Global Attractors for the Three-Dimensional Viscous Primitive Equations of Large-Scale Atmosphere in Log-Pressure Coordinate

Abstract

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 large-scale 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, local-in-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 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 −Δ₃𝑇. But in this paper, thanks to the appearance of the weighted function 𝑒^{−𝑧/𝐻𝑠}, we have to first estimate 𝑇_𝑧 in 𝐿² and then ∇₂𝑇 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 three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate [3]: 𝜕𝑣+()𝜕𝑡𝑣⋅∇𝑣+𝑤𝜕𝑣𝜕𝑧+𝑓₀^→𝑘×𝑣+∇Φ+𝐿₁𝑣=0,𝜕Φ=𝜕𝑧𝑅𝑇𝐻_𝑠,∇⋅𝑒^{−𝑧/𝐻𝑠}𝑣+𝜕𝑒𝜕𝑧^{−𝑧/𝐻𝑠}𝑤=0,𝜕𝑇𝜕𝑡+𝑣⋅∇𝑇+𝑤𝜕𝑇𝜕𝑧+𝐿₂𝑇=𝑄, (1)(2)(3)(4) in the domain 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=2Ω₁sin𝜈₀ is the Coriolis parameter, ^→𝑘 is the vertical unit vector, 𝑅 is a positive constant, and 𝑄 is the heat source. The operators 𝐿₁ and 𝐿₂ are given by 𝐿₁1=−Re₁1Δ−Re₂𝜕²𝜕𝑧²,𝐿₂1=−𝑅𝑡₁1Δ−𝑅𝑡₂𝜕²𝜕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=−𝐻_𝑠log(𝑝/𝑝_∗) is used as a vertical coordinate, where 𝐻_𝑠 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=𝜕²_𝑥+𝜕²_𝑦 the horizontal Laplacian.

We denote the different parts of the boundary 𝜕Ω by Γ_𝑢=𝑥,𝑦,𝑧∈,ΓΩ∶𝑧=0_𝑏=𝑥,𝑦,𝑧∈,ΓΩ∶𝑧=−ℎ_𝑙=𝑥,𝑦,𝑧∈Ω∶𝑥,𝑦∈𝜕𝑀,−ℎ≤𝑧≤0. (7)

We equip (1)–(4) with the following boundary conditions, with nonslip and nonflux on the side walls and bottom [4]: 𝜕𝑣𝜕𝑧_Γ𝑢=0,𝑤|_Γ𝑢1=0,𝑅𝑡₂𝜕𝑇𝜕𝑧+𝛼𝑇_Γ𝑢=0,𝜕𝑣𝜕𝑧_Γ𝑏=0,𝑤|_Γ𝑏=0,𝜕𝑇𝜕𝑧_Γ𝑏=0,𝑣⋅^→𝑛_Γ𝑙=0,𝜕𝑣𝜕^→𝑛×^→𝑛_Γ𝑙=0,𝜕𝑇𝜕^→𝑛_Γ𝑙=0, (8) where 𝛼 is a positive constant.

In addition, (1)–(8) are supplemented with the following initial conditions: 𝑣𝑥,𝑦,𝑧,0=𝑣₀,𝑇𝑥,𝑦,𝑧𝑥,𝑦,𝑧,0=𝑇₀𝑥,𝑦,𝑧. (9) Additionally, assume that ℎ satisfy 9ℎ²4𝐻²_𝑠+4ℎ𝛼𝐻²_𝑠𝑅𝑡₂𝐻<1,_2𝑠Re₂𝜆>1. (10)

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.

2. Mathematical Setting of (1)–(10)

2.1. Reformulation of (1)–(9)

Integrating the continuity equation (3) and taking the boundary conditions for 𝑤 into account, we obtain 𝑤𝑥,𝑦,𝑧,𝑡=−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁,_0−ℎ𝑒^{−𝜁/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁=0. (11)

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 Φ_𝑠∶𝑀⟶ℝ, (12) which is the geopotential of the atmosphere on 𝑧=0.

Then, Φ𝑥,𝑦,𝑧,𝑡=Φ_𝑠+𝑅𝑥,𝑦,𝑡𝐻_𝑠^𝑧₀𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁. (13) Therefore, (1)–(4) can be rewritten as follows: 𝜕𝑣+()𝜕𝑡𝑣⋅∇𝑣−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣𝜕𝑧+∇Φ_𝑠𝑥,𝑦,𝑡+𝑓₀^→𝑘+𝑅×𝑣𝐻_𝑠^𝑧₀∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁+𝐿₁𝑣=0,𝜕𝑇−𝜕𝑡+𝑣⋅∇𝑇_𝑧−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧+𝐿₂𝑇=𝑄, (14)(15) with the following boundary conditions: 𝜕𝑣𝜕𝑧_Γ𝑏=0,𝜕𝑣𝜕𝑧_Γ𝑢=0,𝑣⋅^→𝑛_Γ𝑙=0,𝜕𝑣𝜕^→𝑛×^→𝑛_Γ𝑙=0,𝜕𝑇𝜕𝑧_Γ𝑏1=0,𝑅𝑡₂𝜕𝑇𝜕𝑧+𝛼𝑇_Γ𝑢=0,𝜕𝑇𝜕^→𝑛_Γ𝑙=0, (16)(17) and the initial data 𝑣𝑥,𝑦,𝑧,0=𝑣₀,𝑇𝑥,𝑦,𝑧𝑥,𝑦,𝑧,0=𝑇₀𝑥,𝑦,𝑧. (18) At last, we will divide (14) into two systems with respect to 𝑣 and 𝑣 [4], where 𝑣 and 𝑣 are defined by 1𝑣=𝐴⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}𝑣𝑥,𝑦,𝑧𝑑𝑧,𝑣=𝑣−𝑣, (19) where 𝐴=⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}𝑑𝑧=𝐻_𝑠𝑒^ℎ/𝐻𝑠−1, (20) and we have 𝑣=0.

Therefore, 𝑣 satisfies the following systems: 𝜕𝑣+(𝜕𝑡𝑣⋅∇)1𝑣−Re₁Δ𝑣+∇Φ_𝑠𝑥,𝑦,𝑡+(∇⋅𝑣)𝑣+(𝑣⋅∇)𝑣+𝑓₀^→𝑘×𝑣+𝑅𝐻_𝑠^𝑧₀1∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁−𝐻_𝑠Re₂𝜕𝑣𝜕𝑧=0, (21) with the boundary conditions ∇⋅𝑣=0,𝑣⋅^→𝑛_Γ𝑙𝜕=0,𝑣𝜕^→𝑛×^→𝑛_Γ𝑙=0, (22) and 𝑣 satisfies the following systems𝜕𝑣+()𝜕𝑡𝑣⋅∇𝑣−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}𝑣𝜕𝑣∇⋅𝑥,𝑦,𝜁,𝑡𝑑𝜁+𝑅𝜕𝑧𝐻_𝑠^𝑧₀∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁+𝑓₀^→𝑘×𝑣+𝐿₁𝑣+(𝑣⋅∇)𝑣+()𝑣−𝑣⋅∇(𝑣)()𝑅∇⋅𝑣+𝑣⋅∇𝑣+𝐻_𝑠^𝑧₀1∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁−𝐻_𝑠Re₂𝜕𝑣𝜕𝑧=0, (23)

with the boundary conditions 𝜕𝑣𝜕𝑧_Γ𝑢𝜕𝑣=0,𝜕𝑧_Γ𝑏=0,𝑣⋅^→𝑛_Γ𝑙𝜕𝑣=0,𝜕^→𝑛×^→𝑛_Γ𝑙=0. (24)

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: 𝒱₁=𝑣∈𝐶^∞Ω²∶𝜕𝑣𝜕𝑧_Γ𝑢=0,𝜕𝑣𝜕𝑧_Γ𝑏=0,𝑣⋅^→𝑛_Γ𝑙=0,𝜕𝑣𝜕^→𝑛×^→𝑛_Γ𝑙=0,⁰_−ℎ𝑒^{−𝑧/𝐻𝑠},𝒱∇⋅𝑣𝑥,𝑦,𝑧𝑑𝑧=0₂=𝑇∈𝐶^∞Ω∶𝜕𝑇𝜕𝑧_Γ𝑏1=0,𝑅𝑡₂𝜕𝑇𝜕𝑧+𝛼𝑇_Γ𝑢=0,𝜕𝑇𝜕^→𝑛_Γ𝑙=0. (25) Let 𝐿^𝑝 (or 𝐻^𝑠) be the weighted Hilbert spaces of 𝐿^𝑝(or 𝐻^𝑠) functions or 𝐿^𝑝(or 𝐻^𝑠)-vector valued functions on Ω and denote the usual 𝐿^𝑝(Ω)-norms by ‖⋅‖_𝐿^𝑝_(Ω), where 1≤𝑝≤∞(or 𝑠≥0). Denote the inner product in 𝐿² by (⋅,⋅) and the norm in 𝐿^𝑝 by ‖⋅‖_𝑝, respectively, given by ℎ₁,ℎ₂=_Ω𝑒^{−𝑧/𝐻𝑠}ℎ₁⋅ℎ₂‖ℎ‖𝑑𝑥𝑑𝑦𝑑𝑧,_𝑝𝑝=_Ω𝑒^{−𝑧/𝐻𝑠}|ℎ|^𝑝𝑑𝑥𝑑𝑦𝑑𝑧, (26) for any ℎ₁,ℎ₂∈𝐿² and ℎ∈𝐿^𝑝(1≤𝑝≤∞).

Similarly, define 𝑣₁,𝑣₂=1Re_1Ω𝑒^{−𝑧/𝐻𝑠}∇𝑣₁⋅∇𝑣₂+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕𝑣₁⋅𝜕𝑧𝜕𝑣₂𝑇𝜕𝑧𝑑𝑥𝑑𝑦𝑑𝑧,₁,𝑇₂=1𝑅𝑡_1Ω𝑒^{−𝑧/𝐻𝑠}∇𝑇₁⋅∇𝑇₂+1𝑑𝑥𝑑𝑦𝑑𝑧𝑅𝑡_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕𝑇₁𝜕𝑧𝜕𝑇₂𝜕𝑧𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_Γ𝑢𝑇₁𝑇₂𝑑𝑥𝑑𝑦, (27) 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)), (28) for any 𝑣∈𝒱₁ and 𝑇∈𝒱₂, respectively, and let 𝑉=𝑉₁×𝑉₂.

Next, we recall some results used in the sequel.

Lemma 1 (see [19]). (i) for 𝑢∈𝐻¹(𝑀), ‖𝑢‖_𝐿⁴_(𝑀)≤𝐶‖𝑢‖_𝐿^1/2_2(𝑀)‖𝑢‖_𝐻^1/2_1(𝑀),‖𝑢‖_𝐿⁶_(𝑀)≤𝐶‖𝑢‖_𝐿^2/3_2(𝑀)‖𝑢‖_𝐻^1/3_1(𝑀),‖𝑢‖_𝐿⁸_(𝑀)≤𝐶‖𝑢‖_𝐿^1/2_4(𝑀)‖𝑢‖_𝐻^1/2_1(𝑀). (29)
(ii) for 𝑢∈𝑉_𝑖(𝑖=1,2), ‖𝑢‖₄≤𝐶‖𝑢‖₂^1/4‖𝑢‖^3/4. (30)

Proposition 2. There exists a positive constant 𝜆 such that ‖𝑣‖²≥𝜆‖𝑣‖²₂, (31) for any 𝑣∈𝑉₁.

Lemma 3 (see [4, 6]). Let 𝑣,𝑣₁∈𝑉₁, 𝑇∈𝑉₂, and 𝑝≥1. Then, we have(1)∫_Ω[(𝑣⋅∇)𝑣₁−(∫^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣(𝑥,𝑦,𝜁)𝑑𝜁)(𝜕𝑣₁/𝜕𝑧)]⋅(𝑒^{−𝑧/𝐻𝑠}|𝑣₁|^𝑝−1𝑣₁)=0, (2)∫_Ω[𝑣⋅∇𝑇−(∫^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣(𝑥,𝑦,𝜁)𝑑𝜁)(𝜕𝑇/𝜕𝑧)]𝑒^{−𝑧/𝐻𝑠}|𝑇|^𝑝−1𝑇=0,(3)∫_Ω𝑒^{−𝑧/𝐻𝑠}∇Φ_𝑠(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)∈𝐿^𝑝(𝑋,𝜇)(1≤𝑝≤∞) for a.e.𝑦∈𝑌 and ∫_𝑌𝜌(𝑦)‖𝑓(⋅,𝑦)‖_𝐿^𝑝_(𝑋,𝜇)𝑑𝜈(𝑦)<∞, then _𝑌𝜌𝑦𝑓⋅,𝑦𝑑𝜈𝑦_{𝐿𝑝(𝑋,𝜇)}≤_𝑌𝜌𝑦𝑓⋅,𝑦_{𝐿𝑝(𝑋,𝜇)}𝑑𝜈𝑦. (32)

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 (𝑣,𝑇)∈𝐿^∞(0,𝜏;𝑉)∩𝐿²0,𝜏;𝐻²,𝑣_𝑡,𝑇_𝑡∈𝐿¹0,𝜏;𝐿², (33) and they satisfy 𝑣(𝑡)+,𝜙_𝑡𝑡0Ω𝑒^{−𝑧/𝐻𝑠}(𝑣⋅∇)𝑣−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}×∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣𝜕𝑧+𝑓₀^→𝑘𝑅×𝑣+𝐻_𝑠^𝑧₀∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁⋅𝜙+^𝑡_𝑡0=1𝑣,𝜙𝐻_𝑠Re₂^𝑡_𝑡0𝜕_𝑧𝑣,𝜙+𝑣𝑡₀,𝑇(𝑡)+,𝜙,𝜓_𝑡𝑡0Ω𝑒^{−𝑧/𝐻𝑠}𝑣⋅∇𝑇−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}×∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧𝜓+^𝑡_𝑡0=𝑇,𝜓_𝑡𝑡0+1𝑄,𝜓𝐻_𝑠𝑅𝑡₂^𝑡_𝑡0𝜕_𝑧𝑇,𝜓+𝑇𝑡₀,𝜓, (34) for every 𝜙∈𝒱₁ and 𝜓∈𝒱₂, almost every 𝑡₀,𝑡∈[0,𝜏].

3. Some a Priori Estimates for (1)–(10)

We start with the following general existence and uniqueness of solutions which can be obtained by the standard Fatou-Galerkin 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 (𝑣,𝑇)∈𝐿^∞(0,𝜏;𝑉)∩𝐿²0,𝜏;𝐻² (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, (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).

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 12𝑑‖𝑇‖𝑑𝑡²²+‖𝑇‖²≤‖𝑄‖₂‖𝑇‖₂+1𝐻_𝑠𝑅𝑡₂𝜕𝑇𝜕𝑧₂‖𝑇‖₂. (37) Employing Hölder’s inequality yields 𝑑‖𝑇‖𝑑𝑡²²+‖𝑇‖²≤2𝐻²_𝑠𝑅𝑡₂‖𝑇‖²₂+𝐻²_𝑠𝑅𝑡₂‖𝑄‖²₂. (38) Thanks to ‖𝑇‖²₂≤2ℎ‖𝑇(𝑧=0)‖²_𝐿2(𝑀)+ℎ²𝜕𝑇𝜕𝑧²₂+ℎ²4𝐻²_𝑠‖𝑇‖²₂, (39) we obtain ‖𝑇‖²₂ℎ²𝑅𝑡₂≤+2ℎ/𝛼4𝐻²_𝑠4𝐻²_𝑠−ℎ²1𝑅𝑡₂𝜕𝑇𝜕𝑧²₂+𝛼‖𝑇(𝑧=0)‖²_𝐿2(𝑀). (40) Setting 𝛿=1−8(𝑅𝑡₂ℎ²+2ℎ/𝛼)/𝑅𝑡₂(4𝐻²_𝑠−ℎ²) and 𝛿₁=𝛿(4𝐻²_𝑠−ℎ²)/4𝐻²_𝑠(𝑅𝑡₂ℎ²+2ℎ/𝛼), we find that 𝑑‖𝑇‖𝑑𝑡²²+𝛿‖𝑇‖²≤𝐻²_𝑠𝑅𝑡₂‖𝑄‖²₂,𝑑‖𝑇‖𝑑𝑡²²+𝛿₁‖𝑇‖²₂≤𝐻²_𝑠𝑅𝑡₂‖𝑄‖²₂, (41) which implies that ‖𝑇(𝑡)‖²₂≤𝑇₀²₂𝑒^−𝛿1𝑡+𝐻²_𝑠𝑅𝑡₂𝛿₁‖𝑄‖²₂1−𝑒^−𝛿1𝑡. (42) Therefore, there exist three positive constants 𝜌₁,𝜌₂,𝑇₁ such that ‖𝑇(𝑡)‖²₂≤𝜌₁,_𝑡𝑡+1‖𝑇‖²𝑑𝜏≤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 12𝑑‖𝑣‖𝑑𝑡²²+‖𝑣‖²≤𝐶‖𝑇‖₂‖∇𝑣‖₂+1𝐻_𝑠Re₂𝜕𝑣𝜕𝑧₂‖𝑣‖₂. (45) Setting 𝛿₃=1−1/Re₂𝜆𝐻²_𝑠 and 𝛿₄=𝜆−1/Re₂𝐻²_𝑠, we obtain 𝑑‖𝑣‖𝑑𝑡²²+𝛿₃‖𝑣‖²≤𝐶‖𝑇‖²₂,𝑑‖𝑣‖𝑑𝑡²²+𝛿₄‖𝑣‖²₂≤𝐶‖𝑇‖²₂. (46) From the classical Gronwall inequality, we have ‖𝑣(𝑡)‖²₂≤𝑣₀²₂𝑒^−𝛿4𝑡+𝐶𝜌₁, (47) for all 𝑡≥𝑇₁, which implies that there exist three positive constants 𝜌₃,𝜌₄,𝑇₂≥𝑇₁ such that ‖𝑣(𝑡)‖²₂≤𝜌₃,_𝑡𝑡+1‖𝑣‖²𝑑𝜏≤𝜌₄, (48)(49) for all 𝑡≥𝑇₂.

3.3. 𝐿⁶(Ω) Estimates of 𝑇

Taking the 𝐿²(Ω) inner product of (15) with |𝑇|⁴𝑇 and using Lemma 3, we have 16𝑑‖𝑇‖𝑑𝑡⁶⁶+59|𝑇|³²≤|𝑇|^35/3_10/3‖𝑄‖₂+1𝐻_𝑠𝑅𝑡₂|𝑇|^35/3_10/3𝑇_𝑧2. (50) From the inequality ‖𝜑‖_10/3≤𝐶‖𝜑‖₂^2/5‖𝜑‖^3/5, we get 𝑑‖𝑇‖𝑑𝑡⁶⁶+2|𝑇|³²≤𝐶𝑇_𝑧²₂‖𝑇‖⁴₆+𝐶‖𝑄‖²₂‖𝑇‖⁴₆, (51) which implies that 𝑑‖𝑇‖𝑑𝑡²⁶≤𝐶𝑇_𝑧²₂+𝐶‖𝑄‖²₂. (52) Therefore, by virtue of the uniform Gronwall lemma, from (44) and ‖𝑇‖₆≤𝐶‖𝑇‖_𝐻¹_(Ω), we deduce that ‖𝑇‖²₆≤𝜌₅, (53) for any 𝑡≥𝑇₂+1.

3.4. (𝐿⁶(Ω))² Estimates of 𝑣

Multiplying (23) by |𝑣|⁴𝑣, integrating over Ω, and using Lemma 3, we deduce that 16𝑑‖𝑣‖𝑑𝑡⁶⁶+1Re_1Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧𝑣²|𝑣|⁴4𝑑𝑥𝑑𝑦𝑑𝑧+9Re₁×_Ω𝑒^{−𝑧/𝐻𝑠}∇|𝑣|³²+4𝑑𝑥𝑑𝑦𝑑𝑧9Re_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧|𝑣|³²≤1𝑑𝑥𝑑𝑦𝑑𝑧Re₂𝐻_𝑠𝑣_𝑧2|𝑣|^35/3_10/3+𝐶‖𝑣‖₄‖𝑣‖₆^3/2|𝑣|^31/2×_Ω|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2+𝐶‖∇𝑇‖₂‖𝑣‖²₆|𝑣|³+_Ω𝑒^{−𝑧/𝐻𝑠}(𝑣⋅∇)𝑣+(∇⋅𝑣)𝑣⋅|𝑣|⁴𝑣𝑑𝑥𝑑𝑦𝑑𝑧. (54) It follows from the Sobolev interpolation inequality and Lemma 4 that _Ω𝑒^{−𝑧/𝐻𝑠}(𝑣⋅∇)𝑣+(∇⋅𝑣)𝑣⋅|𝑣|⁴𝑣𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_𝑀∫⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣||𝑣|⁴×𝑑𝑧_0−ℎ𝑒^{−𝑧/𝐻𝑠}|𝑣|²𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶_𝑀⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴𝑑𝑧^1/2×⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|𝑣|⁴𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶_Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2×_𝑀⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|𝑣|⁴𝑑𝑧²𝑑𝑥𝑑𝑦^1/2,_𝑀0−ℎ𝑒^{−𝑧/𝐻𝑠}|𝑣|⁴𝑑𝑧²𝑑𝑥𝑑𝑦^1/2≤⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}_𝑀|𝑣|⁸𝑑𝑥𝑑𝑦^1/2𝑑𝑧≤𝐶⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}‖𝑣‖³_𝐿6(𝑀)‖𝑣‖_𝐻¹_(𝑀)‖𝑣‖𝑑𝑧≤𝐶³₆‖𝑣‖, (55) which implies that _Ω𝑒^{−𝑧/𝐻𝑠}(𝑣⋅∇)𝑣+(∇⋅𝑣)𝑣⋅|𝑣|⁴𝑣𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2‖𝑣‖³₆‖𝑣‖, (56) where we use the inequality ‖𝑣‖⁸_𝐿8(𝑀)≤𝐶‖𝑣‖⁶_𝐿6(𝑀)‖𝑣‖²_𝐻1(𝑀).

We deduce from (54) and (56) that 𝑑‖𝑣‖𝑑𝑡²⁶≤𝐶‖𝑣‖²+‖𝑣‖⁴₄‖𝑣‖²₆+𝐶𝑣_𝑧²₂+𝐶‖∇𝑇‖²₂. (57) Therefore, it follows from (44)–(49) and the uniform Gronwall lemma that ‖𝑣‖²₆≤𝜌₆,_𝑡𝑡+1Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏≤𝜌₇, (58)(59) for any 𝑡≥𝑇₂+2.

3.5. (𝐻¹(𝑀))² Estimates of 𝑣

Taking the 𝐿²(Ω) inner product of (21) with −Δ𝑣 and combining the boundary conditions (22), we deduce that 12𝑑‖∇𝑑𝑡𝑣‖²_𝐿2(𝑀)+1Re_1𝑀|Δ𝑣|²≤1𝑑𝑥𝑑𝑦Re₂𝐻_𝑠𝑣_𝑧2‖Δ𝑣‖₂+𝐶‖𝑣‖₄‖∇𝑣‖₄‖Δ𝑣‖₂+𝐶_𝑀⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣||𝑣|𝑑𝑧|Δ𝑣|‖𝑑𝑥𝑑𝑦+𝐶𝑣‖₂‖Δ𝑣‖₂+𝐶‖∇𝑇‖₂‖Δ𝑣‖₂. (60) By using the Hölder inequality, we get _𝑀⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣||𝑣|𝑑𝑧|Δ𝑣|𝑑𝑥𝑑𝑦≤_𝑀⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣|𝑑𝑧^1/2×⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣||𝑣|²𝑑𝑧^1/2|Δ𝑣|≤𝑑𝑥𝑑𝑦_𝑀0−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣|𝑑𝑧²𝑑𝑥𝑑𝑦^1/4×_𝑀⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}|∇𝑣||𝑣|²𝑑𝑧²𝑑𝑥𝑑𝑦^1/4‖Δ𝑣‖₂≤⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}_𝑀|∇𝑣|²𝑑𝑥𝑑𝑦^1/2𝑑𝑧^1/2×⁰_−ℎ𝑒^{−𝑧/𝐻𝑠}_𝑀|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦^1/2𝑑𝑧^1/2‖Δ𝑣‖₂≤𝐶‖∇𝑣‖₂^1/2_Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/4‖Δ𝑣‖₂. (61) It follows from (60) and (61) that 𝑑‖∇𝑑𝑡𝑣‖²_𝐿2(𝑀)+1Re_1𝑀|Δ𝑣|²‖𝑑𝑥𝑑𝑦≤𝐶𝑣‖²_𝐿2(𝑀)‖∇𝑣‖⁴_𝐿2(𝑀)+𝐶𝑣_𝑧²₂+𝐶‖∇𝑇‖²₂+𝐶‖∇𝑣‖²₂+𝐶_Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑣|²|𝑣|⁴𝑑𝑥𝑑𝑦𝑑𝑧+𝐶‖𝑣‖²₂. (62)

In view of (44)–(49) and (59) and the uniform Gronwall lemma, we obtain ‖∇𝑣‖²_𝐿2(𝑀)≤𝜌₈, (63) for any 𝑡≥𝑇₂+3.

3.6. (𝐿²(Ω))² Estimates of 𝑣_𝑧

Denote that 𝑢=𝑣_𝑧. It is clear that 𝑢 satisfies the following equation obtained by differentiating (14) with respect to 𝑧: 𝜕𝑢𝜕𝑡+𝐿₁()𝑢−𝑢+𝑣⋅∇_𝑧−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑢−1𝜕𝑧𝐻_𝑠^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}𝑢+()()∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝑢⋅∇𝑣−∇⋅𝑣𝑢+𝑓₀^→𝑘𝑅×𝑢+𝐻_𝑠∇𝑇=0, (64) with the boundary conditions 𝑢|_Γ𝑢=0,𝑢|_Γ𝑏=0,𝑢⋅^→𝑛_Γ𝑙=0,𝜕𝑢𝜕^→𝑛×^→𝑛_Γ𝑙=0. (65) Multiplying (64) by 𝑢, integrating over Ω, and using Lemma 3, we get 12𝑑‖𝑢‖𝑑𝑡²²+‖𝑢‖²≤1Re₂𝐻_𝑠𝑢_𝑧2‖𝑢‖₂+𝐶‖𝑣‖₆‖𝑢‖₃‖∇𝑢‖₂+𝐶‖𝑇‖₂‖∇𝑢‖₂. (66) It follows from the Young inequality and the Sobolev embedding theorem that 𝑑‖𝑢‖𝑑𝑡²²+‖𝑢‖²≤𝐶1+‖𝑣‖⁴₆‖𝑢‖²₂+𝐶‖𝑇‖²₂. (67)

Since ‖𝑣‖₆≤𝐶ℎ^−1/3‖𝑣‖₂+𝐶ℎ^1/6‖∇𝑣‖₂+‖𝑣‖₆, (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 𝑣_𝑧²₂≤𝜌₁₀,_𝑡𝑡+1𝑣_𝑧²𝑑𝜏≤𝜌₁₁, (70)(71) 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 12𝑑‖‖𝑑𝑡∇𝑣²₂+1Re_1Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}∇𝑣_𝑧²≤1𝑑𝑥𝑑𝑦𝑑𝑧Re₂𝐻_𝑠𝑣_𝑧2‖Δ𝑣‖₂+𝐶𝑣_𝑧2^1/2∇𝑣_𝑧2^1/2‖∇𝑣‖₂^1/2‖Δ𝑣‖₂^3/2+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 𝑑‖‖𝑑𝑡∇𝑣²₂+1Re_1Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}∇𝑣_𝑧²‖𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶⁴₆+𝑣_𝑧²₂∇𝑣_𝑧²₂‖∇𝑣‖²₂+𝐶‖∇𝑇‖²₂+𝐶‖𝑣‖²₂+𝐶𝑣_𝑧²₂. (73)

By virtue of (44)–(49), (69)–(71), and the uniform Gronwall lemma, we get ‖∇𝑣‖²₂≤𝜌₁₂,1Re_1𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏Re_2𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}∇𝑣_𝑧²𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏≤𝜌₁₃, (74)(75) for any 𝑡≥𝑇₂+5.

3.8. 𝐿²(Ω) Estimates of 𝑇_𝑧

Denote 𝜃=𝑇_𝑧. It is clear that 𝜃 satisfies the following equation obtained by differentiating (15) with respect to 𝑧: 𝜕𝜃𝜕𝑡+𝐿₂𝜃+𝑣⋅∇𝜃−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝜃−1𝜕𝑧𝐻_𝑠^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜃+𝑣_𝑧⋅∇𝑇−(∇⋅𝑣)𝜃=𝜕_𝑧𝑄, (76) with the boundary conditions 1𝑅𝑡₂𝜃+𝛼𝑇_Γ𝑢=0,𝜃|_Γ𝑏=0,𝜕𝜃𝜕^→𝑛_Γ𝑙=0. (77) Multiplying (76) by 𝜃, integrating over Ω, and using Lemma 3, we deduce that 12𝑑‖𝜃‖𝑑𝑡²²+1𝑅𝑡_1Ω𝑒^{−𝑧/𝐻𝑠}|∇𝜃|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑅𝑡_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧𝜃²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_Γ𝑢𝜕𝜃≤1𝜕𝑧𝑇𝑑𝑥𝑑𝑦𝑅𝑡₂𝐻_𝑠𝜃_𝑧2‖𝜃‖₂+𝐶‖𝑣‖₆‖𝜃‖₃‖∇𝜃‖₂+𝐶𝑄_𝑧2‖𝜃‖₂+𝐶∇𝑣_𝑧2‖𝑇‖₆‖𝜃‖₃+𝐶𝑣_𝑧3‖𝑇‖₆‖∇𝜃‖₂. (78) We deduce from the Young inequality that 𝛼_Γ𝑢𝜕𝜃𝜕𝑧𝑇𝑑𝑥𝑑𝑦=𝛼𝑅𝑡_2Γ𝑢𝜕𝑇1𝜕𝑡+𝑣⋅∇𝑇−𝑅𝑡₁=Δ𝑇−𝑄𝑇𝑑𝑥𝑑𝑦𝛼𝑅𝑡₂2𝑑𝑑𝑡_Γ𝑢|𝑇|²𝑑𝑥𝑑𝑦+𝛼𝑅𝑡_2Γ𝑢()+𝑣⋅∇𝑇𝑇𝑑𝑥𝑑𝑦𝛼𝑅𝑡₂𝑅𝑡_1Γ𝑢|∇𝑇|²𝑑𝑥𝑑𝑦−𝛼𝑅𝑡_2Γ𝑢𝑄𝑇𝑑𝑥𝑑𝑦,𝛼𝑅𝑡_2Γ𝑢(𝑣⋅∇𝑇)𝑇𝑑𝑥𝑑𝑦−𝛼𝑅𝑡_2Γ𝑢‖𝑣‖𝑄𝑇𝑑𝑥𝑑𝑦≤𝐶_𝐿⁴_(Γ𝑢)‖∇𝑇‖_𝐿²_(Γ𝑢)‖𝑇‖_𝐿⁴_(Γ𝑢)+𝐶‖𝑄‖_𝐿²_(Γ𝑢)‖𝑇‖_𝐿²_(Γ𝑢)≤𝐶‖𝑣‖‖∇𝑇‖_𝐿^3/2_2(Γ𝑢)‖𝑇‖_𝐿^1/2_2(Γ𝑢)+𝐶‖𝑄‖‖𝑇‖_𝐿²_(Γ𝑢). (79) It follows from (78) and (79) that 𝑑‖𝜃‖𝑑𝑡²²+𝛼𝑅𝑡₂‖𝑇‖²_𝐿2Γ𝑢+1𝑅𝑡_1Ω𝑒^{−𝑧/𝐻𝑠}|∇𝜃|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑅𝑡_2Ω𝑒^{−𝑧/𝐻𝑠}𝜃_𝑧²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼𝑅𝑡₂𝑅𝑡_1Γ𝑢|∇𝑇|²‖𝑣‖𝑑𝑥𝑑𝑦≤𝐶1+⁴₆+‖𝑇‖⁴₆+‖𝑣‖⁴‖𝜃‖²₂+𝛼𝑅𝑡₂‖𝑇‖²_𝐿2Γ𝑢+𝐶‖𝑄‖²+𝑣_𝑧²+𝑣_𝑧²₂‖𝑇‖⁴₆. (80)

As a result of (44), (53), (69)–(71), and (74), we get 𝑇_𝑧²₂+𝛼𝑅𝑡₂‖𝑇‖²_{𝐿2(Γ𝑢)}≤𝜌₁₄,1𝑅𝑡_1𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}∇𝑇_𝑧²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏𝑅𝑡_2𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}𝜕²_𝑧𝑇²+𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏𝛼𝑅𝑡₂𝑅𝑡_1𝑡^𝑡+1_Γ𝑢|∇𝑇|²𝑑𝑥𝑑𝑦𝑑𝜏≤𝜌₁₅, (81)(82) for any 𝑡≥𝑇₂+6.

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 12𝑑‖‖𝑑𝑡∇𝑇²₂+1𝑅𝑡_1Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝑇|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑅𝑡_2Ω∇𝑇_𝑧²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_Γ𝑢|∇𝑇|²≤1𝑑𝑥𝑑𝑦𝑅𝑡₂𝐻_𝑠𝑇_𝑧2‖Δ𝑇‖₂+𝐶‖𝑣‖₆‖∇𝑇‖₃‖Δ𝑇‖₂+𝐶‖𝑄‖₂‖Δ𝑇‖₂+𝐶‖∇𝑣‖₂^1/2‖Δ𝑣‖₂^1/2𝑇_𝑧2^1/2∇𝑇_𝑧2^1/2‖Δ𝑇‖₂. (83) We derive from the Young inequality that 𝑑‖‖𝑑𝑡∇𝑇²₂+1𝑅𝑡_1Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝑇|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑅𝑡_2Ω𝑒^{−𝑧/𝐻𝑠}∇𝑇_𝑧²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_𝑀|∇𝑇(𝑧=0)|²‖𝑣‖𝑑𝑥𝑑𝑦≤𝐶⁴₆‖∇𝑇‖²₂+𝐶‖∇𝑣‖²₂‖Δ𝑣‖²₂𝑇_𝑧²₂+𝐶‖𝑄‖²₂+𝐶𝑇_𝑧²₂. (84)

Using (44), (69), (74), (75), and (81), we get ‖∇𝑇‖²₂≤𝜌₁₆,1𝑅𝑡_1𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝑇|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏𝑅𝑡_2𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}∇𝑇_𝑧²𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏+𝛼_𝑡^𝑡+1_𝑀|∇𝑇(𝑧=0)|²𝑑𝑥𝑑𝑦𝑑𝜏≤𝜌₁₇, (85)(86) for any 𝑡≥𝑇₂+7.

3.10. (𝐿⁶(Ω))² Estimates of 𝝏_𝑧𝑣

Taking the 𝐿²(Ω) inner product of (64) with |𝑢|⁴𝑢, combining the boundary conditions (65), and using Lemma 3, we deduce that 16𝑑‖𝑢‖𝑑𝑡⁶⁶+1Re_1Ω𝑒^{−𝑧/𝐻𝑠}|∇𝑢|²|𝑢|⁴+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧𝑢²|𝑢|⁴+4𝑑𝑥𝑑𝑦𝑑𝑧9Re_1Ω𝑒^{−𝑧/𝐻𝑠}∇|𝑢|³²+4𝑑𝑥𝑑𝑦𝑑𝑧9Re_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧|𝑢|³²≤1𝑑𝑥𝑑𝑦𝑑𝑧Re₂𝐻_𝑠𝑢_𝑧2|𝑢|^35/3_10/3+𝐶‖𝑣‖₆|𝑢|³₃∇|𝑢|³₂+𝐶‖∇𝑇‖₂|𝑢|^35/3_10/3+𝐶‖𝑣‖₆|𝑢|³₃|∇𝑢||𝑢|²₂. (87) From the Young inequality, we obtain 𝑑‖𝑢‖𝑑𝑡²⁶≤𝐶‖𝑣‖⁴₆‖𝑢‖²₆+𝐶‖∇𝑇‖²₂+𝐶𝑢_𝑧²₂. (88)

Thanks to (44), (69), and (71), we have 𝑣_𝑧²₆≤𝜌₁₈, (89) for any 𝑡≥𝑇₂+8.

3.11. 𝐿⁶(Ω) Estimates of 𝝏_𝑧𝑇

Taking the 𝐿²(Ω) inner product of (76) with |𝜃|⁴𝜃, combining the boundary conditions (77), and using Lemma 3, we deduce that 16𝑑‖𝜃‖𝑑𝑡⁶⁶+59𝑅𝑡_1Ω𝑒^{−𝑧/𝐻𝑠}∇|𝜃|³²+5𝑑𝑥𝑑𝑦𝑑𝑧9𝑅𝑡_2Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧|𝜃|³²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼⁵𝑅𝑡⁴_2Γ𝑢𝜕𝜃|𝑇|𝜕𝑧⁴≤1𝑇𝑑𝑥𝑑𝑦𝑅𝑡₂𝐻_𝑠𝜃_𝑧2|𝜃|^35/3_10/3+𝐶‖𝑣‖₆|𝜃|³₃∇|𝜃|³₂+𝐶‖𝑣‖₆|𝜃|³₃|∇𝜃||𝜃|²₂+𝑄_𝑧2|𝜃|^35/3_10/3+−_Ω𝑒^{−𝑧/𝐻𝑠}𝑣_𝑧⋅∇𝑇|𝜃|⁴𝜃𝑑𝑥𝑑𝑦𝑑𝑧. (90)

It follows from the Hölder inequality that −_Ω𝑒^{−𝑧/𝐻𝑠}𝑣_𝑧⋅∇𝑇|𝜃|⁴𝜃𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_Ω𝑒^{−𝑧/𝐻𝑠}𝑣_𝑧|∇𝑇||𝜃|⁵𝑣𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_𝑧6‖∇𝑇‖₃|𝜃|^35/3_10/3≤𝐶𝑣_𝑧6‖∇𝑇‖₂^1/2‖∇𝑇‖_𝐻^1/2_1(Ω)|𝜃|³₂^2/3|𝜃|³_𝐻1(Ω),𝛼⁵𝑅𝑡⁴_2Γ𝑢𝜕𝜃|𝑇|𝜕𝑧⁴𝑇𝑑𝑥𝑑𝑦=𝛼⁵𝑅𝑡⁵_2Γ𝑢𝜕𝑇1𝜕𝑡+𝑣⋅∇𝑇−𝑅𝑡₁Δ𝑇−𝑄|𝑇|⁴=𝛼𝑇𝑑𝑥𝑑𝑦₅𝑅𝑡⁵₂6𝑑𝑑𝑡_Γ𝑢|𝑇|⁶𝑑𝑥𝑑𝑦+𝛼⁵𝑅𝑡⁵_2Γ𝑢(𝑣⋅∇𝑇)|𝑇|⁴+𝑇𝑑𝑥𝑑𝑦5𝛼⁵𝑅𝑡⁵₂9𝑅𝑡_1Γ𝑢∇|𝑇|³²𝑑𝑥𝑑𝑦−𝛼⁵𝑅𝑡⁵_2Γ𝑢𝑄|𝑇|⁴𝛼𝑇𝑑𝑥𝑑𝑦,₅𝑅𝑡⁵_2Γ𝑢(𝑣⋅∇𝑇)|𝑇|⁴𝑇𝑑𝑥𝑑𝑦−𝛼⁵𝑅𝑡⁵_2Γ𝑢𝑄|𝑇|⁴‖𝑣‖𝑇𝑑𝑥𝑑𝑦≤𝐶_𝐿⁴_(Γ𝑢)∇|𝑇|³_{𝐿2(Γ𝑢)}|𝑇|³_{𝐿4(Γ𝑢)}+𝐶‖𝑄‖_𝐿²_(Γ𝑢)|𝑇|³_𝐿^5/3_10/3(Γ𝑢)≤𝐶‖𝑣‖∇|𝑇|³_𝐿^3/2_2(Γ𝑢)‖𝑇‖_𝐿^3/2_6(Γ𝑢)+𝐶‖𝑄‖|𝑇|³_{𝐿2(Γ𝑢)}|𝑇|³_𝐻^2/3_1(Γ𝑢). (91) We derive from the Young inequality that 𝑑‖𝜃‖𝑑𝑡²⁶+𝛼𝑅𝑡₂‖𝑇‖²_𝐿6Γ𝑢≤𝐶1+‖𝑣‖⁴₆+‖𝑣‖⁴‖𝜃‖²₆+𝛼𝑅𝑡₂‖𝑇‖²_𝐿6Γ𝑢+𝐶𝜃_𝑧²₂+𝐶‖𝑄‖²+𝐶𝑣_𝑧⁴₆+𝐶‖∇𝑇‖²₂‖∇𝑇‖²_𝐻1(Ω). (92)

As a consequence of the uniform Gronwall lemma, we get 𝑇_𝑧²₆+𝛼𝑅𝑡₂‖𝑇‖²_{𝐿6(Γ𝑢)}≤𝜌₁₉, (93) for any 𝑡≥𝑇₂+9.

3.12. 𝐻 Estimates of (𝝏_𝑡𝑣,𝝏_𝑡𝑇)

Denote that 𝜋=𝜕_𝑡𝑣,  𝜉=𝜕_𝑡𝑇. It is clear that 𝜋,𝜉 satisfy the following equations obtained by differentiating (14) and (15) with respect to 𝑡, respectively, 𝜕𝜋𝜕𝑡+𝐿₁()𝜋−𝜋+𝑣⋅∇_𝑧−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝜋𝜕𝑧+𝑓₀^→𝑘×𝜋+∇𝜕_𝑡Φ_𝑠+()𝑣−𝑥,𝑦,𝑡𝜋⋅∇_𝑧−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝜋𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣+𝑅𝜕𝑧𝐻_𝑠^𝑧₀∇𝜉𝑥,𝑦,𝜁,𝑡𝑑𝜁=0,𝜕𝜉𝜕𝑡+𝐿₂𝜉+𝑣⋅∇𝜉−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝜉𝜕𝑧+𝜋⋅∇𝑇−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝜋𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧=0, (94)(95) with the boundary conditions 𝜕𝜋𝜕𝑧_Γ𝑢=0,𝜕𝜋𝜕𝑧_Γ𝑏=0,𝜋⋅^→𝑛_Γ𝑙=0,𝜕𝜋𝜕^→𝑛×^→𝑛_Γ𝑙1=0,𝑅𝑡₂𝜕𝜉𝜕𝑧+𝛼𝜉_Γ𝑢=0,𝜕𝜉𝜕𝑧_Γ𝑏=0,𝜕𝜉𝜕^→𝑛_Γ𝑙=0, (96)(97) Multiplying (94) and (95), respectively, by 𝜋,  𝜉, integrating over Ω, and using Lemma 3, we deduce that 12𝑑‖𝜋‖𝑑𝑡²²+‖𝜋‖²≤1Re₂𝐻_𝑠𝜋_𝑧2‖𝜋‖₂+𝐶‖𝑣‖₆‖𝜋‖₃‖∇𝜋‖₂+𝐶𝑣_𝑧6‖𝜋‖₃‖∇𝜋‖₂+𝐶‖∇𝜋‖₂𝜉₂,12𝑑𝜉𝑑𝑡²²+𝜉²≤1𝑅𝑡₂𝐻_𝑠𝜉_𝑧2𝜉₂+𝐶‖𝑇‖₆‖𝜋‖₃∇𝜉₂+𝐶‖𝑇‖₆𝜉₃‖∇𝜋‖₂+𝐶𝑇_𝑧6𝜉₃‖∇𝜋‖₂. (98) It follows from the Young inequality that 𝑑𝜉𝑑𝑡²²+‖𝜋‖²₂≤𝐶1+‖𝑇‖⁴₆+‖𝑣‖⁴₆+𝑇_𝑧⁴₆+𝑣_𝑧⁴₆𝜉²₂+‖𝜋‖²₂. (99) Multiplying (15) by 𝜉 and integrating over Ω, we obtain 𝜉²₂≤‖𝑄‖₂𝜉₂+𝐶‖𝑣‖₆‖∇𝑇‖₃𝜉₂+𝐿₂𝑇₂𝜉₂+𝐶𝑇_𝑧6‖∇𝑣‖₃𝜉₂. (100) Similarly, we have ‖𝜋‖²₂≤𝐶‖𝑣‖₆‖∇𝑣‖₃‖𝜋‖₂+𝐶𝑣_𝑧6‖∇𝑣‖₃‖𝜋‖₂+𝐶‖𝑣‖₂‖𝜋‖₂+𝐶‖∇𝑇‖₂‖𝜋‖₂+𝐿₁𝑣₂‖𝜋‖₂. (101) We derive from the Young inequality that ‖𝜋‖²₂+𝜉²₂≤𝐶1+‖𝑣‖⁴₆+𝑣_𝑧⁴₆+𝑇_𝑧⁴₆‖∇𝑇‖²₂+‖∇𝑣‖²₂+𝐶‖𝑄‖²₂+𝐶𝐿₂𝑇²₂+𝐶𝐿₁𝑣²₂+𝐶‖𝑣‖²₂. (102) By use of the uniform Gronwall inequality, we have 𝑣_𝑡²₂+𝑇_𝑡²₂≤𝜌₂₀, (103) for any 𝑡≥𝑇₂+10.

Moreover, we have _𝑡^𝑡+1𝑣_𝑡²𝑑𝜏+_𝑡^𝑡+1𝑇_𝑡²𝑑𝜏≤𝜌₂₁, (104) for any 𝑡≥𝑇₂+10.

3.13. 𝐻 Estimates of (𝐿₁𝑣,𝐿₂𝑇)

Multiplying (15) by 𝐿₂𝑇 and integrating over Ω, we obtain 𝐿₂𝑇²₂≤𝐶‖𝑣‖₆‖∇𝑇‖₃𝐿₂𝑇₂+𝐶𝑇_𝑧6‖∇𝑣‖₃𝐿₂𝑇₂+‖𝑄‖₂𝐿₂𝑇₂+𝜕_𝑡𝑇₂𝐿₂𝑇₂. (105) Similarly, we have 𝐿₁𝑣²₂≤𝐶‖𝑣‖₆‖∇𝑣‖₃𝐿₁𝑣₂+𝐶𝑣_𝑧6‖∇𝑣‖₃𝐿₁𝑣₂+𝐶‖𝑣‖₂𝐿₁𝑣₂+𝐶‖∇𝑇‖₂𝐿₁𝑣₂+𝑣_𝑡2𝐿₁𝑣₂. (106) It follows from the Young inequality that 𝐿₁𝑣²₂+𝐿₂𝑇²₂≤𝐶1+‖𝑣‖⁴₆+𝑇_𝑧⁴₆+𝑣_𝑧⁴₆‖∇𝑇‖²₂+‖∇𝑣‖²₂+𝐶𝑣_𝑡²₂+𝑇_𝑡²₂+𝐶‖𝑣‖²₂+‖𝑄‖²₂. (107) By virtue of (48), (69), (74), (85), (89), (93), and (103), we have 𝐿₁𝑣²₂+𝐿₂𝑇²₂≤𝜌₂₂, (108) for any 𝑡≥𝑇₂+10.

3.14. 𝐿² Estimates of (𝛁𝑣_𝑡,𝛁𝑇_𝑡)

Taking the 𝐻₁ inner product of (94) with −Δ𝜋 and combining the boundary conditions (96), we have 12𝑑‖‖𝑑𝑡∇𝜋²₂+1Re_1Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝜋|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}∇𝜋_𝑧²≤1𝑑𝑥𝑑𝑦𝑑𝑧Re₂𝐻_𝑠𝜋_𝑧2‖Δ𝜋‖₂+𝐶‖𝑣‖₆‖∇𝜋‖₃‖Δ𝜋‖₂+𝐶‖∇𝑣‖₂^1/2‖Δ𝑣‖₂^1/2𝜋_𝑧2^1/2∇𝜋_𝑧2^1/2‖Δ𝜋‖₂+𝐶‖𝜋‖₂‖Δ𝜋‖₂+𝐶‖𝜋‖₆‖∇𝑣‖₃‖Δ𝜋‖₂+𝐶‖∇𝜋‖₂^1/2𝑣_𝑧2^1/2∇𝑣_𝑧2^1/2‖Δ𝜋‖₂^3/2+𝐶∇𝜉₂‖Δ𝜋‖₂. (109) Therefore, in view of the Young inequality, we obtain 𝑑‖‖𝑑𝑡∇𝜋²₂+1Re_1Ω𝑒^{−𝑧/𝐻𝑠}|Δ𝜋|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝑒^{−𝑧/𝐻𝑠}∇𝜋_𝑧²‖𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶⁴₆+𝑣_𝑧²₂∇𝑣_𝑧²₂‖∇𝜋‖²₂+𝐶‖𝜋‖²₂+𝐶∇𝜉²₂+𝐶𝜋_𝑧²₂+𝐶‖∇𝑣‖²₂‖Δ𝑣‖²₂𝜋_𝑧²₂+𝐶‖𝜋‖²‖∇𝑣‖²₃. (110) Employing the uniform Gronwall inequality, we get ∇𝑣_𝑡²₂≤𝜌₂₂, (111) for any 𝑡≥𝑇₂+11.

Moreover, we have 1Re_1𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}Δ𝑣_𝑡²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏Re_2𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}∇𝜕_𝑧𝑣_𝑡²𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏≤𝜌₂₃. (112) Similarly, we can also obtain ∇𝑇_𝑡²₂≤𝜌₂₄,1Re_1𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}Δ𝑇_𝑡²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏Re_2𝑡^𝑡+1_Ω𝑒^{−𝑧/𝐻𝑠}∇𝜕_𝑧𝑇_𝑡²𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏≤𝜌₂₅, (113) for any 𝑡≥𝑇₂+12.

3.15. 𝐿² Estimates of (𝝏_𝑧𝑣_𝑡,𝝏_𝑧𝑇_𝑡)

Denote that 𝛽=𝜋_𝑧,  𝛾=𝜉_𝑧. It is clear that 𝛽,  𝛾 satisfy the following equations obtained by differentiating (94) and (95) with respect to 𝑧, respectively, 𝜕𝛽𝜕𝑡+𝐿₁𝛽+𝑣_𝑧()𝛽−⋅∇𝜋+𝑣⋅∇_𝑧−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝛽−1𝜕𝑧𝐻_𝑠^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}𝛽−()∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁∇⋅𝑣𝛽+𝑓₀^→𝑘×𝛽+𝛽⋅∇𝑣+(𝜋⋅∇)𝑣_𝑧−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝜋𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣_𝑧−1𝜕𝑧𝐻_𝑠^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝜋𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣−()𝑣𝜕𝑧∇⋅𝜋_𝑧+𝑅𝐻_𝑠∇𝜉=0,𝜕𝛾𝜕𝑡+𝐿₂𝛾+𝑣_𝑧−⋅∇𝜉+𝑣⋅∇𝛾_𝑧−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝛾−()𝑇𝜕𝑧∇⋅𝜋_𝑧−1𝐻_𝑠^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}𝛾−()∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁∇⋅𝑣𝛾+𝛽⋅∇𝑇+𝜋⋅∇𝑇_𝑧−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝜋𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇_𝑧−1𝜕𝑧𝐻_𝑠^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝜋𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧=0, (114)(115) with the boundary conditions 𝛽_Γ𝑢=0,𝛽_Γ𝑏=0,𝛽⋅^→𝑛_Γ𝑙=0,𝜕𝛽𝜕^→𝑛×^→𝑛_Γ𝑙1=0,𝑅𝑡₂𝛾+𝛼𝜉_Γ𝑢=0,𝛾_Γ𝑏=0,𝜕𝛾𝜕^→𝑛_Γ𝑙=0. (116)(117) Taking the 𝐿² inner product of (114) with 𝛽 and combining the boundary conditions (116), we have 12𝑑𝛽𝑑𝑡²²+𝛽²≤1Re₂𝐻_𝑠𝛽_𝑧2𝛽₂+𝐶𝑣_𝑧6‖∇𝜋‖₃𝛽₂+𝐶‖𝑣‖₆𝛽₃∇𝛽₂+𝐶‖𝜋‖₃∇𝑣_𝑧2𝛽₆+𝐶‖∇𝜋‖₃𝜕_𝑧𝑣_𝑧2𝛽₆+𝐶𝜉₂∇𝛽₂. (118) Therefore, in view of the Young inequality, we obtain 𝑑𝛽𝑑𝑡²²+𝛽²≤𝐶1+‖𝑣‖⁴₆𝛽²₂+𝐶𝑣_𝑧²₆‖∇𝜋‖²₃+𝐶‖𝜋‖²₃∇𝑣_𝑧²₂+𝐶‖∇𝜋‖²₃𝜕_𝑧𝑣_𝑧²₂+𝐶𝜉²₂. (119) Employing the uniform Gronwall inequality, we get 𝜕_𝑧𝑣_𝑡²₂≤𝜌₂₆, (120) for any 𝑡≥𝑇₂+13.

Similarly, we can also obtain _Ω𝑒^{−𝑧/𝐻𝑠}𝜕_𝑧𝑇_𝑡²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼𝑅𝑡_2Γ𝑢𝑇_𝑡²𝑑𝑥𝑑𝑦≤𝜌₂₇, (121) for any 𝑡≥𝑇₂+14.

From (43), (48), (70), (74), (81), (85) and (108), we get the following results.

Theorem 7. Assume that 𝑄∈𝐻¹(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=𝜏_1,𝐵 which depend on the norm of 𝐵, such that ‖(𝑣(𝑡),𝑇(𝑡))‖²_𝐻2=𝑆(𝑡)𝑣₀,𝑇₀²_𝐻2=‖𝑣(𝑡)‖²_𝐻2+‖𝑇(𝑡)‖²_𝐻2≤ℛ₁, (122) 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=𝜏_2,𝐵, depending on the norm of 𝐵, such that ‖(𝑣(𝑡),𝑇(𝑡))‖²_𝑉=𝑆(𝑡)𝑣₀,𝑇₀²_𝑉=‖𝑣(𝑡)‖²_𝑉1+‖𝑇(𝑡)‖²_𝑉2≤ℛ₂, (123) 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 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-to-weak continuous semigroup on 𝑋, if {𝑆(𝑡)}_𝑡≥0 satisfies that (i)𝑆(0)=Id (the identity), (ii)𝑆(𝑡)𝑆(𝑠)=𝑆(𝑡+𝑠), for all 𝑡,𝑠≥0,(iii)𝑆(𝑡_𝑛)𝑥_𝑛⇀𝑆(𝑡)𝑥 if 𝑡_𝑛→𝑡 and 𝑥_𝑛→𝑥 in 𝑋.

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 norm-to-weak continuous semigroup on 𝑋 if and only if {𝑆(𝑡)}_𝑡≥0 maps compact subsets of ℝ⁺×𝑋 into bounded sets of 𝑋.

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≥𝑡0𝑆(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≥0𝑡≥𝑠}𝑆(𝑡)𝐵. (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 𝒜_𝑉.

To prove the existence of the (𝑉,𝐻³)-absorbing set, we need the following lemma.

Lemma 16 (see [7]). Let 𝑣∈𝑉₁,Φ_𝑠∈𝐿²(𝑀) be a solution to the following Stokes problem: 𝐿₁𝑣+∇Φ_𝑠=𝑔,_0−ℎ∇⋅𝑣𝑥,𝑦,𝑧𝑑𝑧=0,𝜕𝑣𝜕𝑧_𝑧=−ℎ=0,𝜕𝑣𝜕𝑧_𝑧=0=0. (125) If 𝑔∈(𝑊^𝑚,𝛼(Ω))²,  1<𝛼<∞,𝑚≥0, then 𝑣∈𝑉₁∩𝑊^𝑚+2,𝛼(Ω)²,Φ_𝑠∈𝑊^𝑚+1,𝛼(𝑀). (126)

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).

Theorem 17. 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 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 ‖(𝑣(𝑡),𝑇(𝑡))‖²_𝐻3=𝑆(𝑡)𝑣₀,𝑇₀²_𝐻3=‖𝑣(𝑡)‖²_𝐻3+‖𝑇(𝑡)‖²_𝐻3≤ℛ₃, (127) for any 𝑡≥𝜏₁, where 𝜏₁ is specified in Theorem 7.

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,  𝑣_𝑧∈𝐿^∞(𝜏₁,𝒯;𝐿⁶),  𝑇_𝑧∈𝐿^∞(𝜏₁,𝒯;𝐿⁶),  𝑣_𝑡∈𝐿^∞(𝜏₁,𝒯;𝑉₁),𝑇_𝑡∈𝐿^∞(𝜏₁,𝒯;𝑉₂),(𝑣,𝑇)∈𝐿^∞(𝜏₁,𝒯;𝐻²), where 𝜏₁ is specified in Theorem 7. Then, 𝐿₁𝑣+∇Φ_𝑠=𝑔, (128) where −𝑔=𝜕𝑣+()𝜕𝑡𝑣⋅∇𝑣−^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣+𝑅𝜕𝑧𝐻_𝑠^𝑧₀∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁+𝑓₀^→𝑘×𝑣. (129) Since 𝑣∈𝐿^∞(𝜏₁,𝒯;𝐻²) and 𝑣_𝑧∈𝐿^∞(𝜏₁,𝒯;𝐿⁶), for any 𝑢∈𝐿^3/2, we have _Ω𝑒^{−𝑧/𝐻𝑠𝑧}_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣‖‖𝜕𝑧⋅𝑢𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶∇𝑣₆𝑣_𝑧6‖𝑢‖_3/2, (130) which implies that ^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣𝜕𝑧∈𝐿^∞𝜏₁,𝒯;𝐿³. (131) Since (𝑣,𝑇) is a strong solution to (14)–(18) and 𝑣_𝑡∈𝐿^∞(𝜏₁,𝒯;𝑉₁), it is clear that 𝜕𝑣/𝜕𝑡+(𝑣⋅∇)𝑣−(𝑅/𝐻_𝑠)∫^𝑧₀∇𝑇(𝑥,𝑦,𝜁,𝑡)𝑑𝜁+𝑓₀^→𝑘×𝑣∈𝐿^∞(𝜏₁,𝒯;𝐿³). Thus, 𝑔∈𝐿^∞𝜏₁,𝒯;𝐿³. (132) By the use of Lemma 16, we get 𝑣∈𝐿^∞𝜏₁,𝒯;𝑉₁∩𝑊^2,3(Ω)². (133) Using the Sobolev embedding theorem (𝑊^2,3(Ω))²⊂(𝑊^1,12(Ω))², we know that 𝑣∈𝐿^∞𝜏₁,𝒯;𝑉₁∩𝑊^1,12(Ω)². (134) Now, since 𝑣∈𝐿^∞(𝜏₁,𝒯;𝑉₁∩(𝑊^1,12(Ω))²) and 𝑣_𝑧∈𝐿^∞(𝜏₁,𝒯;𝐿⁶), for any 𝑢∈𝐿^4/3, we have _Ω𝑒^{−𝑧/𝐻𝑠𝑧}_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣‖‖𝜕𝑧⋅𝑢𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶∇𝑣₁₂𝑣_𝑧6‖𝑢‖_4/3, (135) which implies that ^𝑧_−ℎ𝑒^{(𝑧−𝜁)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣𝜕𝑧∈𝐿^∞𝜏₁,𝒯;𝐿⁴. (136) Therefore, we find that 𝑔∈𝐿^∞𝜏₁,𝒯;𝐿⁴. (137) From Lemma 16, we know that 𝑣∈𝐿^∞𝜏₁,𝒯;𝑉₁∩𝑊^2,4(Ω)², (138) which implies that ^𝑧_−ℎ𝑒^{(𝑧−𝜉)/𝐻𝑠}∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣𝜕𝑧∈𝐿^∞𝜏₁,𝒯;𝐻¹. (139) Since 𝑣_𝑡∈𝐿^∞(𝜏₁,𝒯;𝑉₁) and 𝑣∈𝐿^∞(𝜏₁,𝒯;𝑉₁∩(𝑊^2,4(Ω))²), it is clear that 𝜕𝑣/𝜕𝑡+(𝑣⋅∇)𝑣−(𝑅/𝐻_𝑠)∫^𝑧₀∇𝑇(𝑥,𝑦,𝜁,𝑡)𝑑𝜁+𝑓₀^→𝑘×𝑣∈𝐿^∞(𝜏₁,𝒯;𝐻¹). Thus, 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. (142) This complete the proof of Theorem 17.

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,𝐻^2+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 𝑣_𝑛∈𝑉₁ such that 𝑣_𝑛²₂=1,𝑣_𝑛²≤1𝑛. (A.1) Let 𝑣_𝑛²_𝐻1=𝑣_𝑛²₂+𝑣_𝑛². (A.2) Hence, 𝑣_𝑛²_𝐻11≤1+𝑛≤2, (A.3) which implies that there exists a subsequence of {𝑣_𝑛}_𝑛≥1 (still denoted by {𝑣_𝑛}_𝑛≥1) such that 𝑣_𝑛⟶𝑣,weaklyin𝐻¹,𝑣_𝑛⟶𝑣,stronglyin𝐿². (A.4) Therefore, we have ‖𝑣‖²₂=1,‖𝑣‖²_𝐻1≤liminf_𝑛→∞𝑣_𝑛²_𝐻1=1, (A.5) which implies that ‖𝑣‖²=0. (A.6) Combining ‖𝑣‖²=0 with 𝑣⋅^→𝑛=0, we obtain 𝑣=0, (A.7) this is a contradiction.