Pullback Attractor for Nonautonomous Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics

Abstract

We consider the existence of (𝑉,𝑉)-pullback attractor for nonautonomous primitive equations of large-scale ocean and atmosphere dynamics in a three-dimensional bounded cylindrical domain by verifying pullback 𝒟 condition.

1. Introduction

This paper is concerned with the existence of pullback attractor for the following nonautonomous primitive equations of large-scale ocean and atmosphere dynamics: 𝜕𝑣+()𝜕𝑡𝑣⋅∇𝑣−^𝑧_−ℎ∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑣𝜕𝑧+∇𝑝_𝑠−𝑥,𝑦,𝑡_𝑧01∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁+Ro𝑓𝑣^⟂+𝐿₁𝑣=0,𝜕𝑇𝜕𝑡+𝑣⋅∇𝑇−^𝑧_−ℎ∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧+𝐿₂𝑇=𝑄, (1)(2) with the following boundary conditions: 𝜕𝑣𝜕𝑧_𝑧=0=0,𝜕𝑣𝜕𝑧_𝑧=−ℎ=0,𝑣⋅𝑛|_Γ𝑙=0,𝜕𝑣𝜕𝑛×𝑛_Γ𝑙1=0,𝑅𝑡₂𝜕𝑇𝜕𝑧+𝛼𝑇_𝑧=0=0,𝜕𝑇𝜕𝑧_𝑧=−ℎ=0,𝜕𝑇𝜕𝑛_Γ𝑙=0, (3)(4) and the initial data 𝑣𝑥,𝑦,𝑧,𝜏=𝑣₀,𝑇𝑥,𝑦,𝑧𝑥,𝑦,𝑧,𝜏=𝑇₀𝑥,𝑦,𝑧, (5) in the three-dimensional bounded cylindrical domain Ω=𝑀×(−ℎ,0), (6) where 𝑀 is a bounded domain in ℝ² with smooth boundary 𝜕𝑀 and ℎ is a positive constant, the unknown functions 𝑣(𝑥,𝑦,𝑧,𝑡)=(𝑣₁(𝑥,𝑦,𝑧,𝑡),𝑣₂(𝑥,𝑦,𝑧,𝑡)),𝑇(𝑥,𝑦,𝑧,𝑡),𝑝_𝑠(𝑥,𝑦,𝑡) denote the horizontal velocity, the temperature, and the pressure at 𝑧=0, respectively, 𝑣^⟂=(−𝑣₂,𝑣₁),𝑓=𝑓₀+𝛽𝑦 is the Coriolis parameter, Ro is the Rossby number which measures the significant influence of the Earth’s rotation to the dynamical behaviour of the ocean, 𝑄(𝑥,𝑦,𝑧,𝑡) is a given heat source, Γ_𝑙={(𝑥,𝑦,𝑧)∈Ω∶(𝑥,𝑦)∈𝜕𝑀,−ℎ≤𝑧≤0} denotes the lateral boundary of Ω,𝑛 is the normal vector to Γ_𝑙, and 𝛼 is a positive constant related with the turbulent heating on the surface of the ocean. The viscosity and the heat diffusion operators 𝐿₁ and 𝐿₂ are given by 𝐿₁1=−Re₁1Δ−Re₂𝜕²𝜕𝑧²,𝐿₂1=−Rt₁1Δ−Rt₂𝜕²𝜕𝑧², (7) where Re₁,Re₂ are positive constants representing the horizontal and vertical Reynolds numbers, respectively, and Rt₁,Rt₂ are positive constants which stand for the horizontal and vertical eddy diffusivity, respectively. For the sake of simplicity, let ∇=(𝜕_𝑥,𝜕_𝑦) be the horizontal gradient operator, and let Δ=𝜕²_𝑥+𝜕²_𝑦 be the horizontal Laplacian.

Nonautonomous equations appear in many applications in the natural sciences, and so they are of great importance and interest. The long-time behavior of solutions of such equations has been studied extensively in recent years (see [1–12], etc.). The first attempt was to extend the notion of global attractor to the nonautonomous case, leading to the concept of the so-called uniform attractor (see [13]). It is remarkable that the conditions ensuring the existence of the uniform attractor is parallel those for the autonomous case. However, one disadvantage of the uniform attractor is that it does not need to be “invariant,” unlike the global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many nonautonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems. In order to overcome this drawback, the concept of pullback attractor has been introduced for the nonautonomous case. The theory of pullback attractors has been developed for both nonautonomous and random dynamical systems, and it has been shown to be very useful in the understanding of the dynamics of nonautonomous dynamical systems (see [6]).

Large-scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, due to the shallowness of the oceans and atmosphere, that is, the depth of the fluid layer is very small in comparison to the radius of the Earth, the vertical large-scale motion in the oceans and atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result, one can obtain the system (1)–(4), which is known as the primitive equations for large-scale oceans and atmosphere dynamics (see [14–18]). In the case of ocean dynamics, we know that one has to add the diffusion-transport equation of the salinity to the system (1)–(4), but we will omit it here in order to simplify our mathematical presentation. However, we emphasize that our results are equally valid when the salinity effects are taken into account.

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 (see [14–16, 19–26], etc.). 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 [15]. In [16], the authors proved the existence and the uniqueness of local in time strong solutions of the primitive equations of the ocean. The global existence of strong solutions of the three-dimensional primitive equations of large-scale ocean, by assuming that the initial data are small enough, and the local existence of strong solutions of the three-dimensional primitive equations of large-scale ocean for all initial data were proved by Guillén-González et al. in [27]. In [19], the authors proved the maximum principles for the primitive equations of the atmosphere. The existence and uniqueness of strong solutions, global in time, to the primitive equations in thin domains for a large set of initial data whose sizes depend inversely on the thickness were established in [23]. In [28], Temam and Ziane considered the local existence of strong 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 [24]. In [22], the authors proved the existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics. The existence and uniqueness of global strong solutions for the initial boundary value problem of the three-dimensional viscous primitive equations of large-scale ocean were established by the authors in [14]. In [20], the authors 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 𝑉-weak topology. Under the assumption of the initial data (𝑣₀,𝑇₀)∈𝑉∩(𝐻²(Ω))³, the existence of compact global attractor in 𝑉 for the primitive equations of large-scale atmosphere was established by the elliptic regularity theory in [21]. In [26], the authors proved the global existence and uniqueness of 𝑧-weak solutions when the initial conditions satisfy some regularity. The existence of global attractors in 𝑉 and (𝐻²(Ω))³ for three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinates was established in [29]. In [30], the authors proved the existence of (𝑉,𝑉)-global attractor for the primitive equations of large-scale atmosphere and ocean dynamics by use of the Aubin-Lions compactness theorem. Attractor is an important concept in the study of infinite-dimensional dynamical systems; however, the existence of pullback attractor for the three-dimensional nonautonomous viscous primitive equations of large-scale atmosphere and ocean dynamics remains unsolvable. In this paper, we prove the existence of pullback attractor for nonautonomous equations (1)–(5) by verifying pullback 𝒟 condition.

This paper is organized as follows. In Section 2, we introduce the mathematical framework of the system (1)–(5) and recall some auxiliary lemmas used to give some a priori estimates of strong solutions for (1)–(5) in Section 3, which are used to obtain the existence of (𝑉,𝑉)-pullback absorbing set. Finally, in Section 4, we are devoted to proving the existence of (𝑉,𝑉)-pullback attractor for the three dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics by verifying pullback 𝒟 condition.

Throughout this paper, let 𝑋 be a Banach space endowed with the norm ‖⋅‖_𝑋, let ‖𝑢‖_𝑝 be the 𝐿^𝑝(Ω)-norm of 𝑢, and let 𝐶 be positive constants, which may be different from line to line.

2. Preliminaries

2.1. Functional Spaces and Some Lemmas

To study problem (1)–(5), we introduce some function spaces. Let 𝒱₁=𝑣∈𝐶^∞(Ω)²∶𝜕𝑣𝜕𝑧_𝑧=0=0,𝜕𝑣𝜕𝑧_𝑧=−ℎ=0,𝑣⋅𝑛|_Γ𝑙=0,𝜕𝑣𝜕𝑛×𝑛_Γ𝑙=0,_0−ℎ,𝒱∇⋅𝑣𝑥,𝑦,𝜁𝑑𝜁=0₂=𝑇∈𝐶^∞(Ω)∶1Rt₂𝜕𝑇𝜕𝑧+𝛼𝑇_𝑧=0=0,𝜕𝑇𝜕𝑧_𝑧=−ℎ=0,𝜕𝑇𝜕𝑛_Γ𝑙=0. (8) Denote the closure of 𝒱₁,𝒱₂ by 𝑉₁,𝑉₂ with respect to the following norms, respectively, defined as follows: ‖𝑣‖²=1Re_1Ω|∇𝑣|²1𝑑𝑥𝑑𝑦𝑑𝑧+Re_2Ω𝜕_𝑧𝑣²‖𝑇‖𝑑𝑥𝑑𝑦𝑑𝑧,₂=1Rt_1Ω|∇𝑇|²1𝑑𝑥𝑑𝑦𝑑𝑧+Rt_2Ω𝜕_𝑧𝑇²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_𝑀|𝑇(𝑧=0)|²𝑑𝑥𝑑𝑦, (9) for any 𝑣∈𝒱₁,𝑇∈𝒱₂, and let 𝐻₁= the closure of 𝒱₁ with respect to the norm (𝐿²(Ω))²,𝑉=𝑉₁×𝑉₂,𝐻=𝐻₁×𝐿²(Ω).

Now, we recall some lemmas used in the sequel.

Lemma 1 (see [13, 31]). Let 𝑦(𝑡),𝑎(𝑡),ℎ(𝑡) be three positive locally integrable functions on [𝑡₀,∞), and for some 𝑟>0 and all 𝑡≥𝑡₀, the following inequalities hold: 𝑦^′≤𝑎(𝑡)𝑦+ℎ_{𝑡𝑡+𝑟}(𝑡),𝑦(𝜏)𝑑𝜏≤𝑋,_𝑡^𝑡+𝑟𝑎(𝜏)𝑑𝜏≤𝐴,_{𝑡𝑡+𝑟}ℎ(𝜏)𝑑𝜏≤𝐵. (10) Then, 𝑦(𝑡)≤𝑋𝑟+𝐵𝑒^𝐴, (11) for all 𝑡≥𝑡₀+𝑟.

Lemma 2 (see [14]) (Minkowski inequality). Let (𝑋,𝜇),(𝑌,𝜈) be two measure spaces and 𝑓(𝑥,𝑦) be a measurable function about 𝜇×𝜈 on 𝑋×𝑌. If for 𝑎.𝑒. 𝑦∈𝑌,𝑓(⋅,𝑦)∈𝐿^𝑝(𝑋,𝜇)(1≤𝑝≤∞), and ∫_𝑌‖𝑓(⋅,𝑦)‖_{𝐿𝑝(𝑋,𝜇)}𝑑𝜈(𝑦)<∞, then _𝑌𝑓⋅,𝑦𝑑𝜈𝑦_{𝐿𝑝(𝑋,𝜇)}≤_𝑌𝑓(⋅,𝑦)_{𝐿𝑝(𝑋,𝜇)}𝑑𝜈𝑦. (12)

2.2. New Formulation

In this subsection, we divide (1) into two systems with respect to 𝑣 and 𝑣̃ defined by 𝑣=1𝑥,𝑦ℎ⁰_−ℎ𝑣̃𝑥,𝑦,𝜁𝑑𝜁,𝑣=𝑣−𝑣. (13)

Taking the average of (1) in the 𝑧 direction over the interval (−ℎ,0) and using the boundary conditions (3), we obtain 𝜕𝑣+(𝜕𝑡𝑣⋅∇)𝑣+(𝑣̃⋅∇)𝑣̃+(∇⋅𝑣)̃𝑣̃+∇𝑝_𝑠−1𝑥,𝑦,𝑡Re₁Δ𝑣+1𝑓Ro𝑣^⟂−^𝑧₀∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁=0, (14) with the boundary conditions ∇⋅𝑣=0,𝑣⋅𝑛|_Γ𝑙𝜕=0,𝑣𝜕𝑛×𝑛_Γ𝑙=0. (15) Subtracting (14) from (1), we get 𝜕𝑣̃+(̃)̃𝜕𝑡𝑣⋅∇𝑣−^𝑧_−ℎ𝑣̃𝜕𝑣̃∇⋅𝑥,𝑦,𝜁,𝑡𝑑𝜁+(̃)𝜕𝑧𝑣⋅∇𝑣−(̃̃̃𝑣̃+(𝑣⋅∇)𝑣+(∇⋅𝑣)𝑣⋅∇)𝑣̃−^𝑧₀+∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁_𝑧01∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁+𝑓𝑣̃Ro^⟂+𝐿₁𝑣̃=0, (16) with the boundary conditions 𝜕𝑣̃𝜕𝑧_Γ𝑢𝜕𝑣̃=0,𝜕𝑧_Γ𝑏̃=0,𝑣⋅𝑛|_Γ𝑙𝜕𝑣̃=0,𝜕𝑛×𝑛_Γ𝑙=0. (17)

3. Some a Priori Estimates of Strong Solutions

3.1. The Well Posedness of Strong Solutions

We start with the following general existence and uniqueness of solutions which can be obtained by the standard Fatou-Galerkin methods [31] and the similar methods established in [14]. Here, we only state the result as follows.

Theorem 3. Suppose that 𝑄∈𝐿²_loc(ℝ;𝐿²(Ω)). Then for any 𝜏∈ℝ, any initial data (𝑣₀,𝑇₀)∈𝑉, and any 𝒯>0, there exists a unique solution (𝑣,𝑇)∈𝐶([0,𝒯];𝑉) of (1)–(5). Furthermore, the solution is continuous with respect to the initial data in 𝑉.

By Theorem 3, we can define a family of continuous processes {𝑈(𝑡,𝜏)∶−∞<𝜏≤𝑡<∞} in 𝑉 by 𝑈(𝑡,𝜏)𝑣₀,𝑇₀=(𝑣(𝑡),𝑇(𝑡))∶=𝑣𝑡;𝜏,𝑣₀,𝑇𝑡;𝜏,𝑇₀,∀𝑡≥𝜏, (18) where (𝑣(𝑡),𝑇(𝑡)) is the solution of (1)–(5) with initial data (𝑣(𝜏),𝑇(𝜏))=(𝑣₀,𝑇₀)∈𝑉. That is, a family of mappings 𝑈(𝑡,𝜏)∶𝑉→𝑉 satisfies 𝑈(),𝑈()()𝑈()𝜏,𝜏=ididentity𝑡,𝜏=𝑈𝑡,𝑟𝑟,𝜏∀𝜏≤𝑟≤𝑡. (19)

3.2. Some a Priori Estimates about Strong Solutions

In this subsection, assume that 0<𝜎≤min{1/(4Rt₂ℎ²+(4ℎ/𝛼)),1/2𝐶_𝑀Re₁} and sup_𝑟∈ℝ∫^𝑟_𝑟−1‖𝑄(𝑠)‖²₂𝑑𝑠<∞; we give some a priori estimates of strong solutions which imply the existence of pullback absorbing set.

3.2.1. 𝐿²(Ω) Estimate of 𝑇

Taking the inner product of (2) with 𝑇 in 𝐿²(Ω) and combining the boundary conditions (4), we get 12𝑑‖𝑇‖𝑑𝑡²²+‖𝑇‖²=−_Ω𝑣⋅∇𝑇−^𝑧_−ℎ×∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧𝑇𝑑𝑥𝑑𝑦𝑑𝑧+_Ω=𝑄𝑇𝑑𝑥𝑑𝑦𝑑𝑧_Ω𝑄𝑇𝑑𝑥𝑑𝑦𝑑𝑧. (20) Integrating by parts and using the boundary conditions (3), we obtain _Ω𝑣⋅∇𝑇−^𝑧_−ℎ∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝜕𝑧𝑇𝑑𝑥𝑑𝑦𝑑𝑧=0. (21) Since ‖𝑇‖²₂≤2ℎ‖𝑇(𝑧=0)‖²_𝐿2(𝑀)+2ℎ²𝜕_𝑧𝑇²₂, (22)it is implied that ‖𝑇‖²₂2Rt₂ℎ²+()≤‖‖2ℎ/𝛼2ℎ𝑇(𝑧=0)²_𝐿2(𝑀)+2ℎ²𝜕_𝑧𝑇²₂2Rt₂ℎ²+()≤12ℎ/𝛼Rt_2Ω𝜕_𝑧𝑇²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_𝑀|𝑇(𝑧=0)|²≤‖𝑇‖𝑑𝑥𝑑𝑦₂. (23) It follows from (20) and (23) that𝑑‖𝑇‖𝑑𝑡²²+‖𝑇‖²≤2Rt₂ℎ²+2ℎ𝛼‖𝑄‖²₂. (24) Using (23) again, we get 𝑑‖𝑇‖𝑑𝑡²²+‖𝑇‖²₂2Rt₂ℎ²+()≤2ℎ/𝛼2Rt₂ℎ²+2ℎ𝛼‖𝑄‖²₂. (25) Therefore, we deduce from (24) and (25) that 𝑑𝑒𝑑𝑡^𝜎𝑡‖𝑇‖²₂+12𝑒^𝜎𝑡‖𝑇‖²=𝑒^𝜎𝑡𝑑‖𝑇‖𝑑𝑡²²+𝜎𝑒^𝜎𝑡‖𝑇‖²₂+12𝑒^𝜎𝑡‖𝑇‖²≤1𝜎−4Rt₂ℎ²+()𝑒4ℎ/𝛼^𝜎𝑡‖𝑇‖²₂+2Rt₂ℎ²+2ℎ𝛼𝑒^𝜎𝑡‖𝑄‖²₂≤2Rt₂ℎ²+2ℎ𝛼𝑒^𝜎𝑡‖𝑄‖²₂. (26) Integrating (26) from 𝜏 to 𝑡, we obtain 𝑒^𝜎𝑡‖𝑇(𝑡)‖²₂+12^𝑡_𝜏𝑒^𝜎𝑠‖𝑇(𝑠)‖²𝑑𝑠≤𝑒^𝜎𝜏𝑇₀²₂+2Rt₂ℎ²+2ℎ𝛼^𝑡_𝜏𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂𝑑𝑠≤𝑒^𝜎𝜏𝑇₀²₂+2Rt₂ℎ²+2ℎ𝛼^𝑡_−∞𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂𝑑𝑠≤𝑒^𝜎𝜏𝑇₀²₂+𝑒^𝜎𝑡2Rt₂ℎ²+2ℎ𝛼×1+𝜎⁻¹sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠, (27) where we use the inequality ^𝑡_−∞𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂=𝑑𝑠_𝑡𝑡−1𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂𝑑𝑠+^𝑡−1_𝑡−2𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂+𝑑𝑠_{𝑡−2𝑡−3}𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂𝑑𝑠+^𝑡−3_𝑡−4𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂+𝑑𝑠+⋯_{𝑡−𝑛+1𝑡−𝑛}𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂≤𝑒𝑑𝑠+⋯_𝜎𝑡+𝑒^{𝜎(𝑡−1)}+𝑒^{𝜎(𝑡−2)}+𝑒^{𝜎(𝑡−3)}+⋯×sup_{𝑟∈ℝ𝑟}^𝑟+1‖𝑄(𝑠)‖²₂≤𝑒𝑑𝑠_𝜎𝑡1−𝑒^−𝜎sup_{𝑟∈ℝ𝑟}^𝑟+1‖𝑄(𝑠)‖²₂≤𝑑𝑠1+𝜎⁻¹𝑒^𝜎𝑡sup_{𝑟∈ℝ𝑟}^𝑟+1‖𝑄(𝑠)‖²₂𝑑𝑠. (28) That is,‖𝑇(𝑡)‖²₂≤𝑒^{𝜎(𝜏−𝑡)}𝑇₀²₂+2Rt₂ℎ²+2ℎ𝛼×1+𝜎⁻¹sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠. (29) Let 𝜏<𝑡−1, for any 𝑟∈[𝜏,𝑡−1]; integrating (24) from 𝑟 to 𝑟+1 and combining with (29), we obtain _𝑟^𝑟+1‖𝑇(𝑠)‖²≤𝑑𝑠2Rt₂ℎ²+2ℎ𝛼_𝑟^𝑟+1‖𝑄(𝑠)‖²₂𝑑𝑠+‖𝑇(𝑟)‖²₂≤𝑒^{𝜎(𝜏−𝑟)}𝑇₀²₂+2Rt₂ℎ²+2ℎ𝛼×2+𝜎⁻¹sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠. (30)

3.2.2. 𝐻₁ Estimates of 𝑣

Multiplying (1) by 𝑣 and integrating by parts over Ω and using the boundary conditions (3), we deduce 12𝑑‖𝑣‖𝑑𝑡²²+‖𝑣‖²=_Ω^𝑧₀‖𝑇‖∇𝑇𝑥,𝑦,𝜁,𝑡𝑑𝜁⋅𝑣𝑑𝑥𝑑𝑦𝑑𝑧≤ℎ₂‖∇𝑣‖₂≤12Re_1Ω|∇𝑣|²𝑑𝑥𝑑𝑦𝑑𝑧+Re₁ℎ²‖𝑇‖²₂2. (31) That is, 𝑑‖𝑣‖𝑑𝑡²²+‖𝑣‖²≤Re₁ℎ²‖𝑇‖²₂. (32) It follows from ‖𝑣‖²₂≤𝐶_𝑀‖∇𝑣‖²₂ that 𝑑‖𝑣‖𝑑𝑡²²+1𝐶_𝑀Re₁‖𝑣‖²₂≤Re₁ℎ²‖𝑇‖²₂. (33) Thanks to (32) and (33), we get 𝑑𝑒𝑑𝑡^𝜎𝑡‖𝑣‖²₂+12𝑒^𝜎𝑡‖𝑣‖²=𝑒^𝜎𝑡𝑑‖𝑣‖𝑑𝑡²²+𝜎𝑒^𝜎𝑡‖𝑣‖²₂+12𝑒^𝜎𝑡‖𝑣‖²≤1𝜎−2𝐶_𝑀Re₁𝑒^𝜎𝑡‖𝑣‖²₂+Re₁ℎ²𝑒^𝜎𝑡‖𝑇‖²₂. (34) Integrating (34) over [𝜏,𝑡] and using (23) and (27), we have 𝑒^𝜎𝑡‖𝑣(𝑡)‖²₂+12^𝑡_𝜏𝑒^𝜎𝑠‖𝑣‖²≤𝑒^𝜎𝜏𝑣₀²₂+Re₁ℎ^2𝑡_𝜏𝑒^𝜎𝑠‖𝑇(𝑠)‖²₂𝑑𝑠≤𝑒^𝜎𝜏𝑣₀²₂+Re₁ℎ²2Rt₂ℎ²+2ℎ𝛼×^𝑡_𝜏𝑒^𝜎𝑠‖𝑇(𝑠)‖²𝑑𝑠≤𝑒^𝜎𝜏𝑣₀²₂+2Re₁ℎ²2Rt₂ℎ²+2ℎ𝛼×𝑒^𝜎𝜏𝑇₀²₂+𝑒^𝜎𝑡𝜌₁, (35) where 𝜌₁=2Rt₂ℎ²+2ℎ𝛼1+𝜎⁻¹sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠. (36) That is, ‖𝑣(𝑡)‖²₂≤𝑒^{𝜎(𝜏−𝑡)}𝑣₀²₂+2Re₁ℎ²2Rt₂ℎ²+2ℎ𝛼×𝑒^{𝜎(𝜏−𝑡)}𝑇₀²₂+𝜌₁. (37) Let 𝜏<𝑡−1, for any 𝑟∈[𝜏,𝑡−1]; integrating (32) from 𝑟 to 𝑟+1 and combining with (23), (30), and (37), we obtain _𝑟^𝑟+1‖𝑣(𝑠)‖²≤‖‖𝑑𝑠𝑣(𝑟)²₂+Re₁ℎ²_𝑟^𝑟+1‖𝑇(𝑠)‖²₂≤‖𝑣(𝑟𝑑𝑠)‖²₂+Re₁ℎ²2Rt₂ℎ²+2ℎ𝛼×_𝑟^𝑟+1‖𝑇(𝑠)‖²𝑑𝑠≤𝑒^{𝜎(𝜏−𝑟)}𝑣₀²₂+4Re₁ℎ²2Rt₂ℎ²+2ℎ𝛼×𝑒^{𝜎(𝜏−𝑟)}𝑇₀²₂+𝜌₁. (38)

3.2.3. 𝐿⁶(Ω) Estimates of 𝑇

Taking the inner product of (2) with |𝑇|⁴𝑇 in 𝐿²(Ω) and using the boundary conditions (4), we have 16𝑑‖𝑇‖𝑑𝑡⁶⁶+59Rt_1Ω∇|𝑇|³²+5𝑑𝑥𝑑𝑦𝑑𝑧9Rt_2Ω𝜕_𝑧|𝑇|³²𝑑𝑥𝑑𝑦𝑑𝑧+𝛼_𝑀|𝑇(𝑧=0)|⁶≤𝑇|𝑑𝑥𝑑𝑦|^35/3_10/3‖𝑄‖₂≤𝐶‖𝑄‖₂|𝑇|³₂^2/3|𝑇|³_𝐻1(Ω)=𝐶‖𝑄‖₂‖𝑇‖²₆|𝑇|³_𝐻1(Ω). (39) Using the Young inequality, we obtain 𝑑𝑒𝑑𝑡^𝜎𝑡‖𝑇‖²₆≤𝜎𝑒^𝜎𝑡‖𝑇‖²₆+𝐶𝑒^𝜎𝑡‖𝑄‖²₂. (40) Integrating (40) from 𝜏 to 𝑡 and combining with (27) and ‖𝑇‖₆≤𝐶‖𝑇‖, we get 𝑒^𝜎𝑡‖𝑇‖²₆≤𝑒^𝜎𝜏𝑇₀²₆+𝜎^𝑡_𝜏𝑒^𝜎𝑠‖𝑇(𝑠)‖²₆𝑑𝑠+𝐶^𝑡_𝜏𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂𝑑𝑠≤𝑒^𝜎𝜏𝑇₀²+𝐶^𝑡_𝜏𝑒^𝜎𝑠‖𝑇(𝑠)‖²𝑑𝑠+𝐶^𝑡_𝜏𝑒^𝜎𝑠‖𝑄(𝑠)‖²₂𝑑𝑠≤𝐶𝑒^𝜎𝜏𝑇₀²₂+𝐶𝑒^𝜎𝑡2Rt₂ℎ²+2ℎ𝛼+11+𝜎⁻¹×sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠. (41) Therefore, we have ‖𝑇‖²₆≤𝐶𝑒^{𝜎(𝜏−𝑡)}𝑇₀²₂+𝐶2Rt₂ℎ²+2ℎ𝛼+11+𝜎⁻¹×sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠. (42)

3.2.4. (𝐿⁶(Ω))² Estimates of 𝑣̃

Multiplying (16) by |𝑣|̃⁴𝑣̃ and integrating by parts over Ω, we deduce 16𝑑‖𝑣‖̃𝑑𝑡⁶⁶+1Re_1Ω|∇𝑣|̃²|𝑣|̃⁴+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝜕_𝑧𝑣̃²|𝑣|̃⁴+4𝑑𝑥𝑑𝑦𝑑𝑧9Re_1Ω∇|𝑣|̃³²+4𝑑𝑥𝑑𝑦𝑑𝑧9Re_2Ω𝜕_𝑧|𝑣|̃³²𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_Ω|𝑣||∇𝑣|̃|𝑣|̃⁵𝑑𝑥𝑑𝑦𝑑𝑧+𝐶_𝑀⁰_−ℎ|𝑇|𝑑𝑧⁰_−ℎ|∇𝑣|̃|𝑣|̃⁴𝑑𝑧𝑑𝑥𝑑𝑦+𝐶_𝑀⁰_−ℎ|𝑣|̃²𝑑𝑧⁰_−ℎ|∇𝑣|̃|𝑣|̃⁴𝑑𝑧𝑑𝑥𝑑𝑦. (43) It follows from interpolation inequality and Lemma 2 that _Ω|𝑣||∇𝑣|̃|𝑣|̃⁵≤𝑑𝑥𝑑𝑦𝑑𝑧_𝑀|𝑣|⁰_−ℎ|∇𝑣|̃²|𝑣|̃⁴𝑑𝑧^1/20_−ℎ|𝑣|̃⁶𝑑𝑧^1/2≤𝑑𝑥𝑑𝑦_𝑀|𝑣|⁴𝑑𝑥𝑑𝑦^1/4_Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2×⁰_−ℎ𝑀|𝑣|̃¹²𝑑𝑥𝑑𝑦^1/2𝑑𝑧^1/2, (44)_𝑀|𝑣|̃¹²𝑑𝑥𝑑𝑦=_𝑀|𝑣|̃³⁴𝑑𝑥𝑑𝑦≤𝐶_𝑀|𝑣|̃⁶𝑑𝑥𝑑𝑦_𝑀∇|𝑣|̃³²𝑑𝑥𝑑𝑦, (45) which implies that ⁰_−ℎ𝑀|𝑣|̃¹²𝑑𝑥𝑑𝑦^1/2𝑑𝑧^1/2≤𝐶_Ω|𝑣|̃⁶𝑑𝑥𝑑𝑦𝑑𝑧^1/4_Ω∇|𝑣|̃³²𝑑𝑥𝑑𝑦𝑑𝑧^1/4. (46) We deduce from (44) and (46) that _Ω|𝑣||∇𝑣|̃|𝑣|̃⁵‖𝑣‖̃𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶₆^3/2‖𝑣‖₂^1/2‖∇𝑣‖₂^1/2_Ω∇|𝑣|̃³²𝑑𝑥𝑑𝑦𝑑𝑧^1/4×_Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2. (47) Repeating the similar process with the previously mentioned, we have _𝑀⁰_−ℎ|𝑇|𝑑𝑧⁰_−ℎ|∇𝑣|̃|𝑣|̃⁴‖𝑇‖𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶_6Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2‖𝑣‖̃²₆,_𝑀0−ℎ|𝑣|̃²𝑑𝑧⁰_−ℎ|∇𝑣|̃|𝑣|̃⁴𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶_Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/2‖𝑣‖̃³₆‖𝑣‖̃_𝐻¹_(Ω), (48) where we use the inequality ‖𝑣‖̃⁸_𝐿8(𝑀)≤𝐶‖𝑣‖̃⁶_𝐿6(𝑀)‖𝑣‖̃²_𝐻1(𝑀).

We deduce from (43) and (47)–(48) that 𝑑‖𝑣‖̃𝑑𝑡⁶⁶+2Re_1Ω|∇𝑣|̃²|𝑣|̃⁴+2𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝜕_𝑧𝑣̃²|𝑣|̃⁴+2𝑑𝑥𝑑𝑦𝑑𝑧Re_1Ω∇|𝑣|̃³²+2𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝜕_𝑧|𝑣|̃³²𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶‖²₂‖∇𝑣‖²₂+‖𝑣‖̃²_𝐻1(Ω)‖𝑣‖̃⁶₆+𝐶‖𝑇‖²₆‖𝑣‖̃⁴₆, (49) which implies that 𝑑𝑒𝑑𝑡^𝜎𝑡‖𝑣‖̃²₆≤𝐶𝑒^𝜎𝑡1+‖𝑣‖²₂‖∇𝑣‖²₂+‖𝑣‖̃²_𝐻1(Ω)‖𝑣‖̃²₆+𝐶𝑒^𝜎𝑡‖𝑇‖²₆. (50) Therefore, it follows from (35)–(38), (42), and Lemma 1 that there exists a positive constant 𝜌₂,𝜌₃ independent of 𝑡 and a constant 𝜏₁=𝜏₁(𝑡) such that ‖𝑣‖̃²₆≤𝜌₂,_𝑡𝑡+1𝑒^𝜎𝑠_Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑠≤𝜌₃, (51)(52) for any 𝜏≤𝜏₁. 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.

3.2.5. (𝐻¹(𝑀))² Estimates of   𝑣

Taking the 𝐿²(Ω) inner product of (14) with −Δ𝑣 and combining the boundary conditions (15), we deduce 12𝑑‖∇𝑑𝑡𝑣‖²_𝐿2(𝑀)+1Re_1𝑀|Δ𝑣|²𝑑𝑥𝑑𝑦≤𝐶_𝑀|𝑣||∇𝑣||Δ𝑣|𝑑𝑥𝑑𝑦+𝐶_𝑀⁰_−ℎ|∇𝑣|̃|𝑣|̃𝑑𝑧|Δ𝑣|𝑑𝑥𝑑𝑦+𝐶_𝑀|𝑣||Δ𝑣|𝑑𝑥𝑑𝑦+𝐶_𝑀⁰_−ℎ|∇𝑇|𝑑𝑧|Δ𝑣|𝑑𝑥𝑑𝑦. (53) We give the estimates of each term of the right-hand side of (53) as follows: _𝑀|𝑣||∇𝑣||Δ𝑣|𝑑𝑥𝑑𝑦≤𝐶_𝑀|𝑣|⁴𝑑𝑥𝑑𝑦^1/4_𝑀|∇𝑣|⁴𝑑𝑥𝑑𝑦^1/4×_𝑀|Δ𝑣|²𝑑𝑥𝑑𝑦^1/2≤𝐶_𝑀|𝑣|²𝑑𝑥𝑑𝑦^1/4_𝑀|∇𝑣|²𝑑𝑥𝑑𝑦^1/2×_𝑀|Δ𝑣|²𝑑𝑥𝑑𝑦^3/4,_𝑀0−ℎ|𝑣|̃|∇𝑣|̃𝑑𝑧|Δ𝑣|𝑑𝑥𝑑𝑦≤𝐶_Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧^1/4‖∇𝑣‖̃₂^1/2×_𝑀|Δ𝑣|²𝑑𝑥𝑑𝑦^1/2,_𝑀|𝑣||Δ𝑣|𝑑𝑥𝑑𝑦≤𝐶_𝑀|𝑣|²𝑑𝑥𝑑𝑦^1/2_𝑀|Δ𝑣|²𝑑𝑥𝑑𝑦^1/2,_𝑀0−ℎ|∇𝑇|𝑑𝑧|Δ𝑣|𝑑𝑥𝑑𝑦≤𝐶_Ω|∇𝑇|²𝑑𝑥𝑑𝑦𝑑𝑧^1/2_𝑀|Δ𝑣|²𝑑𝑥𝑑𝑦^1/2. (54) It follows from (53)–(54) that 𝑑𝑒𝑑𝑡^𝜎𝑡‖∇𝑣‖²_𝐿2(𝑀)+1Re₁𝑒^𝜎𝑡_𝑀|Δ𝑣|²‖𝑑𝑥𝑑𝑦≤𝐶1+𝑣‖²_𝐿2(𝑀)‖∇𝑣‖²_𝐿2(𝑀)𝑒^𝜎𝑡‖∇𝑣‖²_𝐿2(𝑀)+𝐶𝑒^𝜎𝑡‖∇𝑇‖²₂+𝐶𝑒^𝜎𝑡‖𝑣‖²₂+𝐶𝑒^𝜎𝑡_Ω|∇𝑣|̃²|𝑣|̃⁴𝑑𝑥𝑑𝑦𝑑𝑧+𝐶𝑒^𝜎𝑡‖∇𝑣‖̃²₂. (55) In view of (27), (35)–(38), (52), and Lemma 1, we obtain ‖∇𝑣‖²_𝐿2(𝑀)≤𝜌₄, (56) for any 𝜏≤𝜏₁−1.

3.2.6. (𝐿²(Ω))² Estimates of   𝑣_𝑧

Denote 𝑢=𝑣_𝑧. It is clear that 𝑢 satisfies the following equation obtained by differentiating (1) with respect to 𝑧: 𝜕𝑢𝜕𝑡+𝐿₁𝑢+(𝑣⋅∇)𝑢−^𝑧_−ℎ∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑢+()()1𝜕𝑧𝑢⋅∇𝑣−∇⋅𝑣𝑢+Ro𝑓𝑢^⟂−∇𝑇=0, (57) with the boundary conditions 𝑢|_𝑧=0=0,𝑢|_𝑧=−ℎ𝑛|=0,𝑢⋅_Γ𝑙=0,𝜕𝑢𝜕𝑛×𝑛_Γ𝑙=0. (58) Multiplying (57) by 𝑢, integrating over Ω, and using the boundary conditions (58), we get 12𝑑‖𝑢‖𝑑𝑡²²+1Re_1Ω|∇𝑢|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω𝜕_𝑧𝑢²𝑑𝑥𝑑𝑦𝑑𝑧=−_Ω[(𝑢⋅∇)𝑣−(∇⋅𝑣)𝑢−∇𝑇]⋅𝑢𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_Ω|𝑣||𝑢||∇𝑢|𝑑𝑥𝑑𝑦𝑑𝑧+_Ω|𝑇||∇𝑢|𝑑𝑥𝑑𝑦𝑑𝑧. (59) We estimate the right-hand side of (59) term by term as follows: _Ω|𝑇||∇𝑢|𝑑𝑥𝑑𝑦𝑑𝑧≤‖𝑇‖₂‖∇𝑢‖₂,_Ω|𝑣||𝑢||∇𝑢|𝑑𝑥𝑑𝑦𝑑𝑧≤‖𝑣‖₆‖𝑢‖₃‖∇𝑢‖₂≤𝐶‖𝑣‖₆‖𝑢‖₂^1/2‖∇𝑢‖₂+𝜕_𝑧𝑢₂^3/2. (60)

It follows from (59)–(60) that 𝑑‖𝑢‖𝑑𝑡²²+1Re_1Ω|∇𝑢|²1𝑑𝑥𝑑𝑦𝑑𝑧+Re_2Ω𝜕_𝑧𝑢²‖𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶⁴₆‖𝑢‖²₂+𝐶‖𝑇‖²₂. (61) Therefore, we have 𝑑𝑒𝑑𝑡^𝜎𝑡‖𝑢‖²₂+1Re₁𝑒^𝜎𝑡_Ω|∇𝑢|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re₂𝑒^𝜎𝑡_Ω𝜕_𝑧𝑢²‖𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶1+⁴₆𝑒^𝜎𝑡‖𝑢‖²₂+𝐶𝑒^𝜎𝑡‖𝑇‖²₂. (62)

It is shown in [30] that ‖𝑣‖₆≤𝐶ℎ^−1/3‖𝑣‖₂+𝐶ℎ^1/6‖∇𝑣‖₂+‖𝑣‖̃₆, (63) which implies that ‖𝑣‖²₆≤𝜌₅, (64) for any 𝜏≤𝜏₁−1.

By virtue of Lemma 1, from (23), (27)–(29), and (35), we deduce 𝜕_𝑧𝑣²₂≤𝜌₆,1Re_1𝑡^𝑡+1_Ω∇𝑣_𝑧²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑠Re_2𝑡^𝑡+1_Ω𝜕_𝑧𝑣_𝑧²𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑠≤𝜌₇, (65)(66) for any 𝜏≤𝜏₁−2.

3.2.7. (𝐿²(Ω))² Estimates of   ∇𝑣

Taking the 𝐿²(Ω) inner product of (1) with −Δ𝑣 and combining the boundary conditions (3), we deduce 12𝑑‖‖𝑑𝑡∇𝑣²₂+1Re_1Ω|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω∇𝜕_𝑧𝑣²𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶_𝑀⁰_−ℎ|∇𝑣|𝑑𝑧⁰_−ℎ𝜕_𝑧𝑣|Δ𝑣|𝑑𝑧𝑑𝑥𝑑𝑦+𝐶_𝑀⁰_−ℎ|∇𝑇|𝑑𝑧⁰_−ℎ|Δ𝑣|𝑑𝑧𝑑𝑥𝑑𝑦+𝐶_Ω|𝑣||∇𝑣||Δ𝑣|𝑑𝑥𝑑𝑦𝑑𝑧+𝐶_Ω|𝑣||Δ𝑣|𝑑𝑥𝑑𝑦𝑑𝑧. (67)

We estimate each term in the right-hand side of (67) as follows _Ω|𝑣|‖𝑣‖||∇𝑣||Δ𝑣𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶₆‖∇𝑣‖₃‖Δ𝑣‖₂≤𝐶‖𝑣‖₆‖∇𝑣‖₂^1/2∇𝜕_𝑧𝑣₂+‖Δ𝑣‖₂^3/2,_𝑀0−ℎ|∇𝑣|𝑑𝑧⁰_−ℎ𝜕_𝑧𝑣|𝑣|Δ𝑣𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶_𝑧2^1/2∇𝑣_𝑧2^1/2‖∇𝑣‖₂^1/2‖Δ𝑣‖₂^3/2,_𝑀0−ℎ|∇𝑇|𝑑𝑧⁰_−ℎ|Δ𝑣|𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶‖∇𝑇‖₂‖Δ𝑣‖₂,_Ω|𝑣||Δ𝑣|𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶‖𝑣‖₂‖Δ𝑣‖₂. (68)

We derive from (67)–(68) that 𝑑‖‖𝑑𝑡∇𝑣²₂+1Re_1Ω|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re_2Ω∇𝜕_𝑧𝑣²‖𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶⁴₆+𝜕_𝑧𝑣²₂∇𝜕_𝑧𝑣²₂‖∇𝑣‖²₂+𝐶‖∇𝑇‖²₂+𝐶‖𝑣‖²₂. (69) Therefore, we have 𝑑𝑒𝑑𝑡^𝜎𝑡‖∇𝑣‖²₂+1Re₁𝑒^𝜎𝑡_Ω|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧Re₂𝑒^𝜎𝑡_Ω∇𝜕_𝑧𝑣²‖𝑣‖𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶1+⁴₆+𝜕_𝑧𝑣²₂∇𝜕_𝑧𝑣²₂𝑒^𝜎𝑡‖∇𝑣‖²₂+𝐶𝑒^𝜎𝑡‖∇𝑇‖²₂+𝐶𝑒^𝜎𝑡‖𝑣‖²₂. (70)

By virtue of (27), (35), (64)–(66), and Lemma 1, we get ‖∇𝑣‖²₂≤𝜌₈,1Re_1𝑡^𝑡+1_Ω|Δ𝑣|²+1𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏Re_2𝑡^𝑡+1_Ω∇𝜕_𝑧𝑣²𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝜏≤𝜌₉, (71)(72) for any 𝜏≤𝜏₁−3.

3.2.8. 𝐻¹(Ω) Estimates of 𝑇

Multiplying (2) by 𝐿₂𝑇, integrating over Ω, and combining the boundary conditions (4), we deduce 12𝑑‖𝑇‖𝑑𝑡²+𝐿₂𝑇²₂≤𝐶_Ω|𝑣||∇𝑇|𝐿₂𝑇+𝐶_Ω^𝑧_−ℎ∇⋅𝑣𝑥,𝑦,𝜁,𝑡𝑑𝜁𝜕𝑇𝐿𝜕𝑧²𝑇+_Ω|𝑄|𝐿₂𝑇. (73)

We give the estimates of each term in the right-hand side of (73) as follows _Ω|𝑣||∇𝑇|𝐿₂𝑇𝑑𝑥𝑑𝑦𝑑𝑧≤‖𝑣‖₆‖∇𝑇‖₃𝐿₂𝑇₂≤𝐶‖𝑣‖₆‖∇𝑇‖₂^1/2𝐿₂𝑇₂^3/2,_𝑀0−ℎ|∇𝑣|𝑑𝑧⁰_−ℎ𝜕_𝑧𝑇𝐿₂𝑇‖‖𝑑𝑧𝑑𝑥𝑑𝑦≤𝐶∇𝑣₂^1/2‖Δ𝑣‖₂^1/2𝜕_𝑧𝑇₂^1/2𝐿₂𝑇₂^3/2,_Ω|𝑄|𝐿₂𝑇𝑑𝑥𝑑𝑦𝑑𝑧≤𝐶‖𝑄‖₂𝐿₂𝑇₂. (74)

We derive from (73)–(74) that 𝑑𝑒𝑑𝑡^𝜎𝑡‖𝑇‖²≤𝐶1+‖𝑣‖⁴₆+‖∇𝑣‖²₂‖Δ𝑣‖²₂𝑒^𝜎𝑡‖𝑇‖²+𝐶𝑒^𝜎𝑡‖𝑄‖²₂. (75)

Using (27), (64), (71), (72), and Lemma 1, we get ‖𝑇‖²≤𝜌₁₀,_𝑡𝑡+1𝐿₂𝑇(𝑠)²₂𝑑𝑠≤𝜌₁₁, (76)(77) for any 𝜏≤𝜏₁−4.

4. The Existence of the Pullback Attractor

In this section, we recall some definitions and lemmas about pullback attractor and prove the existence of (𝑉,𝑉)-pullback attractor.

Definition 4 (see [13, 32]). Let 𝑋 be a complete metric space. A two-parameter family of mappings {𝑈(𝑡,𝜏)}_𝜏≤𝑡 is said to be a continuous process in 𝑋 if (i)𝑈(𝜏,𝜏)=Id for any 𝜏∈ℝ,(ii)𝑈(𝑡,𝑟)𝑈(𝑟,𝜏)=𝑈(𝑡,𝜏),𝑡≥𝑟≥𝜏, (iii)𝑈(𝑡,𝜏)𝑥_𝑛→𝑈(𝑡,𝜏)𝑥, if 𝑥_𝑛→𝑥 in 𝑋.

Definition 5 (see [11, 13, 33]). Let 𝑋,𝑌 be two Banach spaces. Then, the family 𝒜={𝐴(𝑡)∶𝑡∈ℝ} is said to be a (𝑋,𝑌)-pullback attractor for {𝑈(𝑡,𝜏)}_𝑡≥𝜏, if (i)𝐴(𝑡) is closed in 𝑋 and compact in 𝑌 for any 𝑡∈ℝ,(ii)𝑈(𝑡,𝜏)𝐴(𝜏)=𝐴(𝑡) for any 𝜏≤𝑡 (invariance property),(iii)it pullback attracts every bounded subsets of 𝑋 in the topology of 𝑌 ((𝑋,𝑌)-pullback attracting), that is, for any nonempty class of bounded subsets {𝐷(𝑡)∶𝑡∈ℝ}⊂𝑋, lim_{𝜏→−∞}dist_𝑌(𝑈(𝑡,𝜏)𝐷(𝜏),𝐴(𝑡))=0, (78)for any 𝑡∈ℝ.

Definition 6 (see [31, 33]). The process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 is said to be pullback asymptotically compact if for any 𝑡∈ℝ and any nonempty class of bounded subsets {𝐷(𝑡)∶𝑡∈ℝ}⊂𝑋, any sequence 𝜏_𝑛→−∞, and any sequence 𝑥_𝑛∈𝐷(𝜏_𝑛), the sequence {𝑈(𝑡,𝜏_𝑛)𝑥_𝑛} is relatively compact in 𝑋.

Lemma 7 (see [33]). Let {𝑈(𝑡,𝜏)}_𝑡≥𝜏 be a continuous process such that {𝑈(𝑡,𝜏)} is pullback asymptotically compact. If there exists a family of pullback absorbing sets 𝐵={𝐵(𝑡)∶𝑡∈ℝ}, then {𝑈(𝑡,𝜏)}_𝑡≥𝜏 has a unique pullback attractor 𝒜={𝒜(𝑡)∶𝑡∈ℝ} and 𝒜(𝑡)=_{𝑠≤𝑡𝜏≤𝑠}𝑈(𝑡,𝜏)𝐵(𝜏). (79)

Definition 8 (see [33]). A process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 is said to satisfy pullback 𝒟 condition (PDC), if for any fixed 𝑡∈ℝ, any bounded subset 𝐷 of a Banach space 𝑋 and any 𝜂>0, there exist a time 𝜏₀=𝜏₀(𝐷,𝜂,𝑡)≤𝑡 and a finite-dimensional subspace 𝑋₁ of 𝑋 such that (i)𝑃(⋃_𝜏≤𝜏0𝑈(𝑡,𝜏)𝐷) is bounded, (ii)(Id−𝑃)𝑦≤𝜂 for any 𝑦∈⋃_𝜏≤𝜏0𝑈(𝑡,𝜏)𝑥 and any 𝑥∈𝐷, where 𝑃∶𝑋→𝑋₁ is a bounded projection and Id is the identity.

Lemma 9 (see [33]). A process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 satisfying pullback 𝒟 condition (PDC) is pullback asymptotically compact.

From (29), (37), (65), (71), and (76), we get the following theorem.

Theorem 10. Assume that 𝑄∈𝐿²_loc(ℝ;𝐿²(Ω)) satisfies sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠<∞. (80) Then, for any 0<𝜎≤min{1/(4𝑅𝑡₂ℎ²+(4ℎ/𝛼)),1/2𝐶_𝑀𝑅𝑒₁}, there exists a (𝑉,𝑉)-pullback absorbing set for the processes {𝑈(𝑡,𝜏)}_𝑡≥𝜏 associated with (1)–(5).

Thanks to the knowledge of functional analysis, we know that the operator 𝐴=(𝐿₁,𝐿₂)∶𝑉→𝑉^′ with domain 𝒟(𝐴)=𝑉∩𝐻²(Ω)³ (81) is a positive self-adjoint operator with compact inverse. Therefore, the space 𝐻 possesses an orthonormal basis {(𝜔_𝑘,𝜂_𝑘)}^∞_𝑘=1 of eigenfunctions of the operator 𝐴 such that 𝐿₁𝜔_𝑘,𝐿₂𝜂_𝑘=𝜆_𝑘𝜔_𝑘,𝜇_𝑘𝜂_𝑘, (82) where 0<𝜆₁≤𝜆₂≤⋯,lim_𝑘→∞𝜆_𝑘=∞,0<𝜇₁≤𝜇₂≤⋯,lim_𝑘→∞𝜇_𝑘=∞. (83) Let 𝐻_𝑚=span{(𝜔₁,𝜂₁),(𝜔₂,𝜂₂),…,(𝜔_𝑚,𝜂_𝑚)}, and let 𝑃_𝑚∶𝐻→𝐻_𝑚 be the 𝐿²(Ω) orthogonal projection onto 𝐻_𝑚.

In the following, we prove that the process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 associated with (1)–(5) is pullback asymptotically compact in 𝑉 by verifying the pullback 𝒟 condition.

Theorem 11. Assume that 𝑄∈𝐿²_loc(ℝ;𝐿²(Ω)) satisfies sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠<∞. (84) Then, for any 0<𝜎≤min{1/(4𝑅𝑡₂ℎ²+(4ℎ/𝛼)),1/2𝐶_𝑀𝑅𝑒₁}, the process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 associated with (1)–(5) is pullback asymptotically compact in 𝑉.

Proof. Let 𝐵 be a bounded subset of 𝐻(Ω),(𝑣(𝑡),𝑇(𝑡))=𝑈(𝑡,𝜏)(𝑣_𝜏,𝑇_𝜏)=(𝑣₁+𝑣₂,𝑇₁+𝑇₂), where (𝑣_𝜏,𝑇_𝜏)∈𝐵,(𝑣₁,𝑇₁)=𝑃_𝑚(𝑣,𝑇), and (𝑣₂,𝑇₂)=(Id−𝑃_𝑚)(𝑣,𝑇)=𝑄_𝑚(𝑣,𝑇).
Taking the inner product of (1) with 𝐿₁𝑣₂, we have 12𝑑𝑣𝑑𝑡²²+𝐿₁𝑣₂²₂≤‖𝑣‖₆‖∇𝑣‖₃𝐿₁𝑣₂₂+𝐶‖∇𝑣‖₂^1/2‖Δ𝑣‖₂^1/2𝜕𝑣𝜕𝑧₂^1/2∇𝑣_𝑧2^1/2𝐿₁𝑣₂₂+𝐶‖∇𝑇‖₂𝐿₁𝑣₂₂+𝐶‖𝑣‖₂𝐿₁𝑣₂₂. (85) From the Young inequality, we obtain 𝑑𝑣𝑑𝑡²²+𝜆_𝑚𝑣₂²≤𝐶‖𝑣‖²₆‖∇𝑣‖²₃+𝐶‖∇𝑣‖²₂‖Δ𝑣‖²₂+𝐶𝑣_𝑧²₂∇𝑣_𝑧²₂+𝐶‖∇𝑇‖²₂+𝐶‖𝑣‖²₂. (86) By virtue of Lemma 1, we get 𝑣₂(𝑡+𝑟)²≤𝑒^−𝜆𝑚_𝑡^𝑡+1𝑣₂(𝑠)²𝑑𝑠+_𝑡^𝑡+1𝑏(𝑠)𝑑𝑠, (87) where 𝑏(𝑠)=𝐶‖𝑣‖²₆‖∇𝑣‖²₃+𝐶‖∇𝑣‖²₂‖Δ𝑣‖²₂+𝐶𝑣_𝑧²₂∇𝑣_𝑧²₂+𝐶‖∇𝑇‖²₂+𝐶‖𝑣‖²₂. (88) Therefore, we obtain 𝑣₂(𝑡)²≤𝜖2, (89) for any 𝜖>0,𝑡∈ℝ, and any 𝜏≤𝜏₂−1, if 𝑚₁ is sufficiently large.
Similarly, we get 𝑇₂(𝑡)²≤𝜖2, (90) for any 𝜖>0,𝑡∈ℝ, and any 𝜏≤𝜏₂−2, if 𝑚₂ is sufficiently large. Hence, 𝑣₂²+𝑇₂(𝑡)²≤𝜖, (91) for any 𝜖>0,𝑡∈ℝ, and any 𝜏≤𝜏₂−2, if 𝑚≥max{𝑚₁,𝑚₂}.

From Theorems 10 and 11 and Lemma 7, we obtain the following main Theorem.

Theorem 12. Assume that 𝑄∈𝐿²_loc(ℝ;𝐿²(Ω)) satisfies sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑄(𝑠)‖²₂𝑑𝑠<∞. (92) Then, for any 0<𝜎≤min{1/(4𝑅𝑡₂ℎ²+(4ℎ/𝛼)),1/2𝐶_𝑀𝑅𝑒₁}, the process {𝑈(𝑡,𝜏)}_𝑡≥𝜏 associated with the solutions of (1)–(5) possesses a (𝑉,𝑉)-pullback attractor 𝒜={𝒜(𝑡)∶𝑡∈ℝ}.

Acknowledgment

The authors of this paper would like to express their sincere thanks to the reviewer for valuable comments and suggestions.