑧
1 𝜕̃𝑣
= 0,
+ 𝑓0 𝑘⃗ × 𝑣 +
∫ ∇𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁 −
𝐻𝑠 0
𝐻𝑠 Re2 𝜕𝑧
𝑧
𝜕̃
𝑣
𝜕̃
𝑣
+ (̃
𝑣 ⋅ ∇) 𝑣̃ − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣̃ (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑡
𝜕𝑧
−ℎ
𝑧
𝑧
(20)
𝜕𝑣
1
Δ𝑣 + ∇Φ𝑠 (𝑥, 𝑦, 𝑡) + (∇ ⋅ 𝑣̃) 𝑣̃ + (̃𝑣 ⋅ ∇) 𝑣̃
+ (𝑣 ⋅ ∇) 𝑣 −
𝜕𝑡
Re1
∇ ⋅ 𝑣 = 0,
𝑧
𝜕𝑣
𝜕𝑣
+ (𝑣 ⋅ ∇) 𝑣 − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑡
𝜕𝑧
−ℎ
− (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
−ℎ
with the boundary conditions
Therefore, (1)–(4) can be rewritten as follows:
+
𝐴 = ∫ 𝑒−𝑧/𝐻𝑠 𝑑𝑧 = 𝐻𝑠 (𝑒ℎ/𝐻𝑠 − 1) ,
and we have 𝑣̃ = 0.
Therefore, 𝑣 satisfies the following systems:
−ℎ
Φ𝑠 : 𝑀 󳨀→ R,
(19)
where
(11)
0
1 0 −𝑧/𝐻𝑠
𝑣 (𝑥, 𝑦, 𝑧) 𝑑𝑧,
∫ 𝑒
𝐴 −ℎ
𝑣̃ = 𝑣 − 𝑣,
∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁,
∫ 𝑒−𝜁/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁 = 0.
(18)
𝑇 (𝑥, 𝑦, 𝑧, 0) = 𝑇0 (𝑥, 𝑦, 𝑧) .
(15)
𝑣
1 𝜕̃
𝑅 𝑧
= 0,
∫ ∇𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁−
𝐻𝑠 0
𝐻𝑠 Re2 𝜕𝑧
(23)
with the boundary conditions
𝜕̃𝑣 󵄨󵄨󵄨󵄨
𝜕̃𝑣 󵄨󵄨󵄨󵄨
𝑣̃ ⋅ 𝑛|⃗ Γ𝑙 = 0,
󵄨󵄨 = 0,
󵄨 = 0,
𝜕𝑧 󵄨󵄨Γ𝑢
𝜕𝑧 󵄨󵄨󵄨Γ𝑏
󵄨󵄨
𝜕̃𝑣
󵄨
× 𝑛󵄨󵄨󵄨⃗ = 0.
󵄨󵄨Γ𝑙
𝜕𝑛 ⃗
(24)
4
Abstract and Applied Analysis
2.2. Some Function Spaces. In this subsection, we will quote
briefly some notations and function spaces used in this paper.
First of all, we introduce the notations for some function
spaces on Ω as follows:
Lemma 1 (see [19]). (i) for 𝑢 ∈ 𝐻1 (𝑀),
,
‖𝑢‖𝐿4 (𝑀) ≤ 𝐶‖𝑢‖1/2
‖𝑢‖1/2
𝐿2 (𝑀)
𝐻1 (𝑀)
,
‖𝑢‖𝐿6 (𝑀) ≤ 𝐶‖𝑢‖2/3
‖𝑢‖1/3
𝐿2 (𝑀)
𝐻1 (𝑀)
𝜕𝑣 󵄨󵄨󵄨󵄨
𝜕𝑣 󵄨󵄨󵄨󵄨
V1 = {𝑣 ∈ (𝐶 (Ω)) :
󵄨󵄨 = 0,
󵄨 = 0, 𝑣 ⋅ 𝑛|⃗ Γ𝑙 = 0,
𝜕𝑧 󵄨󵄨Γ𝑢
𝜕𝑧 󵄨󵄨󵄨Γ𝑏
2
∞
.
‖𝑢‖1/2
‖𝑢‖𝐿8 (𝑀) ≤ 𝐶‖𝑢‖1/2
𝐿4 (𝑀)
𝐻1 (𝑀)
󵄨󵄨
0
𝜕𝑣
󵄨
× 𝑛󵄨󵄨󵄨⃗ = 0, ∫ 𝑒−𝑧/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝑧) 𝑑𝑧 = 0} ,
󵄨󵄨Γ𝑙
𝜕𝑛 ⃗
−ℎ
V2 = {𝑇 ∈ 𝐶∞ (Ω) :
(
(ii) for 𝑢 ∈ 𝑉𝑖 (𝑖 = 1, 2),
𝜕𝑇 󵄨󵄨󵄨󵄨
󵄨 = 0,
𝜕𝑧 󵄨󵄨󵄨Γ𝑏
3/4
‖𝑢‖4 ≤ 𝐶‖𝑢‖1/4
2 ‖𝑢‖ .
󵄨󵄨
1 𝜕𝑇
𝜕𝑇 󵄨󵄨󵄨󵄨
󵄨
+ 𝛼𝑇)󵄨󵄨󵄨 = 0,
󵄨 = 0} .
󵄨󵄨Γ
𝑅𝑡2 𝜕𝑧
𝜕𝑛⃗ 󵄨󵄨󵄨Γ𝑙
𝑢
𝑠
𝑝
(26)
−𝑧/𝐻𝑠
=∫ 𝑒
Ω
𝑝
|ℎ| 𝑑𝑥 𝑑𝑦 𝑑𝑧,
((𝑇1 , 𝑇2 )) =
+
(1) ∫Ω [(𝑣 ⋅ ∇)𝑣1 − (∫−ℎ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣(𝑥, 𝑦, 𝜁)𝑑𝜁)(𝜕𝑣1 /𝜕𝑧)] ⋅
(𝑒−𝑧/𝐻𝑠 |𝑣1 |𝑝−1 𝑣1 ) = 0,
(3) ∫Ω 𝑒−𝑧/𝐻𝑠 ∇Φ𝑠 (𝑥, 𝑦) ⋅ 𝑣(𝑥, 𝑦, 𝑧) = 0.
Lemma 4 (see [4], (Minkowski inequality)). Let (𝑋, 𝜇), (𝑌, 𝜈)
be two measure spaces and 𝑓(𝑥, 𝑦) a measurable function
about 𝜇 × 𝜈 on 𝑋 × 𝑌, and 𝜌(𝑦) is a measurable positive
function. If 𝑓(⋅, 𝑦) ∈ 𝐿𝑝 (𝑋, 𝜇) (1 ≤ 𝑝 ≤ ∞) for a.e.𝑦 ∈ 𝑌
and ∫𝑌 𝜌(𝑦)‖𝑓(⋅, 𝑦)‖𝐿𝑝 (𝑋,𝜇) 𝑑𝜈(𝑦) < ∞, then
𝜕𝑣 𝜕𝑣
1
∫ 𝑒−𝑧/𝐻𝑠 1 ⋅ 2 𝑑𝑥 𝑑𝑦 𝑑𝑧,
Re2 Ω
𝜕𝑧 𝜕𝑧
1
∫ 𝑒−𝑧/𝐻𝑠 ∇𝑇1 ⋅ ∇𝑇2 𝑑𝑥 𝑑𝑦 𝑑𝑧
𝑅𝑡1 Ω
Lemma 3 (see [4, 6]). Let 𝑣, 𝑣1 ∈ 𝑉1 , 𝑇 ∈ 𝑉2 , and 𝑝 ≥ 1. Then,
we have
𝑧
1
∫ 𝑒−𝑧/𝐻𝑠 ∇𝑣1 ⋅ ∇𝑣2 𝑑𝑥 𝑑𝑦 𝑑𝑧
Re1 Ω
+
for any 𝑣 ∈ 𝑉1 .
(2) ∫Ω [𝑣 ⋅ ∇𝑇 − (∫−ℎ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣(𝑥, 𝑦, 𝜁)𝑑𝜁)(𝜕𝑇/
𝜕𝑧)]𝑒−𝑧/𝐻𝑠 |𝑇|𝑝−1 𝑇 = 0,
for any ℎ1 , ℎ2 ∈ 𝐿2 and ℎ ∈ 𝐿𝑝 (1 ≤ 𝑝 ≤ ∞).
Similarly, define
((𝑣1 , 𝑣2 )) =
(31)
𝑧
(ℎ1 , ℎ2 ) = ∫ 𝑒−𝑧/𝐻𝑠 ℎ1 ⋅ ℎ2 𝑑𝑥 𝑑𝑦 𝑑𝑧,
Ω
‖𝑣‖2 ≥ 𝜆‖𝑣‖22 ,
𝑠
Let 𝐿 (or 𝐻 ) be the weighted Hilbert spaces of 𝐿 (or 𝐻 )
functions or 𝐿𝑝 (or 𝐻𝑠 )-vector valued functions on Ω and
denote the usual 𝐿𝑝 (Ω)-norms by ‖ ⋅ ‖𝐿𝑝 (Ω) , where 1 ≤ 𝑝 ≤
∞(or 𝑠 ≥ 0). Denote the inner product in 𝐿2 by (⋅, ⋅) and the
norm in 𝐿𝑝 by ‖ ⋅ ‖𝑝 , respectively, given by
‖ℎ‖𝑝𝑝
(30)
Proposition 2. There exists a positive constant 𝜆 such that
(25)
𝑝
(29)
(27)
𝜕𝑇 𝜕𝑇
1
∫ 𝑒−𝑧/𝐻𝑠 1 2 𝑑𝑥 𝑑𝑦 𝑑𝑧
𝑅𝑡2 Ω
𝜕𝑧 𝜕𝑧
󵄩
󵄩󵄩
󵄩󵄩∫ 𝜌 (𝑦) 𝑓 (⋅, 𝑦) 𝑑𝜈 (𝑦)󵄩󵄩󵄩
󵄩󵄩 𝑝
󵄩󵄩
󵄩𝐿 (𝑋,𝜇)
󵄩 𝑌
󵄩
󵄩
≤ ∫ 𝜌 (𝑦) 󵄩󵄩󵄩𝑓 (⋅, 𝑦)󵄩󵄩󵄩𝐿𝑝 (𝑋,𝜇) 𝑑𝜈 (𝑦) .
𝑌
(32)
+ 𝛼 ∫ 𝑇1 𝑇2 𝑑𝑥 𝑑𝑦,
Γ𝑢
for any 𝑣1 , 𝑣2 ∈ V1 and 𝑇1 , 𝑇2 ∈ V2 , respectively.
Let 𝑉1 , 𝑉2 be the closure of V1 , V2 with respect to the
following norms:
‖𝑣‖2 = ((𝑣, 𝑣)) ,
‖𝑇‖2 = ((𝑇, 𝑇)) ,
Finally, we will recall the definition of strong solutions
and state it as follows.
Definition 5. Assume that (𝑣0 , 𝑇0 ) ∈ 𝑉 and 𝑄 ∈ 𝐻1 and let 𝜏
be any fixed positive time. (𝑣, 𝑇) is called a strong solution of
(14)–(18) on [0, 𝜏], if
(28)
for any 𝑣 ∈ V1 and 𝑇 ∈ V2 , respectively, and let 𝑉 = 𝑉1 × 𝑉2 .
Next, we recall some results used in the sequel.
(𝑣, 𝑇) ∈ 𝐿∞ (0, 𝜏; 𝑉) ∩ 𝐿2 (0, 𝜏; 𝐻2 ) ,
(𝑣𝑡 , 𝑇𝑡 ) ∈ 𝐿1 (0, 𝜏; 𝐿2 ) ,
(33)
Abstract and Applied Analysis
5
and they satisfy
(𝑣 (𝑡) , 𝜙)
𝑡
𝑧
+∫ ∫ 𝑒−𝑧/𝐻𝑠 [ (𝑣⋅∇) 𝑣−(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇⋅𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝑡0
−ℎ
Ω
×
𝜕𝑣
𝑅 𝑧
⃗
+𝑓0 𝑘×𝑣+
∫ ∇𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁]
𝜕𝑧
𝐻𝑠 0
𝑡
⋅ 𝜙 + ∫ ((𝑣, 𝜙))
𝑡0
𝑧
+ ∫ ∫ 𝑒−𝑧/𝐻𝑠 [𝑣⋅∇𝑇 − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇⋅𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
−ℎ
Ω
×
𝑡
= ∫ (𝑄, 𝜓) +
𝑡0
𝑡
𝜕𝑇
] 𝜓 + ∫ ((𝑇, 𝜓))
𝜕𝑧
𝑡0
(38)
1
∫ (𝜕 𝑇, 𝜓) + (𝑇 (𝑡0 ) , 𝜓) ,
𝐻𝑠 𝑅𝑡2 𝑡0 𝑧
(34)
for every 𝜙 ∈ V1 and 𝜓 ∈ V2 , almost every 𝑡0 , 𝑡 ∈ [0, 𝜏].
We start with the following general existence and uniqueness
of solutions which can be obtained by the standard FatouGalerkin methods [20–22] and similar methods in [4]. Here,
we only state it as follows.
Theorem 6. Assume that (𝑣0 , 𝑇0 ) ∈ 𝑉 and 𝑄 ∈ 𝐻1 and let
𝜏 be any fixed positive time. Then, there exists a unique strong
solution (𝑣, 𝑇) of (14)–(18) on [0, 𝜏], which satisfies
2
2
(𝑣, 𝑇) ∈ 𝐿 (0, 𝜏; 𝑉) ∩ 𝐿 (0, 𝜏; 𝐻 )
(35)
and depends continuously on the initial data (𝑣0 , 𝑇0 ); that is,
the mapping (𝑣0 , 𝑇0 ) → (𝑣, 𝑇) is continuous in 𝑉.
By Theorem 6, we can define the operator semigroup
{𝑆(𝑡)}𝑡≥0 in 𝑉 as
𝑆 (⋅) (𝑣0 , 𝑇0 ) : R+ × 𝑉 󳨀→ 𝑉,
‖𝑇‖22
ℎ2 𝑅𝑡2 + 2ℎ/𝛼
4𝐻𝑠2
1 󵄩󵄩󵄩󵄩 𝜕𝑇 󵄩󵄩󵄩󵄩2
2
(
󵄩 󵄩 + 𝛼‖𝑇 (𝑧 = 0)‖𝐿2 (𝑀) ) .
4𝐻𝑠2 − ℎ2 𝑅𝑡2 󵄩󵄩󵄩 𝜕𝑧 󵄩󵄩󵄩2
(36)
which is (𝑉, 𝑉)-continuous.
In this section, we will give some a priori estimates which
imply the existence of the (𝑋, 𝑍)-absorbing set 𝐵0 for (1)–(10).
That is, for any bounded subset 𝐵 ⊂ 𝑋, there exists 𝑇 = 𝑇(𝐵)
such that 𝑆(𝑡)𝐵 ⊂ 𝐵0 for any 𝑡 ≥ 𝑇, where {𝑆(𝑡)}𝑡≥0 is a
semigroup on Banach space 𝑋 generated by (1)–(10).
(40)
Setting 𝛿 = 1 − 8(𝑅𝑡2 ℎ2 + 2ℎ/𝛼)/𝑅𝑡2 (4𝐻𝑠2 − ℎ2 ) and 𝛿1 =
𝛿(4𝐻𝑠2 − ℎ2 )/4𝐻𝑠2 (𝑅𝑡2 ℎ2 + 2ℎ/𝛼), we find that
𝑑
‖𝑇‖22 + 𝛿‖𝑇‖2 ≤ 𝐻𝑠2 𝑅𝑡2 ‖𝑄‖22 ,
𝑑𝑡
3. Some a Priori Estimates for (1)–(10)
(39)
we obtain
≤
𝑡
∞
𝑑
2
‖𝑇‖22 + ‖𝑇‖2 ≤ 2
‖𝑇‖22 + 𝐻𝑠2 𝑅𝑡2 ‖𝑄‖22 .
𝑑𝑡
𝐻𝑠 𝑅𝑡2
󵄩󵄩 𝜕𝑇 󵄩󵄩2
ℎ2
󵄩 󵄩
‖𝑇‖22 ≤ 2ℎ‖𝑇 (𝑧 = 0)‖2𝐿2 (𝑀) + ℎ2 󵄩󵄩󵄩 󵄩󵄩󵄩 +
‖𝑇‖22 ,
󵄩󵄩 𝜕𝑧 󵄩󵄩2 4𝐻𝑠2
(𝑇 (𝑡) , 𝜓)
𝑡0
Employing Hölder’s inequality yields
Thanks to
𝑡
1
=
∫ (𝜕𝑧 𝑣, 𝜙) + (𝑣 (𝑡0 ) , 𝜙) ,
𝐻𝑠 Re2 𝑡0
𝑡
3.1. 𝐿2 Estimates of 𝑇. Taking the 𝐿2 inner product of
(15) with 𝑇, using the boundary conditions (17) and using
Lemma 3, we deduce that
1 󵄩󵄩󵄩󵄩 𝜕𝑇 󵄩󵄩󵄩󵄩
1 𝑑
‖𝑇‖22 + ‖𝑇‖2 ≤ ‖𝑄‖2 ‖𝑇‖2 +
󵄩 󵄩 ‖𝑇‖2 .
2 𝑑𝑡
𝐻𝑠 𝑅𝑡2 󵄩󵄩󵄩 𝜕𝑧 󵄩󵄩󵄩2
(37)
𝑑
‖𝑇‖22 + 𝛿1 ‖𝑇‖22 ≤ 𝐻𝑠2 𝑅𝑡2 ‖𝑄‖22 ,
𝑑𝑡
(41)
which implies that
𝐻2 𝑅𝑡2
󵄩 󵄩2
‖𝑇 (𝑡)‖22 ≤ 󵄩󵄩󵄩𝑇0 󵄩󵄩󵄩2 𝑒−𝛿1 𝑡 + 𝑠
‖𝑄‖22 (1 − 𝑒−𝛿1 𝑡 ) .
𝛿1
(42)
Therefore, there exist three positive constants 𝜌1 , 𝜌2 , 𝑇1 such
that
‖𝑇 (𝑡)‖22 ≤ 𝜌1 ,
𝑡+1
∫
𝑡
‖𝑇‖2 𝑑𝜏 ≤ 𝜌2 ,
(43)
(44)
for any 𝑡 ≥ 𝑇1 . For brevity, we omit writing out explicitly
these bounds here, and we also omit writing out other similar
bounds in our future discussion for all other uniform a priori
estimates.
3.2. 𝐿2 Estimates of 𝑣. Multiplying (14) by 𝑣, integrating over
Ω, and using Lemma 3, we deduce that
1 𝑑
1 󵄩󵄩󵄩󵄩 𝜕𝑣 󵄩󵄩󵄩󵄩
‖𝑣‖22 + ‖𝑣‖2 ≤ 𝐶‖𝑇‖2 ‖∇𝑣‖2 +
󵄩 󵄩 ‖𝑣‖ .
2 𝑑𝑡
𝐻𝑠 Re2 󵄩󵄩󵄩 𝜕𝑧 󵄩󵄩󵄩2 2
(45)
6
Abstract and Applied Analysis
Setting 𝛿3 = 1 − 1/Re2 𝜆𝐻𝑠2 and 𝛿4 = 𝜆 − 1/Re2 𝐻𝑠2 , we obtain
𝑑
‖𝑣‖2 + 𝛿3 ‖𝑣‖2 ≤ 𝐶‖𝑇‖22 ,
𝑑𝑡 2
𝑑
‖𝑣‖2 + 𝛿4 ‖𝑣‖22 ≤ 𝐶‖𝑇‖22 .
𝑑𝑡 2
(46)
3.4. (𝐿6 (Ω))2 Estimates of 𝑣̃. Multiplying (23) by |̃𝑣|4 𝑣̃, integrating over Ω, and using Lemma 3, we deduce that
1
1 𝑑
𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧
∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣|2 |̃
‖̃𝑣‖6 +
6 𝑑𝑡 6 Re1 Ω
+
1
4
󵄨 󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨𝜕𝑧 𝑣̃󵄨󵄨󵄨 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧 +
Re2 Ω
9Re1
󵄨
󵄨2
× ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨∇|̃𝑣|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
From the classical Gronwall inequality, we have
Ω
󵄩 󵄩2
‖𝑣 (𝑡)‖22 ≤ 󵄩󵄩󵄩𝑣0 󵄩󵄩󵄩2 𝑒−𝛿4 𝑡 + 𝐶𝜌1 ,
for all 𝑡 ≥ 𝑇1 , which implies that there exist three positive
constants 𝜌3 ,𝜌4 ,𝑇2 ≥ 𝑇1 such that
‖𝑣 (𝑡)‖22 ≤ 𝜌3 ,
∫
𝑡+1
𝑡
2
‖𝑣‖ 𝑑𝜏 ≤ 𝜌4 ,
+
(47)
≤
4
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨𝜕𝑧 |̃𝑣|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
9Re2 Ω
1 󵄩󵄩 󵄩󵄩 󵄩󵄩 3 󵄩󵄩5/3
󵄩󵄩 3 󵄩󵄩1/2
󵄩|̃𝑣| 󵄩󵄩
+ 𝐶‖𝑣‖4 ‖̃𝑣‖3/2
󵄩𝑣̃ 󵄩 󵄩󵄩|̃𝑣| 󵄩󵄩
6 󵄩
󵄩 󵄩
Re2 𝐻𝑠 󵄩 𝑧 󵄩2 󵄩 󵄩10/3
× (∫ |∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧)
Ω
(48)
(49)
(54)
1/2
󵄩 󵄩
+ 𝐶‖∇𝑇‖2 ‖̃𝑣‖26 󵄩󵄩󵄩󵄩|̃𝑣|3 󵄩󵄩󵄩󵄩
+ ∫ 𝑒−𝑧/𝐻𝑠 (̃𝑣 ⋅ ∇) 𝑣̃ + (∇ ⋅ 𝑣̃) 𝑣̃ ⋅ |̃𝑣|4 𝑣̃𝑑𝑥 𝑑𝑦 𝑑𝑧.
Ω
It follows from the Sobolev interpolation inequality and
Lemma 4 that
for all 𝑡 ≥ 𝑇2 .
3.3. 𝐿6 (Ω) Estimates of 𝑇. Taking the 𝐿2 (Ω) inner product of
(15) with |𝑇|4 𝑇 and using Lemma 3, we have
𝑣 ⋅ ∇) 𝑣̃ + (∇ ⋅ 𝑣̃) 𝑣̃ ⋅ |̃𝑣|4 𝑣̃𝑑𝑥 𝑑𝑦 𝑑𝑧
∫ 𝑒−𝑧/𝐻𝑠 (̃
Ω
0
(∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣| |̃𝑣|4 𝑑𝑧)
≤ 𝐶∫
1 𝑑
5󵄩
1 󵄩󵄩 3 󵄩󵄩5/3 󵄩󵄩 󵄩󵄩
󵄩2 󵄩
󵄩5/3
󵄩󵄩|𝑇| 󵄩󵄩 󵄩󵄩𝑇𝑧 󵄩󵄩2 .
‖𝑇‖66 + 󵄩󵄩󵄩󵄩|𝑇|3 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩|𝑇|3 󵄩󵄩󵄩󵄩10/3 ‖𝑄‖2 +
󵄩10/3
6 𝑑𝑡
9
𝐻𝑠 𝑅𝑡2 󵄩
(50)
𝑀
−ℎ
0
× (∫ 𝑒−𝑧/𝐻𝑠 |̃𝑣|2 𝑑𝑧) 𝑑𝑥 𝑑𝑦
−ℎ
1/2
0
≤ 𝐶 ∫ (∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣|2 |̃𝑣|4 𝑑𝑧)
3/5
From the inequality ‖𝜑‖10/3 ≤ 𝐶‖𝜑‖2/5
2 ‖𝜑‖ , we get
−ℎ
𝑀
0
𝑑
󵄩2
󵄩
󵄩 󵄩2
‖𝑇‖66 + 2󵄩󵄩󵄩󵄩|𝑇|3 󵄩󵄩󵄩󵄩 ≤ 𝐶󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩2 ‖𝑇‖46 + 𝐶‖𝑄‖22 ‖𝑇‖46 ,
𝑑𝑡
× (∫ 𝑒−𝑧/𝐻𝑠 |̃𝑣|4 𝑑𝑧) 𝑑𝑥 𝑑𝑦
(51)
−ℎ
≤ 𝐶(∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧)
1/2
𝑑
󵄩 󵄩2
‖𝑇‖26 ≤ 𝐶󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩2 + 𝐶‖𝑄‖22 .
𝑑𝑡
1/2
2
0
× (∫ (∫ 𝑒−𝑧/𝐻𝑠 |̃𝑣|4 𝑑𝑧) 𝑑𝑥 𝑑𝑦)
𝑀
(52)
−ℎ
2
0
1/2
(∫ (∫ 𝑒−𝑧/𝐻𝑠 |̃𝑣|4 𝑑𝑧) 𝑑𝑥 𝑑𝑦)
Therefore, by virtue of the uniform Gronwall lemma, from
(44) and ‖𝑇‖6 ≤ 𝐶‖𝑇‖𝐻1 (Ω) , we deduce that
𝑀
−ℎ
0
≤ ∫ 𝑒−𝑧/𝐻𝑠 (∫ |̃𝑣|8 𝑑𝑥 𝑑𝑦)
−ℎ
‖𝑇‖26 ≤ 𝜌5 ,
(53)
𝑀
0
1/2
𝑑𝑧
≤ 𝐶 ∫ 𝑒−𝑧/𝐻𝑠 ‖̃
𝑣‖3𝐿6 (𝑀) ‖̃𝑣‖𝐻1 (𝑀) 𝑑𝑧
−ℎ
for any 𝑡 ≥ 𝑇2 + 1.
(55)
Ω
which implies that
≤ 𝐶‖̃𝑣‖36 ‖̃𝑣‖ ,
,
Abstract and Applied Analysis
7
0
which implies that
|∇̃𝑣| 𝑑𝑧) 𝑑𝑥 𝑑𝑦)
≤ (∫ (∫ 𝑒
−ℎ
𝑀
𝑣 ⋅ ∇) 𝑣̃ + (∇ ⋅ 𝑣̃) 𝑣̃ ⋅ |̃𝑣|4 𝑣̃ 𝑑𝑥 𝑑𝑦 𝑑𝑧
∫ 𝑒−𝑧/𝐻𝑠 (̃
Ω
≤ 𝐶(∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧)
1/2
Ω
0
(56)
𝑣‖ ,
‖̃𝑣‖36 ‖̃
0
−ℎ
−𝑧/𝐻𝑠
≤ (∫ 𝑒
−ℎ
where we use the inequality ‖̃𝑣‖8𝐿8 (𝑀) ≤ 𝐶‖̃𝑣‖6𝐿6 (𝑀) ‖̃𝑣‖2𝐻1 (𝑀) .
We deduce from (54) and (56) that
−𝑧/𝐻𝑠
× (∫ (∫ 𝑒
𝑀
∫
𝑡+1
(∫ |∇̃𝑣| 𝑑𝑥 𝑑𝑦)
𝑡
Ω
1/2
𝑀
(59)
‖Δ𝑣‖2
1/2
𝑑𝑧)
0
× (∫ 𝑒−𝑧/𝐻𝑠 (∫ |∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦)
1/2
𝑀
1/2
𝑑𝑧)
1/4
Ω
(57)
(58)
∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏 ≤ 𝜌7 ,
|∇̃𝑣| |̃𝑣| 𝑑𝑧) 𝑑𝑥 𝑑𝑦)
−𝑧/𝐻𝑠
≤ 𝐶‖∇̃𝑣‖1/2
|∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧)
2 (∫ 𝑒
Therefore, it follows from (44)–(49) and the uniform Gronwall lemma that
𝑣‖26 ≤ 𝜌6 ,
‖̃
1/4
2
2
2
−ℎ
𝑑
󵄩 󵄩2
𝑣‖2 ≤ 𝐶 (‖̃𝑣‖2 + ‖𝑣‖44 ) ‖̃𝑣‖26 + 𝐶󵄩󵄩󵄩𝑣̃𝑧 󵄩󵄩󵄩2 + 𝐶‖∇𝑇‖22 .
‖̃
𝑑𝑡 6
1/4
2
−𝑧/𝐻𝑠
‖Δ𝑣‖2
‖Δ𝑣‖2 .
(61)
It follows from (60) and (61) that
𝑑
1
∫ |Δ𝑣|2 𝑑𝑥 𝑑𝑦
‖∇𝑣‖2𝐿2 (𝑀) +
𝑑𝑡
Re1 𝑀
󵄩 󵄩2
≤ 𝐶‖𝑣‖2𝐿2 (𝑀) ‖∇𝑣‖4𝐿2 (𝑀) + 𝐶󵄩󵄩󵄩𝑣̃𝑧 󵄩󵄩󵄩2 + 𝐶‖∇𝑇‖22 + 𝐶‖∇̃𝑣‖22
+ 𝐶 (∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣|2 |̃𝑣|4 𝑑𝑥 𝑑𝑦 𝑑𝑧) + 𝐶‖𝑣‖22 .
Ω
(62)
for any 𝑡 ≥ 𝑇2 + 2.
2
1
2
3.5. (𝐻 (𝑀)) Estimates of 𝑣. Taking the 𝐿 (Ω) inner product of (21) with −Δ𝑣 and combining the boundary conditions
(22), we deduce that
0
𝜕𝑢
+ 𝐿 1 𝑢 + (𝑣 ⋅ ∇) 𝑢
𝜕𝑡
+ 𝐶 ∫ (∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣| |̃𝑣| 𝑑𝑧) |Δ𝑣| 𝑑𝑥 𝑑𝑦
−ℎ
𝑧
+ 𝐶‖𝑣‖2 ‖Δ𝑣‖2 + 𝐶‖∇𝑇‖2 ‖Δ𝑣‖2 .
(60)
By using the Hölder inequality, we get
− (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
−ℎ
𝑣| 𝑑𝑧) |Δ𝑣| 𝑑𝑥 𝑑𝑦
∫ (∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣| |̃
𝑀
−ℎ
(64)
𝑅
∇𝑇 = 0,
𝐻𝑠
with the boundary conditions
1/2
0
≤ ∫ (∫ 𝑒−𝑧/𝐻𝑠 |∇̃𝑣| 𝑑𝑧)
𝑀
𝜕𝑢
𝜕𝑧
𝑧
1
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁) 𝑢
−
𝐻𝑠 −ℎ
+ (𝑢 ⋅ ∇) 𝑣 − (∇ ⋅ 𝑣) 𝑢 + 𝑓0 𝑘⃗ × 𝑢 +
0
(63)
3.6. (𝐿2 (Ω))2 Estimates of 𝑣𝑧 . Denote that 𝑢 = 𝑣𝑧 . It is clear
that 𝑢 satisfies the following equation obtained by differentiating (14) with respect to 𝑧:
1 󵄩󵄩 󵄩󵄩
󵄩𝑣̃ 󵄩 ‖Δ𝑣‖2 + 𝐶‖𝑣‖4 ‖∇𝑣‖4 ‖Δ𝑣‖2
Re2 𝐻𝑠 󵄩 𝑧 󵄩2
𝑀
‖∇𝑣‖2𝐿2 (𝑀) ≤ 𝜌8 ,
for any 𝑡 ≥ 𝑇2 + 3.
1 𝑑
1
∫ |Δ𝑣|2 𝑑𝑥 𝑑𝑦
‖∇𝑣‖2𝐿2 (𝑀) +
2 𝑑𝑡
Re1 𝑀
≤
In view of (44)–(49) and (59) and the uniform Gronwall
lemma, we obtain
𝑢|Γ𝑢 = 0,
−ℎ
0
−𝑧/𝐻𝑠
× (∫ 𝑒
−ℎ
2
|∇̃𝑣| |̃𝑣| 𝑑𝑧)
1/2
|Δ𝑣| 𝑑𝑥 𝑑𝑦
𝑢 ⋅ 𝑛|⃗ Γ𝑙 = 0,
𝑢|Γ𝑏 = 0,
󵄨󵄨
𝜕𝑢
󵄨
× 𝑛󵄨󵄨󵄨⃗ = 0.
󵄨󵄨Γ𝑙
𝜕𝑛 ⃗
(65)
8
Abstract and Applied Analysis
Multiplying (64) by 𝑢, integrating over Ω, and using
Lemma 3, we get
By virtue of (44)–(49), (69)–(71), and the uniform Gronwall lemma, we get
1 𝑑
‖𝑢‖2 + ‖𝑢‖2
2 𝑑𝑡 2
1 󵄩󵄩 󵄩󵄩
≤
󵄩𝑢 󵄩 ‖𝑢‖ + 𝐶‖𝑣‖6 ‖𝑢‖3 ‖∇𝑢‖2 + 𝐶‖𝑇‖2 ‖∇𝑢‖2 .
Re2 𝐻𝑠 󵄩 𝑧 󵄩2 2
(66)
‖∇𝑣‖22 ≤ 𝜌12 ,
1 𝑡+1
∫ ∫ 𝑒−𝑧/𝐻𝑠 |Δ𝑣|2 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
Re1 𝑡
Ω
It follows from the Young inequality and the Sobolev embedding theorem that
𝑑
‖𝑢‖2 + ‖𝑢‖2 ≤ 𝐶 (1 + ‖𝑣‖46 ) ‖𝑢‖22 + 𝐶‖𝑇‖22 .
𝑑𝑡 2
(74)
+
1 𝑡+1
󵄨 󵄨2
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝑣𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏 ≤ 𝜌13 ,
Re2 𝑡
Ω
for any 𝑡 ≥ 𝑇2 + 5.
Since
𝑣‖6 ,
‖𝑣‖6 ≤ 𝐶ℎ−1/3 ‖𝑣‖2 + 𝐶ℎ1/6 ‖∇𝑣‖2 + ‖̃
(68)
which implies that
‖𝑣‖26 ≤ 𝜌9 ,
(69)
for any 𝑡 ≥ 𝑇2 + 3.
By virtue of the uniform Gronwall lemma, from (43),
(49), and (69), we deduce that
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑧 󵄩󵄩2 ≤ 𝜌10 ,
𝑡+1
∫
𝑡
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑧 󵄩󵄩 𝑑𝜏 ≤ 𝜌11 ,
3.8. 𝐿2 (Ω) Estimates of 𝑇𝑧 . Denote 𝜃 = 𝑇𝑧 . It is clear that
𝜃 satisfies the following equation obtained by differentiating
(15) with respect to 𝑧:
𝑧
𝜕𝜃
𝜕𝜃
+ 𝐿 2 𝜃 + 𝑣 ⋅ ∇𝜃 − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑡
𝜕𝑧
−ℎ
(70)
−
(71)
𝑧
1
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁) 𝜃
𝐻𝑠 −ℎ
+ 𝑣𝑧 ⋅ ∇𝑇 − (∇ ⋅ 𝑣) 𝜃 = 𝜕𝑧 𝑄,
(76)
for any 𝑡 ≥ 𝑇2 + 4.
3.7. (𝐿2 (Ω))2 Estimates of ∇𝑣. Taking the 𝐿2 (Ω) inner product of (14) with −Δ𝑣 and combining the boundary conditions
(16), we deduce that
1
1 𝑑
∫ 𝑒−𝑧/𝐻𝑠 |Δ𝑣|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖∇𝑣‖22 +
2 𝑑𝑡
Re1 Ω
+
≤
1
󵄨 󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝑣𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
Re2 Ω
1 󵄩󵄩 󵄩󵄩
󵄩 󵄩1/2 󵄩 󵄩1/2
1/2
3/2
󵄩𝑣 󵄩 ‖Δ𝑣‖2 +𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩2 󵄩󵄩󵄩∇𝑣𝑧 󵄩󵄩󵄩2 ‖∇𝑣‖2 ‖Δ𝑣‖2
Re2 𝐻𝑠 󵄩 𝑧 󵄩2
+ 𝐶‖∇𝑇‖2 ‖Δ𝑣‖2 + 𝐶‖𝑣‖6 ‖∇𝑣‖3 ‖Δ𝑣‖2 + 𝐶‖𝑣‖2 ‖Δ𝑣‖2 .
(72)
We derive from (72) and the Young inequality that
with the boundary conditions
(
󵄨󵄨
1
󵄨
𝜃 + 𝛼𝑇)󵄨󵄨󵄨 = 0,
󵄨󵄨Γ
𝑅𝑡2
𝑢
1
󵄨 󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝑣𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
Re2 Ω
󵄩 󵄩2 󵄩 󵄩2
󵄩 󵄩2
≤ 𝐶 (‖𝑣‖46 + 󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩2 󵄩󵄩󵄩∇𝑣𝑧 󵄩󵄩󵄩2 ) ‖∇𝑣‖22 +𝐶‖∇𝑇‖22 +𝐶‖𝑣‖22 +𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩2 .
(73)
𝜃|Γ𝑏 = 0,
𝜕𝜃 󵄨󵄨󵄨󵄨
󵄨 = 0.
𝜕𝑛⃗ 󵄨󵄨󵄨Γ𝑙
(77)
Multiplying (76) by 𝜃, integrating over Ω, and using Lemma 3,
we deduce that
1 𝑑
1
∫ 𝑒−𝑧/𝐻𝑠 |∇𝜃|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖𝜃‖2 +
2 𝑑𝑡 2 𝑅𝑡1 Ω
𝑑
1
∫ 𝑒−𝑧/𝐻𝑠 |Δ𝑣|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖∇𝑣‖22 +
𝑑𝑡
Re1 Ω
+
(75)
(67)
+
≤
𝜕𝜃
1
󵄨 󵄨2
𝑇 𝑑𝑥 𝑑𝑦
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨𝜕𝑧 𝜃󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 + 𝛼 ∫
𝑅𝑡2 Ω
Γ𝑢 𝜕𝑧
1 󵄩󵄩 󵄩󵄩
󵄩 󵄩
󵄩𝜃 󵄩 ‖𝜃‖ + 𝐶‖𝑣‖6 ‖𝜃‖3 ‖∇𝜃‖2 + 𝐶󵄩󵄩󵄩𝑄𝑧 󵄩󵄩󵄩2 ‖𝜃‖2
𝑅𝑡2 𝐻𝑠 󵄩 𝑧 󵄩2 2
󵄩 󵄩
󵄩 󵄩
+ 𝐶󵄩󵄩󵄩∇𝑣𝑧 󵄩󵄩󵄩2 ‖𝑇‖6 ‖𝜃‖3 + 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩3 ‖𝑇‖6 ‖∇𝜃‖2 .
(78)
Abstract and Applied Analysis
9
3.9. 𝐿2 (Ω) Estimates of ∇𝑇. Taking the 𝐿2 (Ω) inner product
of (15) with −Δ𝑇 and combining the boundary conditions
(17), we deduce that
We deduce from the Young inequality that
𝛼∫
Γ𝑢
𝜕𝜃
𝑇 𝑑𝑥 𝑑𝑦
𝜕𝑧
= 𝛼𝑅𝑡2 ∫ (
Γ𝑢
=
1
1 𝑑
∫ 𝑒−𝑧/𝐻𝑠 |Δ𝑇|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖∇𝑇‖22 +
2 𝑑𝑡
𝑅𝑡1 Ω
𝜕𝑇
1
Δ𝑇 − 𝑄) 𝑇 𝑑𝑥 𝑑𝑦
+ 𝑣 ⋅ ∇𝑇 −
𝜕𝑡
𝑅𝑡1
𝛼𝑅𝑡2 𝑑
∫ |𝑇|2 𝑑𝑥 𝑑𝑦 + 𝛼𝑅𝑡2 ∫ (𝑣 ⋅ ∇𝑇) 𝑇 𝑑𝑥 𝑑𝑦
2 𝑑𝑡 Γ𝑢
Γ𝑢
𝛼𝑅𝑡2
∫ |∇𝑇|2 𝑑𝑥 𝑑𝑦 − 𝛼𝑅𝑡2 ∫ 𝑄𝑇 𝑑𝑥 𝑑𝑦,
𝑅𝑡1 Γ𝑢
Γ𝑢
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨𝛼𝑅𝑡2 ∫ (𝑣 ⋅ ∇𝑇) 𝑇 𝑑𝑥 𝑑𝑦 − 𝛼𝑅𝑡2 ∫ 𝑄𝑇 𝑑𝑥 𝑑𝑦󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
Γ𝑢
Γ𝑢
󵄨
󵄨
+
≤
+
1
󵄨
󵄨2
∫ 󵄨󵄨∇𝑇 󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 + 𝛼 ∫ |∇𝑇|2 𝑑𝑥 𝑑𝑦
𝑅𝑡2 Ω 󵄨 𝑧 󵄨
Γ𝑢
1 󵄩󵄩 󵄩󵄩
󵄩𝑇 󵄩 ‖Δ𝑇‖2 + 𝐶‖𝑣‖6 ‖∇𝑇‖3 ‖Δ𝑇‖2 + 𝐶‖𝑄‖2 ‖Δ𝑇‖2
𝑅𝑡2 𝐻𝑠 󵄩 𝑧 󵄩2
1/2 󵄩
󵄩𝑇𝑧 󵄩󵄩󵄩1/2 󵄩󵄩󵄩∇𝑇𝑧 󵄩󵄩󵄩1/2 ‖Δ𝑇‖2 .
+ 𝐶‖∇𝑣‖1/2
2 ‖Δ𝑣‖2 󵄩
󵄩 󵄩2 󵄩
󵄩2
(83)
≤ 𝐶‖𝑣‖𝐿4 (Γ𝑢 ) ‖∇𝑇‖𝐿2 (Γ𝑢 ) ‖𝑇‖𝐿4 (Γ𝑢 ) + 𝐶‖𝑄‖𝐿2 (Γ𝑢 ) ‖𝑇‖𝐿2 (Γ𝑢 )
≤ 𝐶 ‖𝑣‖ ‖∇𝑇‖3/2
+ 𝐶 ‖𝑄‖ ‖𝑇‖𝐿2 (Γ𝑢 ) .
‖𝑇‖1/2
𝐿2 (Γ )
𝐿2 (Γ )
𝑢
𝑢
(79)
𝑑
1
∫ 𝑒−𝑧/𝐻𝑠 |Δ𝑇|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖∇𝑇‖22 +
𝑑𝑡
𝑅𝑡1 Ω
It follows from (78) and (79) that
𝑑
(‖𝜃‖22 + 𝛼𝑅𝑡2 ‖𝑇‖2𝐿2 (Γ ) )
𝑢
𝑑𝑡
+
We derive from the Young inequality that
+
1
∫ 𝑒−𝑧/𝐻𝑠 |∇𝜃|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
𝑅𝑡1 Ω
1
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝑇𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
𝑅𝑡2 Ω
+ 𝛼 ∫ |∇𝑇 (𝑧 = 0)|2 𝑑𝑥 𝑑𝑦
𝑀
𝛼𝑅𝑡2
1
󵄨 󵄨2
+
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨𝜃𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 +
∫ |∇𝑇|2 𝑑𝑥 𝑑𝑦
𝑅𝑡2 Ω
𝑅𝑡1 Γ𝑢
󵄩 󵄩2
󵄩 󵄩2
≤ 𝐶‖𝑣‖46 ‖∇𝑇‖22 + 𝐶‖∇𝑣‖22 ‖Δ𝑣‖22 󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩2 + 𝐶‖𝑄‖22 + 𝐶󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩2 .
(84)
≤ 𝐶 (1 + ‖𝑣‖46 + ‖𝑇‖46 + ‖𝑣‖4 ) (‖𝜃‖22 + 𝛼𝑅𝑡2 ‖𝑇‖2𝐿2 (Γ ) )
𝑢
󵄩 󵄩2 󵄩 󵄩2
+ 𝐶 (‖𝑄‖2 + 󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩2 ‖𝑇‖46 ) .
(80)
Using (44), (69), (74), (75), and (81), we get
As a result of (44), (53), (69)–(71), and (74), we get
󵄩󵄩 󵄩󵄩2
2
󵄩󵄩𝑇𝑧 󵄩󵄩2 + 𝛼𝑅𝑡2 ‖𝑇‖𝐿2 (Γ𝑢 ) ≤ 𝜌14 ,
‖∇𝑇‖22 ≤ 𝜌16 ,
(81)
1 𝑡+1
󵄨
󵄨2
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝑇𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
𝑅𝑡1 𝑡
Ω
1 𝑡+1
󵄨
󵄨2
+
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨𝜕𝑧2 𝑇󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
𝑅𝑡2 𝑡
Ω
𝛼𝑅𝑡2 𝑡+1
+
∫ ∫ |∇𝑇|2 𝑑𝑥 𝑑𝑦 𝑑𝜏 ≤ 𝜌15 ,
𝑅𝑡1 𝑡
Γ𝑢
for any 𝑡 ≥ 𝑇2 + 6.
1 𝑡+1
∫ ∫ 𝑒−𝑧/𝐻𝑠 |Δ𝑇|2 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
𝑅𝑡1 𝑡
Ω
+
(82)
(85)
1 𝑡+1
󵄨
󵄨2
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝑇𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
𝑅𝑡2 𝑡
Ω
𝑡+1
+ 𝛼∫
𝑡
≤ 𝜌17 ,
for any 𝑡 ≥ 𝑇2 + 7.
∫ |∇𝑇 (𝑧 = 0)|2 𝑑𝑥 𝑑𝑦 𝑑𝜏
𝑀
(86)
10
Abstract and Applied Analysis
3.10. (𝐿6 (Ω))2 Estimates of 𝜕𝑧 𝑣. Taking the 𝐿2 (Ω) inner product of (64) with |𝑢|4 𝑢, combining the boundary conditions
(65), and using Lemma 3, we deduce that
1 𝑑
1
∫ 𝑒−𝑧/𝐻𝑠 |∇𝑢|2 |𝑢|4 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖𝑢‖6 +
6 𝑑𝑡 6 Re1 Ω
+
1
󵄨 󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨𝜕𝑧 𝑢󵄨󵄨󵄨 |𝑢|4 𝑑𝑥 𝑑𝑦 𝑑𝑧
Re2 Ω
+
4
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨∇|𝑢|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
9Re1 Ω
+
4
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨𝜕𝑧 |𝑢|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
9Re2 Ω
It follows from the Hölder inequality that
󵄨
󵄨󵄨
󵄨󵄨− ∫ 𝑒−𝑧/𝐻𝑠 (𝑣 ⋅ ∇𝑇) |𝜃|4 𝜃𝑑𝑥 𝑑𝑦 𝑑𝑧󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑧
󵄨
󵄨 Ω
󵄨 󵄨
≤ 𝐶 ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨𝑣𝑧 󵄨󵄨󵄨 |∇𝑇| |𝜃|5 𝑑𝑥 𝑑𝑦 𝑑𝑧
Ω
󵄩 󵄩5/3
󵄩 󵄩
≤ 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖∇𝑇‖3 󵄩󵄩󵄩󵄩|𝜃|3 󵄩󵄩󵄩󵄩10/3
(87)
󵄩 3 󵄩󵄩2/3 󵄩󵄩 3 󵄩󵄩
󵄩 󵄩
1/2 󵄩
󵄩|𝜃| 󵄩󵄩 󵄩󵄩|𝜃| 󵄩󵄩 1 ,
≤ 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖∇𝑇‖1/2
2 ‖∇𝑇‖𝐻1 (Ω) 󵄩
󵄩 󵄩2 󵄩 󵄩𝐻 (Ω)
𝛼5 𝑅𝑡24 ∫
Γ𝑢
1 󵄩󵄩 󵄩󵄩 󵄩󵄩 3 󵄩󵄩5/3
󵄩 󵄩󵄩
󵄩
≤
+ 𝐶‖𝑣‖6 󵄩󵄩󵄩󵄩|𝑢|3 󵄩󵄩󵄩󵄩3 󵄩󵄩󵄩󵄩∇|𝑢|3 󵄩󵄩󵄩󵄩2
󵄩𝑢 󵄩 󵄩󵄩|𝑢| 󵄩󵄩
Re2 𝐻𝑠 󵄩 𝑧 󵄩2 󵄩 󵄩10/3
= 𝛼5 𝑅𝑡25 ∫ (
Γ𝑢
󵄩 󵄩5/3
󵄩 󵄩󵄩
󵄩
+ 𝐶‖∇𝑇‖2 󵄩󵄩󵄩󵄩|𝑢|3 󵄩󵄩󵄩󵄩10/3 + 𝐶‖𝑣‖6 󵄩󵄩󵄩󵄩|𝑢|3 󵄩󵄩󵄩󵄩3 󵄩󵄩󵄩󵄩|∇𝑢| |𝑢|2 󵄩󵄩󵄩󵄩2 .
=
From the Young inequality, we obtain
(88)
Thanks to (44), (69), and (71), we have
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑧 󵄩󵄩6 ≤ 𝜌18 ,
𝜕𝑇
1
Δ𝑇 − 𝑄) |𝑇|4 𝑇𝑑𝑥 𝑑𝑦
+ 𝑣 ⋅ ∇𝑇 −
𝜕𝑡
𝑅𝑡1
𝛼5 𝑅𝑡25 𝑑
∫ |𝑇|6 𝑑𝑥 𝑑𝑦+𝛼5 𝑅𝑡25 ∫ (𝑣⋅∇𝑇) |𝑇|4 𝑇𝑑𝑥 𝑑𝑦
6 𝑑𝑡 Γ𝑢
Γ𝑢
+
𝑑
󵄩 󵄩2
‖𝑢‖26 ≤ 𝐶‖𝑣‖46 ‖𝑢‖26 + 𝐶‖∇𝑇‖22 + 𝐶󵄩󵄩󵄩𝑢𝑧 󵄩󵄩󵄩2 .
𝑑𝑡
𝜕𝜃 4
|𝑇| 𝑇𝑑𝑥 𝑑𝑦
𝜕𝑧
5𝛼5 𝑅𝑡25
󵄨
󵄨2
∫ 󵄨󵄨󵄨∇|𝑇|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 − 𝛼5 𝑅𝑡25 ∫ 𝑄|𝑇|4 𝑇𝑑𝑥 𝑑𝑦,
9𝑅𝑡1 Γ𝑢 󵄨
Γ𝑢
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨 5 5
󵄨󵄨𝛼 𝑅𝑡2 ∫ (𝑣 ⋅ ∇𝑇) |𝑇|4 𝑇𝑑𝑥 𝑑𝑦 − 𝛼5 𝑅𝑡25 ∫ 𝑄|𝑇|4 𝑇𝑑𝑥 𝑑𝑦󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
Γ𝑢
Γ𝑢
󵄨
󵄨
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩5/3
≤ 𝐶‖𝑣‖𝐿4 (Γ𝑢 ) 󵄩󵄩󵄩󵄩∇|𝑇|3 󵄩󵄩󵄩󵄩𝐿2 (Γ ) 󵄩󵄩󵄩󵄩|𝑇|3 󵄩󵄩󵄩󵄩𝐿4 (Γ ) +𝐶‖𝑄‖𝐿2 (Γ𝑢 ) 󵄩󵄩󵄩󵄩|𝑇|3 󵄩󵄩󵄩󵄩𝐿10/3(Γ )
𝑢
𝑢
𝑢
(89)
󵄩3/2
󵄩󵄩 3 󵄩󵄩2/3
󵄩
󵄩󵄩 3 󵄩󵄩
󵄩
󵄩󵄩|𝑇| 󵄩󵄩 1 .
󵄩
≤ 𝐶 ‖𝑣‖ 󵄩󵄩󵄩󵄩∇ |𝑇|3 󵄩󵄩󵄩󵄩𝐿2 (Γ ) ‖𝑇‖3/2
6 (Γ ) +𝐶 ‖𝑄‖ 󵄩|𝑇| 󵄩 2
𝐿
󵄩
󵄩𝐻 (Γ𝑢 )
󵄩
𝐿
(Γ
)
𝑢
𝑢
𝑢 󵄩
(91)
for any 𝑡 ≥ 𝑇2 + 8.
3.11. 𝐿6 (Ω) Estimates of 𝜕𝑧 𝑇. Taking the 𝐿2 (Ω) inner product
of (76) with |𝜃|4 𝜃, combining the boundary conditions (77),
and using Lemma 3, we deduce that
𝑑
(‖𝜃‖26 + 𝛼𝑅𝑡2 ‖𝑇‖2𝐿6 (Γ ) )
𝑢
𝑑𝑡
1 𝑑
5
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨∇|𝜃|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖𝜃‖6 +
6 𝑑𝑡 6 9𝑅𝑡1 Ω
≤ 𝐶 (1 + ‖𝑣‖46 + ‖𝑣‖4 ) (‖𝜃‖26 + 𝛼𝑅𝑡2 ‖𝑇‖2𝐿6 (Γ ) )
𝑢
5
󵄨
󵄨2
+
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨󵄨𝜕𝑧 |𝜃|3 󵄨󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
9𝑅𝑡2 Ω
+ 𝛼5 𝑅𝑡24 ∫
Γ𝑢
≤
We derive from the Young inequality that
󵄩 󵄩2
󵄩 󵄩4
+ 𝐶󵄩󵄩󵄩𝜃𝑧 󵄩󵄩󵄩2 + 𝐶‖𝑄‖2 + 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 + 𝐶‖∇𝑇‖22 ‖∇𝑇‖2𝐻1 (Ω) .
𝜕𝜃 4
|𝑇| 𝑇 𝑑𝑥 𝑑𝑦
𝜕𝑧
(92)
1 󵄩󵄩 󵄩󵄩 󵄩󵄩 3 󵄩󵄩5/3
󵄩 3󵄩 󵄩
3󵄩
󵄩󵄩𝜃𝑧 󵄩󵄩2 󵄩󵄩󵄩|𝜃| 󵄩󵄩󵄩10/3 + 𝐶‖𝑣‖6 󵄩󵄩󵄩󵄩|𝜃| 󵄩󵄩󵄩󵄩3 󵄩󵄩󵄩󵄩∇ |𝜃| 󵄩󵄩󵄩󵄩2
𝑅𝑡2 𝐻𝑠
As a consequence of the uniform Gronwall lemma, we get
󵄩 󵄩󵄩
󵄩 󵄩 󵄩 󵄩 󵄩5/3
+ 𝐶‖𝑣‖6 󵄩󵄩󵄩󵄩|𝜃|3 󵄩󵄩󵄩󵄩3 󵄩󵄩󵄩󵄩|∇𝜃| |𝜃|2 󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝑄𝑧 󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩|𝜃|3 󵄩󵄩󵄩󵄩10/3
󵄨󵄨
󵄨󵄨
+ 󵄨󵄨󵄨󵄨− ∫ 𝑒−𝑧/𝐻𝑠 (𝑣𝑧 ⋅ ∇𝑇) |𝜃|4 𝜃𝑑𝑥 𝑑𝑦 𝑑𝑧󵄨󵄨󵄨󵄨 .
󵄨 Ω
󵄨
(90)
󵄩󵄩 󵄩󵄩2
2
󵄩󵄩𝑇𝑧 󵄩󵄩6 + 𝛼𝑅𝑡2 ‖𝑇‖𝐿6 (Γ𝑢 ) ≤ 𝜌19 ,
for any 𝑡 ≥ 𝑇2 + 9.
(93)
Abstract and Applied Analysis
11
3.12. 𝐻 Estimates of (𝜕𝑡 𝑣, 𝜕𝑡 𝑇). Denote that 𝜋 = 𝜕𝑡 𝑣, 𝜉 = 𝜕𝑡 𝑇.
It is clear that 𝜋, 𝜉 satisfy the following equations obtained by
differentiating (14) and (15) with respect to 𝑡, respectively,
𝜕𝜋
+ 𝐿 1 𝜋 + (𝑣 ⋅ ∇) 𝜋
𝜕𝑡
𝑧
(𝑧−𝜁)/𝐻𝑠
− (∫ 𝑒
−ℎ
(𝑧−𝜁)/𝐻𝑠
− (∫ 𝑒
−ℎ
+
󵄩 󵄩
󵄩 󵄩
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝜉󵄩󵄩2 ≤ ‖𝑄‖2 󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 + 𝐶‖𝑣‖6 ‖∇𝑇‖3 󵄩󵄩󵄩𝜉󵄩󵄩󵄩2
󵄩 󵄩
󵄩
󵄩󵄩 󵄩
󵄩 󵄩
+ 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 + 𝐶󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩6 ‖∇𝑣‖3 󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 .
󵄩 󵄩
‖𝜋‖22 ≤ 𝐶‖𝑣‖6 ‖∇𝑣‖3 ‖𝜋‖2 + 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖∇𝑣‖3 ‖𝜋‖2
(94)
𝜕𝑣
∇ ⋅ 𝜋 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑧
󵄩
󵄩
+ 𝐶‖𝑣‖2 ‖𝜋‖2 + 𝐶‖∇𝑇‖2 ‖𝜋‖2 + 󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 ‖𝜋‖2 .
󵄩 󵄩2
󵄩 󵄩4 󵄩 󵄩4
‖𝜋‖22 + 󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 ≤ 𝐶 (1 + ‖𝑣‖46 + 󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 + 󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩6 ) (‖∇𝑇‖22 + ‖∇𝑣‖22 )
𝑅 𝑧
∫ ∇𝜉 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁 = 0,
𝐻𝑠 0
𝑧
+ 𝜋 ⋅ ∇𝑇 − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝜋 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
−ℎ
󵄩
󵄩2
󵄩
󵄩2
+ 𝐶‖𝑄‖22 + 𝐶󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 + 𝐶󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 + 𝐶‖𝑣‖22 .
(102)
𝜕𝑇
= 0,
𝜕𝑧
(95)
with the boundary conditions
𝜕𝜋 󵄨󵄨󵄨󵄨
𝜕𝜋 󵄨󵄨󵄨󵄨
󵄨󵄨 = 0,
󵄨 = 0,
𝜕𝑧 󵄨󵄨Γ𝑢
𝜕𝑧 󵄨󵄨󵄨Γ𝑏
(96)
󵄨󵄨
𝜕𝜋
󵄨
󵄨
𝜋 ⋅ 𝑛|⃗ Γ𝑙 = 0,
× 𝑛󵄨󵄨⃗ = 0,
󵄨󵄨Γ𝑙
𝜕𝑛 ⃗
󵄨󵄨󵄨
1 𝜕𝜉
𝜕𝜉 󵄨󵄨󵄨󵄨
𝜕𝜉 󵄨󵄨󵄨󵄨
(
+ 𝛼𝜉)󵄨󵄨󵄨 = 0,
󵄨󵄨 = 0,
󵄨 = 0,
𝑅𝑡2 𝜕𝑧
𝜕𝑧 󵄨󵄨Γ𝑏
𝜕𝑛⃗ 󵄨󵄨󵄨Γ𝑙
󵄨󵄨Γ𝑢
(97)
Multiplying (94) and (95), respectively, by 𝜋, 𝜉, integrating
over Ω, and using Lemma 3, we deduce that
1 𝑑
‖𝜋‖22 + ‖𝜋‖2
2 𝑑𝑡
󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑡 󵄩󵄩2 + 󵄩󵄩𝑇𝑡 󵄩󵄩2 ≤ 𝜌20 ,
(103)
for any 𝑡 ≥ 𝑇2 + 10.
Moreover, we have
∫
𝑡+1
𝑡
𝑡+1
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑡 󵄩󵄩 𝑑𝜏 + ∫
𝑡
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑇𝑡 󵄩󵄩 𝑑𝜏 ≤ 𝜌21 ,
(104)
for any 𝑡 ≥ 𝑇2 + 10.
3.13. 𝐻 Estimates of (𝐿 1 𝑣, 𝐿 2 𝑇). Multiplying (15) by 𝐿 2 𝑇 and
integrating over Ω, we obtain
󵄩
󵄩
󵄩2
󵄩
󵄩 󵄩
󵄩
󵄩󵄩
󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 ≤ 𝐶‖𝑣‖6 ‖∇𝑇‖3 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 + 𝐶󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩6 ‖∇𝑣‖3 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2
󵄩
󵄩 󵄩 󵄩󵄩
󵄩
+ ‖𝑄‖2 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝜕𝑡 𝑇󵄩󵄩󵄩2 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 .
(105)
󵄩󵄩
󵄩
󵄩
󵄩2
󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 ≤ 𝐶‖𝑣‖6 ‖∇𝑣‖3 󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 + 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖∇𝑣‖3 󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2
󵄩
󵄩
󵄩
󵄩 󵄩 󵄩󵄩
󵄩
+ 𝐶‖𝑣‖2 󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 + 𝐶‖∇𝑇‖2 󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝑣𝑡 󵄩󵄩󵄩2 󵄩󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 .
1 󵄩󵄩 󵄩󵄩
󵄩𝜋 󵄩 ‖𝜋‖ + 𝐶‖𝑣‖6 ‖𝜋‖3 ‖∇𝜋‖2
Re2 𝐻𝑠 󵄩 𝑧 󵄩2 2
󵄩 󵄩
󵄩 󵄩
+ 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖𝜋‖3 ‖∇𝜋‖2 + 𝐶‖∇𝜋‖2 󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 ,
(98)
(106)
It follows from the Young inequality that
1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩
󵄩 󵄩
󵄩󵄩𝜉𝑧 󵄩󵄩2 󵄩󵄩𝜉󵄩󵄩2 + 𝐶‖𝑇‖6 ‖𝜋‖3 󵄩󵄩󵄩∇𝜉󵄩󵄩󵄩2
𝑅𝑡2 𝐻𝑠
󵄩 󵄩
󵄩 󵄩󵄩 󵄩
+ 𝐶‖𝑇‖6 󵄩󵄩󵄩𝜉󵄩󵄩󵄩3 ‖∇𝜋‖2 + 𝐶󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩6 󵄩󵄩󵄩𝜉󵄩󵄩󵄩3 ‖∇𝜋‖2 .
󵄩2 󵄩
󵄩2
󵄩󵄩
󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2
󵄩 󵄩4 󵄩 󵄩4
≤ 𝐶 (1 + ‖𝑣‖46 + 󵄩󵄩󵄩𝑇𝑧 󵄩󵄩󵄩6 + 󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ) (‖∇𝑇‖22 + ‖∇𝑣‖22 )
󵄩 󵄩2 󵄩 󵄩2
+ 𝐶 (󵄩󵄩󵄩𝑣𝑡 󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝑇𝑡 󵄩󵄩󵄩2 ) + 𝐶 (‖𝑣‖22 + ‖𝑄‖22 ) .
It follows from the Young inequality that
𝑑 󵄩󵄩 󵄩󵄩2
(󵄩𝜉󵄩 + ‖𝜋‖22 )
𝑑𝑡 󵄩 󵄩2
≤ 𝐶 (1 + ‖𝑇‖46 + ‖𝑣‖46 +
By use of the uniform Gronwall inequality, we have
Similarly, we have
1 𝑑 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩𝜉󵄩 + 󵄩𝜉󵄩
2 𝑑𝑡 󵄩 󵄩2 󵄩 󵄩
≤
(101)
We derive from the Young inequality that
𝑧
𝜕𝜉
𝜕𝜉
+ 𝐿 2 𝜉 + 𝑣 ⋅ ∇𝜉 − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑡
𝜕𝑧
−ℎ
≤
(100)
Similarly, we have
𝜕𝜋
∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑧
+ 𝑓0 𝑘⃗ × 𝜋 + ∇𝜕𝑡 Φ𝑠 (𝑥, 𝑦, 𝑡) + (𝜋 ⋅ ∇) 𝑣
𝑧
Multiplying (15) by 𝜉 and integrating over Ω, we obtain
(107)
By virtue of (48), (69), (74), (85), (89), (93), and (103), we have
󵄩󵄩 󵄩󵄩4
󵄩󵄩𝑇𝑧 󵄩󵄩6
+
󵄩󵄩 󵄩󵄩4 󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑧 󵄩󵄩6 ) (󵄩󵄩𝜉󵄩󵄩2
+ ‖𝜋‖22 ) .
(99)
󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩󵄩𝐿 1 𝑣󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐿 2 𝑇󵄩󵄩󵄩2 ≤ 𝜌22 ,
for any 𝑡 ≥ 𝑇2 + 10.
(108)
12
Abstract and Applied Analysis
3.14. 𝐿2 Estimates of (∇𝑣𝑡 , ∇𝑇𝑡 ). Taking the 𝐻1 inner product
of (94) with −Δ𝜋 and combining the boundary conditions
(96), we have
3.15. 𝐿2 Estimates of (𝜕𝑧 𝑣𝑡 , 𝜕𝑧 𝑇𝑡 ). Denote that 𝛽 = 𝜋𝑧 , 𝛾 = 𝜉𝑧 .
It is clear that 𝛽, 𝛾 satisfy the following equations obtained by
differentiating (94) and (95) with respect to 𝑧, respectively,
𝜕𝛽
+ 𝐿 1 𝛽 + (𝑣𝑧 ⋅ ∇) 𝜋 + (𝑣 ⋅ ∇) 𝛽
𝜕𝑡
1 𝑑
1
∫ 𝑒−𝑧/𝐻𝑠 |Δ𝜋|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖∇𝜋‖22 +
2 𝑑𝑡
Re1 Ω
𝑧
1
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝜋𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
Re2 Ω
+
− (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
−ℎ
1 󵄩󵄩 󵄩󵄩
≤
󵄩𝜋 󵄩 ‖Δ𝜋‖2 + 𝐶‖𝑣‖6 ‖∇𝜋‖3 ‖Δ𝜋‖2
Re2 𝐻𝑠 󵄩 𝑧 󵄩2
−
𝜕𝛽
𝜕𝑧
𝑧
1
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁) 𝛽
𝐻𝑠 −ℎ
1/2 󵄩
󵄩𝜋𝑧 󵄩󵄩󵄩1/2 󵄩󵄩󵄩∇𝜋𝑧 󵄩󵄩󵄩1/2 ‖Δ𝜋‖2
+ 𝐶‖∇𝑣‖1/2
2 ‖Δ𝑣‖2 󵄩
󵄩 󵄩2 󵄩
󵄩2
− (∇ ⋅ 𝑣) 𝛽 + 𝑓0 𝑘⃗ × 𝛽 + (𝛽 ⋅ ∇) 𝑣 + (𝜋 ⋅ ∇) 𝑣𝑧
+ 𝐶‖𝜋‖2 ‖Δ𝜋‖2 + 𝐶‖𝜋‖6 ‖∇𝑣‖3 ‖Δ𝜋‖2
− (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝜋 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝑧
−ℎ
󵄩󵄩 󵄩󵄩1/2 󵄩󵄩 󵄩󵄩1/2
󵄩󵄩 󵄩󵄩
3/2
+ 𝐶‖∇𝜋‖1/2
2 󵄩
󵄩𝑣𝑧 󵄩󵄩2 󵄩󵄩∇𝑣𝑧 󵄩󵄩2 ‖Δ𝜋‖2 + 𝐶󵄩󵄩∇𝜉󵄩󵄩2 ‖Δ𝜋‖2 .
−
(109)
𝑑
1
∫ 𝑒−𝑧/𝐻𝑠 |Δ𝜋|2 𝑑𝑥 𝑑𝑦 𝑑𝑧
‖∇𝜋‖22 +
𝑑𝑡
Re1 Ω
+
≤
𝑧
− (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
−ℎ
󵄩 󵄩2 󵄩 󵄩2
󵄩 󵄩2
+ 󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩2 󵄩󵄩󵄩∇𝑣𝑧 󵄩󵄩󵄩2 ) ‖∇𝜋‖22 + 𝐶‖𝜋‖22 + 𝐶󵄩󵄩󵄩∇𝜉󵄩󵄩󵄩2
− (∇ ⋅ 𝜋) 𝑇𝑧 −
󵄩 󵄩2
󵄩 󵄩2
+ 𝐶󵄩󵄩󵄩𝜋𝑧 󵄩󵄩󵄩2 + 𝐶‖∇𝑣‖22 ‖Δ𝑣‖22 󵄩󵄩󵄩𝜋𝑧 󵄩󵄩󵄩2 + 𝐶‖𝜋‖2 ‖∇𝑣‖23 .
𝑧
Employing the uniform Gronwall inequality, we get
𝑡+1
1
∫
Re1 𝑡
−ℎ
(111)
󵄨 󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨Δ𝑣𝑡 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
Ω
1 𝑡+1
󵄨
󵄨2
+
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝜕𝑧 𝑣𝑡 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏 ≤ 𝜌23 .
Re2 𝑡
Ω
(112)
Similarly, we can also obtain
󵄩󵄩 󵄩󵄩2
󵄩󵄩∇𝑇𝑡 󵄩󵄩2 ≤ 𝜌24 ,
1 𝑡+1
󵄨 󵄨2
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨Δ𝑇𝑡 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏
Re1 𝑡
Ω
1 𝑡+1
󵄨
󵄨2
+
∫ ∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝜕𝑧 𝑇𝑡 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝜏 ≤ 𝜌25 ,
Re2 𝑡
Ω
(113)
for any 𝑡 ≥ 𝑇2 + 12.
𝑧
1
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁) 𝛾
𝐻𝑠 −ℎ
− (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝜋 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
for any 𝑡 ≥ 𝑇2 + 11.
Moreover, we have
𝜕𝛾
𝜕𝑧
− (∇ ⋅ 𝑣) 𝛾 + 𝛽 ⋅ ∇𝑇 + 𝜋 ⋅ ∇𝑇𝑧
(110)
󵄩󵄩 󵄩󵄩2
󵄩󵄩∇𝑣𝑡 󵄩󵄩2 ≤ 𝜌22 ,
𝑅
∇𝜉 = 0,
𝐻𝑠
𝜕𝛾
+ 𝐿 2 𝛾 + 𝑣𝑧 ⋅ ∇𝜉 + 𝑣 ⋅ ∇𝛾
𝜕𝑡
1
󵄨
󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨∇𝜋𝑧 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧
Re2 Ω
𝐶 (‖𝑣‖46
𝜕𝑣𝑧
𝜕𝑧
𝑧
𝜕𝑣
1
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝜋 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝐻𝑠 −ℎ
𝜕𝑧
− (∇ ⋅ 𝜋) 𝑣𝑧 +
Therefore, in view of the Young inequality, we obtain
(114)
−
𝑧
𝜕𝑇
1
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝜋 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
= 0,
𝐻𝑠 −ℎ
𝜕𝑧
with the boundary conditions
󵄨
󵄨
𝛽󵄨󵄨󵄨Γ𝑢 = 0,
𝛽󵄨󵄨󵄨Γ𝑏 = 0,
(
𝜕𝑇𝑧
𝜕𝑧
󵄨󵄨
1
󵄨
𝛾 + 𝛼𝜉)󵄨󵄨󵄨 = 0,
󵄨󵄨Γ
𝑅𝑡2
𝑢
𝜕𝛽
× 𝑛|⃗ Γ𝑙 = 0,
𝜕𝑛 ⃗
󵄨
𝛾󵄨󵄨󵄨Γ𝑏 = 0,
(115)
󵄨
𝛽 ⋅ 𝑛󵄨󵄨󵄨⃗ Γ𝑙 = 0,
(116)
𝜕𝛾 󵄨󵄨󵄨󵄨
󵄨 = 0.
𝜕𝑛⃗ 󵄨󵄨󵄨Γ𝑙
(117)
Taking the 𝐿2 inner product of (114) with 𝛽 and combining
the boundary conditions (116), we have
1 𝑑 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩𝛽󵄩 + 󵄩𝛽󵄩
2 𝑑𝑡 󵄩 󵄩2 󵄩 󵄩
≤
1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩
󵄩 󵄩
󵄩 󵄩
󵄩𝛽 󵄩 󵄩𝛽󵄩 + 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖∇𝜋‖3 󵄩󵄩󵄩𝛽󵄩󵄩󵄩2
Re2 𝐻𝑠 󵄩 𝑧 󵄩2 󵄩 󵄩2
󵄩 󵄩󵄩 󵄩
󵄩 󵄩󵄩 󵄩
+ 𝐶‖𝑣‖6 󵄩󵄩󵄩𝛽󵄩󵄩󵄩3 󵄩󵄩󵄩∇𝛽󵄩󵄩󵄩2 + 𝐶‖𝜋‖3 󵄩󵄩󵄩∇𝑣𝑧 󵄩󵄩󵄩2 󵄩󵄩󵄩𝛽󵄩󵄩󵄩6
󵄩
󵄩󵄩 󵄩
󵄩 󵄩󵄩 󵄩
+ 𝐶‖∇𝜋‖3 󵄩󵄩󵄩𝜕𝑧 𝑣𝑧 󵄩󵄩󵄩2 󵄩󵄩󵄩𝛽󵄩󵄩󵄩6 + 𝐶󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 󵄩󵄩󵄩∇𝛽󵄩󵄩󵄩2 .
(118)
Abstract and Applied Analysis
13
Definition 9 (see [23]). Let 𝑋 be a Banach space and {𝑆(𝑡)}𝑡≥0
a family operators on 𝑋. We say that {𝑆(𝑡)}𝑡≥0 is a norm-toweak continuous semigroup on 𝑋, if {𝑆(𝑡)}𝑡≥0 satisfies that
Therefore, in view of the Young inequality, we obtain
𝑑 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩𝛽󵄩 + 󵄩𝛽󵄩
𝑑𝑡 󵄩 󵄩2 󵄩 󵄩
󵄩 󵄩2
󵄩 󵄩2
≤ 𝐶 (1 + ‖𝑣‖46 ) 󵄩󵄩󵄩𝛽󵄩󵄩󵄩2 + 𝐶󵄩󵄩󵄩𝑣𝑧 󵄩󵄩󵄩6 ‖∇𝜋‖23
(119)
(ii) 𝑆(𝑡)𝑆(𝑠) = 𝑆(𝑡 + 𝑠), for all 𝑡, 𝑠 ≥ 0,
󵄩 󵄩2
󵄩
󵄩2
󵄩 󵄩2
+ 𝐶‖𝜋‖23 󵄩󵄩󵄩∇𝑣𝑧 󵄩󵄩󵄩2 + 𝐶‖∇𝜋‖23 󵄩󵄩󵄩𝜕𝑧 𝑣𝑧 󵄩󵄩󵄩2 + 𝐶󵄩󵄩󵄩𝜉󵄩󵄩󵄩2 .
(iii) 𝑆(𝑡𝑛 )𝑥𝑛 ⇀ 𝑆(𝑡)𝑥 if 𝑡𝑛 → 𝑡 and 𝑥𝑛 → 𝑥 in 𝑋.
Employing the uniform Gronwall inequality, we get
󵄩󵄩
󵄩2
󵄩󵄩𝜕𝑧 𝑣𝑡 󵄩󵄩󵄩2 ≤ 𝜌26 ,
(i) 𝑆(0) = Id (the identity),
(120)
for any 𝑡 ≥ 𝑇2 + 13.
Similarly, we can also obtain
󵄨
󵄨2
󵄨 󵄨2
∫ 𝑒−𝑧/𝐻𝑠 󵄨󵄨󵄨𝜕𝑧 𝑇𝑡 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 𝑑𝑧 + 𝛼𝑅𝑡2 ∫ 󵄨󵄨󵄨𝑇𝑡 󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 ≤ 𝜌27 ,
Ω
Γ
Lemma 10 (see [24]). Let 𝑋, 𝑌 be two Banach spaces and 𝑋∗ ,
𝑌∗ be the dual spaces of 𝑋, 𝑌, respectively. If 𝑋 is dense in
𝑌, the injection 𝑖 : 𝑋 → 𝑌 is continuous and its adjoint
𝑖∗ : 𝑌∗ → 𝑋∗ is dense. Let {𝑆(𝑡)}𝑡≥0 be a semigroup on 𝑋
and 𝑌, respectively, and assume furthermore that {𝑆(𝑡)}𝑡≥0 is
continuous or weak continuous on 𝑌. Then, {𝑆(𝑡)}𝑡≥0 is a normto-weak continuous semigroup on 𝑋 if and only if {𝑆(𝑡)}𝑡≥0
maps compact subsets of R+ × 𝑋 into bounded sets of 𝑋.
𝑢
(121)
for any 𝑡 ≥ 𝑇2 + 14.
From (43), (48), (70), (74), (81), (85) and (108), we get the
following results.
1
Theorem 7. Assume that 𝑄 ∈ 𝐻 (Ω) and (𝑣0 , 𝑇0 ) ∈ 𝑉.
Let {𝑆(𝑡)}𝑡≥0 be a semigroup generated by the initial boundary
problem (14)–(18). Then, there exists an absorbing set in 𝐻2 .
That is, for any bounded subset 𝐵 of 𝑉, there exist a positive
constant R1 and a positive time 𝜏1 = 𝜏1,𝐵 which depend on the
norm of 𝐵, such that
󵄩
󵄩2
‖(𝑣 (𝑡) , 𝑇 (𝑡))‖2𝐻2 = 󵄩󵄩󵄩𝑆 (𝑡) (𝑣0 , 𝑇0 )󵄩󵄩󵄩𝐻2
= ‖𝑣 (𝑡)‖2𝐻2 + ‖𝑇 (𝑡)‖2𝐻2
(122)
≤ R1 ,
for any 𝑡 ≥ 𝜏1 .
Corollary 8. Assume that 𝑄 ∈ 𝐻1 and (𝑣0 , 𝑇0 ) ∈ 𝑉. Let
{𝑆(𝑡)}𝑡≥0 be a semigroup generated by the initial boundary
problem (14)–(18). Then, there exist an absorbing set in 𝑉. That
is, for any bounded subset 𝐵 of 𝑉, there exist a positive constant
R2 and a positive time 𝜏2 = 𝜏2,𝐵 , depending on the norm of 𝐵,
such that
󵄩
󵄩2
‖(𝑣 (𝑡) , 𝑇 (𝑡))‖2𝑉 = 󵄩󵄩󵄩𝑆 (𝑡) (𝑣0 , 𝑇0 )󵄩󵄩󵄩𝑉
= ‖𝑣 (𝑡)‖2𝑉1 + ‖𝑇 (𝑡)‖2𝑉2
(123)
≤ R2 ,
for any 𝑡 ≥ 𝜏2 .
4. The Existence of the Global Attractors
In this section, we will recall some definitions and lemmas
about the global attractor and prove the existence of the global
attractor in 𝐻2 .
Definition 11 (see [20]). Let 𝑋 be a Banach space and let
{𝑆(𝑡)}𝑡≥0 be a continuous semigroup on 𝑋. A subset A in 𝑋 is
called a global attractor if and only if
(1) A is invariant; that is, 𝑆(𝑡)A = A for all 𝑡 ≥ 0,
(2) A is compact in 𝑋,
(3) A attracts each bounded subset 𝐵 in 𝑋.
Definition 12 (see [20]). Let {𝑆(𝑡)}𝑡≥0 be a semigroup on
Banach space 𝑋. {𝑆(𝑡)}𝑡≥0 is called (𝑋, 𝑍) uniformly compact,
if for any bounded (in 𝑋) set 𝐵 ⊂ 𝑋, there exists 𝑡0 = 𝑡0 (𝐵),
such that ∪𝑡≥𝑡0 𝑆(𝑡)𝐵 is relatively compact in 𝑍.
Lemma 13 (see [20, 24]). Let 𝑋 be a Banach space and
{𝑆(𝑡)}𝑡≥0 a 𝐶0 semigroup on 𝑋. Then {𝑆(𝑡)}𝑡≥0 has a global
attractor A in 𝑋 provided that the following conditions hold
true:
(1) {𝑆(𝑡)}𝑡≥0 has a bounded absorbing set 𝐵0 in 𝑋,
(2) {𝑆(𝑡)}𝑡≥0 is uniformly compact.
Lemma 14 (see [20, 24, 25]). Let {𝑆(𝑡)}𝑡≥0 be a norm-to-weak
continuous semigroup such that {𝑆(𝑡)}𝑡≥0 is uniformly compact.
If there exists an absorbing set 𝐵, then {𝑆(𝑡)}𝑡≥0 has a global
attractor A and
A = ⋂ ⋃𝑆 (𝑡) 𝐵.
𝑠≥0 𝑡≥𝑠
(124)
By virtue of the Rellich-Kondrachov theorem, we obtain
the following.
Corollary 15. Assume that 𝑄 ∈ 𝐻1 , (𝑣0 , 𝑇0 ) ∈ 𝑉. Then, the
semigroup {𝑆(𝑡)}𝑡≥0 corresponding to (14)–(18) has a (𝑉, 𝑉)global attractor A𝑉.
To prove the existence of the (𝑉, 𝐻3 )-absorbing set, we
need the following lemma.
14
Abstract and Applied Analysis
Lemma 16 (see [7]). Let 𝑣 ∈ 𝑉1 , Φ𝑠 ∈ 𝐿2 (𝑀) be a solution to
the following Stokes problem:
which implies that
𝑧
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝐿 1 𝑣 + ∇Φ𝑠 = 𝑔,
−ℎ
0
∫ ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝑧) 𝑑𝑧 = 0,
(125)
−ℎ
𝜕𝑣 󵄨󵄨󵄨󵄨
= 0,
󵄨
𝜕𝑧 󵄨󵄨󵄨𝑧=−ℎ
𝜕𝑣 󵄨󵄨󵄨󵄨
󵄨 = 0.
𝜕𝑧 󵄨󵄨󵄨𝑧=0
2
Since (𝑣, 𝑇) is a strong solution to (14)–(18) and 𝑣𝑡 ∈
𝑧
𝐿∞ (𝜏1 , T; 𝑉1 ), it is clear that 𝜕𝑣/𝜕𝑡+(𝑣⋅∇)𝑣−(𝑅/𝐻𝑠 ) ∫0 ∇𝑇(𝑥,
𝑦, 𝜁, 𝑡)𝑑𝜁 + 𝑓 𝑘⃗ × 𝑣 ∈ 𝐿∞ (𝜏 , T; 𝐿3 ). Thus,
0
1
∞
𝑔 ∈ 𝐿 (𝜏1 , T; 𝐿3 ) .
If 𝑔 ∈ (𝑊𝑚,𝛼 (Ω))2 , 1 < 𝛼 < ∞, 𝑚 ≥ 0, then
𝑣 ∈ 𝑉1 ∩ (𝑊𝑚+2,𝛼 (Ω)) ,
Φ𝑠 ∈ 𝑊𝑚+1,𝛼 (𝑀) .
Theorem 17. Assume that 𝑄 ∈ 𝐻 and (𝑣0 , 𝑇0 ) ∈ 𝑉. Let
{𝑆(𝑡)}𝑡≥0 be a semigroup generated by the initial boundary
problem (14)–(18). Then, there exists an absorbing set in 𝐻3 .
That is, for any bounded subset 𝐵 of 𝑉, there exists a positive
constant R3 depending on the norm of 𝐵, such that
󵄩
󵄩2
‖(𝑣 (𝑡) , 𝑇 (𝑡))‖2𝐻3 = 󵄩󵄩󵄩𝑆 (𝑡) (𝑣0 , 𝑇0 )󵄩󵄩󵄩𝐻3
= ‖𝑣 (𝑡)‖2𝐻3 + ‖𝑇 (𝑡)‖2𝐻3
2
𝑣 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ∩ (𝑊2,3 (Ω)) ) .
(126)
1
(127)
2
𝑣 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ∩ (𝑊1,12 (Ω)) ) .
−ℎ
for any 𝑡 ≥ 𝜏1 , where 𝜏1 is specified in Theorem 7.
(128)
𝑔 ∈ 𝐿∞ (𝜏1 , T; 𝐿4 ) .
2
𝑣 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ∩ (𝑊2,4 (Ω)) ) ,
−ℎ
2
∞
6
Since 𝑣 ∈ 𝐿 (𝜏1 , T; 𝐻 ) and 𝑣𝑧 ∈ 𝐿 (𝜏1 , T; 𝐿 ), for any 𝑢 ∈
𝐿3/2 , we have
󵄨󵄨
󵄨󵄨
𝑧
𝜕𝑣
󵄨󵄨
󵄨
−𝑧/𝐻𝑠
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
⋅ 𝑢 𝑑𝑥 𝑑𝑦 𝑑𝑧󵄨󵄨󵄨
󵄨󵄨∫ 𝑒
󵄨󵄨
𝜕𝑧
−ℎ
󵄨󵄨 Ω
󵄩󵄩 󵄩󵄩
≤ 𝐶‖∇𝑣‖6 󵄩󵄩𝑣𝑧 󵄩󵄩6 ‖𝑢‖3/2 ,
(130)
(138)
which implies that
𝑧
(129)
(137)
From Lemma 16, we know that
(∫ 𝑒(𝑧−𝜉)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝑅 𝑧
∫ ∇𝑇 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁 + 𝑓0 𝑘⃗ × 𝑣.
𝐻𝑠 0
𝜕𝑣
∈ 𝐿∞ (𝜏1 , T; 𝐿4 ) .
𝜕𝑧
(136)
Therefore, we find that
where
𝑧
𝜕𝑣
𝜕𝑣
−𝑔 =
+ (𝑣 ⋅ ∇) 𝑣 − (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
𝜕𝑡
𝜕𝑧
−ℎ
(134)
which implies that
𝑧
𝐿 1 𝑣 + ∇Φ𝑠 = 𝑔,
⊂
Now, since 𝑣 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ∩ (𝑊1,12 (Ω))2 ) and 𝑣𝑧 ∈ 𝐿∞ (𝜏1 ,
T; 𝐿6 ), for any 𝑢 ∈ 𝐿4/3 , we have
󵄨󵄨
𝑧
󵄨󵄨󵄨
󵄨󵄨∫ 𝑒−𝑧/𝐻𝑠 (∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁) 𝜕𝑣 ⋅ 𝑢𝑑𝑥 𝑑𝑦 𝑑𝑧󵄨󵄨󵄨
󵄨󵄨 Ω
󵄨󵄨
𝜕𝑧
−ℎ
󵄨
󵄨
󵄩󵄩 󵄩󵄩
≤ 𝐶‖∇𝑣‖12 󵄩󵄩𝑣𝑧 󵄩󵄩6 ‖𝑢‖4/3 ,
(135)
(∫ 𝑒(𝑧−𝜁)/𝐻𝑠 ∇ ⋅ 𝑣 (𝑥, 𝑦, 𝜁, 𝑡) 𝑑𝜁)
Proof. From the results of a priori estimates in the previous section, we know that there exists a strong solution
(𝑣, 𝑇) to (14)–(18), which satisfies for any T > 𝜏1 , 𝑣𝑧 ∈
𝐿∞ (𝜏1 , T; 𝐿6 ), 𝑇𝑧 ∈ 𝐿∞ (𝜏1 , T; 𝐿6 ), 𝑣𝑡 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ), 𝑇𝑡 ∈
𝐿∞ (𝜏1 , T; 𝑉2 ),(𝑣, 𝑇) ∈ 𝐿∞ (𝜏1 , T; 𝐻2 ), where 𝜏1 is specified in
Theorem 7. Then,
(133)
Using the Sobolev embedding theorem (𝑊2,3 (Ω))2
(𝑊1,12 (Ω))2 , we know that
≤ R3 ,
∞
(132)
By the use of Lemma 16, we get
Next, we will prove the existence of absorbing set in 𝐻3 by
the use of Lemma 16, which implies the existence of (𝑉, 𝐻2 )global attractor of the initial boundary problem (14)–(18).
+
𝜕𝑣
∈ 𝐿∞ (𝜏1 , T; 𝐿3 ) .
𝜕𝑧
(131)
𝜕𝑣
∈ 𝐿∞ (𝜏1 , T; 𝐻1 ) .
𝜕𝑧
(139)
Since 𝑣𝑡 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ) and 𝑣 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ∩ (𝑊2,4 (Ω))2 ),
𝑧
it is clear that 𝜕𝑣/𝜕𝑡 + (𝑣 ⋅ ∇)𝑣 − (𝑅/𝐻𝑠 ) ∫0 ∇𝑇(𝑥, 𝑦, 𝜁, 𝑡)𝑑𝜁 +
𝑓 𝑘⃗ × 𝑣 ∈ 𝐿∞ (𝜏 , T; 𝐻1 ). Thus,
0
1
𝑔 ∈ 𝐿∞ (𝜏1 , T; 𝐻1 ) .
(140)
From Lemma 16, we deduce that
𝑣 ∈ 𝐿∞ (𝜏1 , T; 𝑉1 ∩ 𝐻3 ) .
(141)
Employing the elliptic regularity theory, we obtain
𝑇 ∈ 𝐿∞ (𝜏1 , T; 𝑉2 ∩ 𝐻3 ) .
This complete the proof of Theorem 17.
(142)
Abstract and Applied Analysis
15
Since 𝐻2 ⊂ 𝑉 is compact, from Theorem 7, we know that
the semigroup generated by (14)–(18) is compact in 𝑉. From
Theorem 7 and Lemma 10, we deduce that the semigroup
generated by (14)–(18) maps a compact set of 𝐻2 to be
a bounded set of 𝐻2 ; that is, the semigroup generated by
(14)–(18) is norm to weak continuous in 𝐻2 . Furthermore,
𝐻3 ⊂ 𝐻2 compactly, and from Theorem 7 and Lemma 13, we
immediately get the following result.
Theorem 18. Assume that 𝑄 ∈ 𝐻1 (Ω), (𝑣0 , 𝑇0 ) ∈ 𝑉. Then,
the semigroup {𝑆(𝑡)}𝑡≥0 corresponding to (14)–(18) has a
(𝑉, 𝐻2 )-global attractor A.
By using the Rellich-Kondrachov theorem, we can obtain
the existence of the (𝑉, 𝐻2+𝛿 )-global attractor for 𝛿 ∈ [0, 1).
Appendix
Proof of Proposition 2
If this conclusion is not true, then for each natural number 𝑛,
there exists 𝑣𝑛 ∈ 𝑉1 such that
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑛 󵄩󵄩2 = 1,
󵄩󵄩 󵄩󵄩2 1
󵄩󵄩𝑣𝑛 󵄩󵄩 ≤ .
𝑛
(A.1)
Let
󵄩󵄩 󵄩󵄩2
󵄩 󵄩2 󵄩 󵄩2
󵄩󵄩𝑣𝑛 󵄩󵄩𝐻1 = 󵄩󵄩󵄩𝑣𝑛 󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝑣𝑛 󵄩󵄩󵄩 .
(A.2)
1
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝑣𝑛 󵄩󵄩𝐻1 ≤ 1 + ≤ 2,
𝑛
(A.3)
Hence,
which implies that there exists a subsequence of {𝑣𝑛 }𝑛≥1 (still
denoted by {𝑣𝑛 }𝑛≥1 ) such that
𝑣𝑛 󳨀→ 𝑣,
weakly in 𝐻1 ,
𝑣𝑛 󳨀→ 𝑣,
strongly in 𝐿2 .
(A.4)
Therefore, we have
‖𝑣‖22 = 1,
󵄩 󵄩2
inf 󵄩󵄩𝑣 󵄩󵄩 1 = 1,
‖𝑣‖2𝐻1 ≤ lim
𝑛 → ∞ 󵄩 𝑛 󵄩𝐻
(A.5)
which implies that
‖𝑣‖2 = 0.
(A.6)
Combining ‖𝑣‖2 = 0 with 𝑣 ⋅ 𝑛⃗ = 0, we obtain
𝑣 = 0,
(A.7)
this is a contradiction.
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