Global Attractor of Atmospheric Circulation Equations with Humidity Effect

Abstract

Global attractor of atmospheric circulation equations is considered in this paper. Firstly, it is proved that this system possesses a unique global weak solution in 𝐿²(Ω,𝑅⁴). Secondly, by using C-condition, it is obtained that atmospheric circulation equations have a global attractor in 𝐿²(Ω,𝑅⁴).

1. Introduction

This paper is concerned with global attractor of the following initial-boundary problem of atmospheric circulation equations involving unknown functions (𝑢,𝑇,𝑞,𝑝) at (𝑥,𝑡)=(𝑥₁,𝑥₂,𝑡)∈Ω×(0,∞) (Ω=(0,2𝜋)×(0,1) is a period of 𝐶^∞ field (−∞,+∞)×(0,1)): 𝜕𝑢𝜕𝑡=𝑃^𝑟(Δ𝑢𝜕−𝑇∇𝑝−𝜎𝑢)+𝑃_𝑟𝑅𝑇−𝑅𝑞^→𝜅−(𝑢⋅∇)𝑢,(1.1)𝜕𝑡=Δ𝑇+𝑢²𝜕𝑞−(𝑢⋅∇)𝑇+𝑄,(1.2)𝜕𝑡=𝐿^𝑒Δ𝑞d+iv𝑢𝑢₂=−0(𝑢,⋅∇)𝑞+𝐺,((11..34)) where 𝑃_𝑟, 𝑅, 𝑅, and 𝐿_𝑒 are constants, 𝑢=(𝑢₁,𝑢₂), 𝑇, 𝑞, and 𝑝 denote velocity field, temperature, humidity, and pressure, respectively; 𝑄, 𝐺 are known functions, and 𝜎 is constant matrix: 𝜎=⎛⎜⎜⎝𝜎⁰𝜔𝜎𝜔₁⎞⎟⎟⎠.(1.5)

The problems (1.1)–(1.4) are supplemented with the following Dirichlet boundary condition at 𝑥₂=0,1 and periodic condition for 𝑥₁: (𝑢,𝑇,(𝑢𝑞,)𝑇,0𝑞,)𝑥=₂0,𝑥₂=0,1,=(𝑢,𝑇,𝑞)2𝜋,𝑥₂,(1.6) and initial value conditions (𝑢,𝑇,𝑞)=𝑢₀,𝑇₀,𝑞₀,𝑡=0.(1.7)

The partial differential equations (1.1)–(1.7) were firstly presented in atmospheric circulation with humidity effect [1]. Atmospheric circulation is one of the main factors affecting the global climate, so it is very necessary to understand and master its mysteries and laws. Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies. On the contrary, it is also the important result of these physical transports, balance and conversion. Thus, it is of necessity to study the characteristics, formation, preservation, change and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human’s understanding of nature, but also the helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources.

The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments. The atmosphere or the ocean or the couple atmosphere and ocean can be viewed as an initial and boundary value problem [2–5], or an infinite dimensional dynamical system [6–8]. We deduce the atmospheric circulation model (1.1)–(1.7) which is able to show features of atmospheric circulation and is easy to be studied from the very complex atmospheric circulation model based on the actual background and meteorological data, and we present global solutions of atmospheric circulation equations with the use of the 𝑇-weakly continuous operator [1]. In fact, there are numerous papers on this topic [9–13]. Compared with some similar papers, we add humidity function in this paper. We propose firstly the atmospheric circulation equation with humidity function which does not appear in the previous literature.

As far as the theory of infinite-dimensional dynamical system is concerned, we refer to [9–11, 14–18]. In the study of infinite dimensional dynamical system, the long-time behavior of the solution to equations is an important issue. The long-time behavior of the solution to equations can be shown by the global attractor with the finite-dimensional characteristics. Some authors have already studied the existence of the global attractor for some evolution equations [2, 3, 13, 19–21]. The global attractor strictly defined as 𝜔-limit set of ball, which under additional assumptions is nonempty, compact, and invariant [13, 17]. Attractor theory has been intensively investigated within the science, mathematics, and engineering communities. Lü et al. [22–25] apply the current theoretical results or approaches to investigate the global attractor of complex multiscroll chaotic systems. We obtain existence of global attractor for the atmospheric circulation equations from the mathematical perspective in this paper.

The paper is organized as follows. In Section 2, we recall preliminary results. In Section 3, we present uniqueness of the solution to the atmospheric circulation equations. In Section 4, we obtain global attractor of the equations.

‖⋅‖_𝑋 denote norm of the space 𝑋; 𝐶 and 𝐶_𝑖 are variable constants. Let 𝐻={𝜙=(𝑢,𝑇,𝑞)∈𝐿²(Ω,𝑅⁴)∣𝜙 satisfy (1.4), (1.6)}, and 𝐻₁={𝜙=(𝑢,𝑇,𝑞)∈𝐻¹(Ω,𝑅⁴)∣𝜙 satisfy (1.4), (1.6)}.

2. Preliminaries

Let 𝑋 and 𝑋₁ be two Banach spaces, 𝑋₁⊂𝑋 a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on 𝑋, given by 𝑑𝑢𝑑𝑡𝑢=(𝑥𝐿,0𝑢)+=𝐺𝑢₀(𝑢),,(2.1) where 𝑢(𝑡) is an unknown function, 𝐿∶𝑋₁→𝑋 a linear operator, and 𝐺∶𝑋₁→𝑋 a nonlinear operator.

A family of operators 𝑆(𝑡)∶𝑋→𝑋 (𝑡≥0) is called a semigroup generated by (2.1) if it satisfies the following properties:(1)𝑆(𝑡)∶𝑋→𝑋 is a continuous map for any 𝑡≥0;(2)𝑆(0)=𝑖𝑑∶𝑋→𝑋 is the identity;(3)𝑆(𝑡+𝑠)=𝑆(𝑡)⋅𝑆(𝑠), for all 𝑡,𝑠≥0. Then, the solution of (2.1) can be expressed as 𝑢𝑡,𝑢₀=𝑆(𝑡)𝑢₀.(2.2)Next, we introduce the concepts and definitions of invariant sets, global attractors, and 𝜔-limit sets for the semigroup 𝑆(𝑡).

Definition 2.1. Let 𝑆(𝑡) be a semigroup defined on 𝑋. A set Σ⊂𝑋 is called an invariant set of 𝑆(𝑡) if 𝑆(𝑡)Σ=Σ, for all 𝑡≥0. An invariant set Σ is an attractor of 𝑆(𝑡) if Σ is compact, and there exists a neighborhood 𝑈⊂𝑋 of Σ such that for any 𝑢₀∈𝑈, inf_𝑣∈Σ‖‖𝑆(𝑡)𝑢₀−𝑣‖‖_𝑋⟶0,as𝑡⟶∞.(2.3)
In this case, we say that Σ attracts 𝑈. Particularly, if Σ attracts any bounded set of 𝑋, Σ is called a global attractor of 𝑆(𝑡) in 𝑋.
For a set 𝐷⊂𝑋, we define the 𝜔-limit set of 𝐷 as follows: 𝜔(𝐷)=_𝑠≥0_𝑡≥𝑠𝑆(𝑡)𝐷,(2.4) where the closure is taken in the 𝑋-norm. Lemma 2.2 is the classical existence theorem of global attractor by Temam [13].

Lemma 2.2. Let 𝑆(𝑡)∶𝑋→𝑋 be the semigroup generated by (2.1). Assume that the following conditions hold:(1)𝑆(𝑡) has a bounded absorbing set 𝐵⊂𝑋, that is, for any bounded set 𝐴⊂𝑋 there exists a time 𝑡_𝐴≥0 such that 𝑆(𝑡)𝑢₀∈𝐵,  for  all  𝑢₀∈𝐴 and 𝑡>𝑡_𝐴;(2)𝑆(𝑡) is uniformly compact, that is, for any bounded set 𝑈⊂𝑋 and some 𝑇>0 sufficiently large, the set ⋃_𝑡≥𝑇𝑆(𝑡)𝑈 is compact in 𝑋.
Then the 𝜔-limit set 𝒜=𝜔(𝐵) of 𝐵 is a global attractor of (2.1), and 𝒜 is connected providing 𝐵 is connected.

Definition 2.3 (see [19]). We say that 𝑆(𝑡)∶𝑋→𝑋 satisfies 𝐶-condition, if for any bounded set 𝐵⊂𝑋 and 𝜀>0, there exist 𝑡_𝐵>0 and a finite dimensional subspace 𝑋₁⊂𝑋 such that {𝑃𝑆(𝑡)𝐵} is bounded, and ‖(𝐼−𝑃)𝑆(𝑡)𝑢‖_𝑋<𝜀,∀𝑡≥𝑡_𝐵,𝑢∈𝐵,(2.5) where 𝑃∶𝑋→𝑋₁ is a projection.

Lemma 2.4 (see [19]). Let 𝑆(𝑡)∶𝑋→𝑋 (𝑡≥0) be a dynamical systems. If the following conditions are satisfied:(1)there exists a bounded absorbing set 𝐵⊂𝑋;(2)𝑆(𝑡) satisfies 𝐶-condition,
then 𝑆(𝑡) has a global attractor in 𝑋.

From Linear elliptic equation theory, one has the following.

Lemma 2.5. The eigenvalue equation: −Δ𝑇𝑥₁,𝑥₂=𝛽𝑇𝑇𝑥=₁,0𝑥,₂,𝑥₂𝑥₁,𝑥₂∈(0,2𝜋)×(0,1),𝑇0,𝑥₂=0,1,=𝑇2𝜋,𝑥₂(2.6) has eigenvalue {𝛽_𝑘}^∞𝑘=1, and 0<𝛽₁≤𝛽₂≤⋯,𝛽_𝑘⟶∞,as𝑘⟶∞.(2.7)

3. Uniqueness of Global Solution

Theorem 3.1. If 𝜎𝛽₁≥max{(𝑅+1)²,((𝑅−1)²/𝐿_𝑒)}, and 𝛽₁ is the first eigenvalue of elliptic equation (2.6), then the weak solution to (1.1)–(1.7) is unique.

Proof. From [1], (𝑢,𝑇,𝑞)∈𝐿^∞((0,𝑇),𝐻)∩𝐿²((0,𝑇),𝐻₁), 0<𝑇<∞ is the weak solution to (1.1)–(1.7). Then for all (𝑣,𝑆,𝑧)∈𝐻₁, 0≤𝑡≤𝑇, we have 1𝑃_𝑟_Ω𝑢𝑣𝑑𝑥+_Ω𝑇𝑆𝑑𝑥+_Ω𝑞𝑧𝑑𝑥=^𝑡0_Ω−∇𝑢∇𝑣−𝜎𝑢𝑣+𝑅𝑇−𝑅𝑞𝑣²−1𝑃_𝑟+𝑄𝑆(−𝑢⋅𝐿∇_𝑒)𝑢𝑣−∇𝑇∇𝑆+𝑢₂𝑆−(𝑢⋅∇)𝑇𝑆+1∇𝑞∇𝑧+𝑢₂𝑧−(𝑢⋅∇)𝑞𝑧+𝐺𝑧𝑑𝑥𝑑𝑡𝑃_𝑟_Ω𝑢₀𝑣𝑑𝑥+_Ω𝑇₀𝑆𝑑𝑥+_Ω𝑞₀𝑧𝑑𝑥.(3.1)
Set (𝑢¹,𝑇¹,𝑞¹) and (𝑢²,𝑇²,𝑞²) are two weak solutions to (1.1)–(1.7), which satisfy (3.1). Let (𝑢,𝑇,𝑞)=(𝑢¹,𝑇¹,𝑞¹)−(𝑢²,𝑇²,𝑞²). Then, 1𝑃_𝑟_Ω𝑢𝑣𝑑𝑥+_Ω𝑇𝑆𝑑𝑥+_Ω𝑞𝑧𝑑𝑥=^𝑡0_Ω−∇𝑢∇𝑣−𝜎𝑢𝑣+𝑅𝑇−𝑅𝑞𝑣²+1𝑃_𝑟𝑢²⋅∇𝑢²𝑣−1𝑃_𝑟𝑢¹+𝑢₂⋅∇𝑢¹𝑣−∇𝑇∇𝑆−𝐿_𝑒𝑆+𝑢²⋅∇𝑇²𝑆−𝑢¹⋅∇𝑇¹𝑆−𝑢₁∇𝑞∇𝑧+𝑢₂𝑧+𝑢²⋅∇𝑞²𝑧⋅∇𝑞¹𝑧𝑑𝑥𝑑𝑡.(3.2)
Let (𝑣,𝑆,𝑧)=(𝑢,𝑇,𝑞). We obtain from (3.2) the following: 1𝑃_𝑟_Ω=|𝑢|²𝑑𝑥+_Ω||𝑇||²𝑑𝑥+_Ω||𝑞||²𝑑𝑥_𝑡0_Ω−||∇𝑢||²−𝜎𝑢⋅𝑢+𝑅𝑇−𝑅𝑞𝑢₂+1𝑃_𝑟𝑢²⋅∇𝑢²𝑢−1𝑃_𝑟𝑢¹−||∇𝑇||₂⋅∇𝑢¹𝑢+𝑢₂+𝑢₂𝑇+𝑢²⋅∇𝑇²𝑇−𝑢¹⋅∇𝑇¹𝑇−𝐿_𝑒||∇𝑞||²+𝑢₂𝑞≤⋅∇𝑞²𝑞−𝑢¹⋅∇𝑞¹𝑞𝑑𝑥𝑑𝑡_𝑡0_Ω−||∇𝑢||²+−||∇𝑇||²−𝐿_𝑒||∇𝑞||²𝑑𝑥𝑑𝑡_𝑡0_Ω−𝜎|𝑢|²++(𝑅+1)𝑇𝑢₂−𝑅−1𝑞𝑢₂𝑑𝑥𝑑𝑡_𝑡0_Ω𝑃1_𝑟≤(𝑢⋅∇)𝑢²𝑢+(𝑢⋅∇)𝑇²𝑇+(𝑢⋅∇)𝑞²𝑞𝑑𝑥𝑑𝑡_𝑡0_Ω−||∇𝑢||²+−||∇𝑇||²−𝐿_𝑒||∇𝑞||²𝑑𝑥𝑑𝑡_𝑡0_Ω⎡⎢⎢⎢⎣−𝜎|𝑢|₂+𝜎||𝑢₂||²+(𝑅+1)²2𝜎||𝑇||²+𝑅−1²2𝜎||𝑞||²+⎤⎥⎥⎥⎦𝑑𝑥𝑑𝑡_𝑡0√2𝑃_𝑟‖𝑢‖_𝐿²‖‖∇𝑢²‖‖_𝐿2‖∇𝑢‖_𝐿²+√2‖𝑢‖^𝐿1/2₂‖∇𝑢‖^𝐿1/2₂‖‖∇𝑇²‖‖_𝐿2‖𝑇‖^𝐿1/2₂‖∇𝑇‖^𝐿1/2₂+√2‖𝑢‖^𝐿1/2₂‖∇𝑢‖^𝐿1/2₂‖‖∇𝑞²‖‖‖𝑞‖^𝐿1/2₂‖∇𝑞‖^𝐿1/2₂≤𝑑𝑡_𝑡0_Ω−||∇𝑢||²−12||∇𝑇||²−𝐿^𝑒+2||∇𝑞||²𝑑𝑥𝑑𝑡_𝑡0√2𝑃_𝑟‖𝑢‖_𝐿²‖‖∇𝑢²‖‖_𝐿2‖∇𝑢‖_𝐿²+√2‖𝑢‖_𝐿²‖∇𝑢‖‖‖∇𝑇²‖‖_𝐿2+√2‖‖∇𝑇²‖‖_𝐿2‖𝑇‖_𝐿²‖∇𝑇‖_𝐿²+√2‖𝑢‖_𝐿²‖∇𝑢‖_𝐿²‖‖∇𝑞²‖‖_𝐿2+√2‖𝑞‖_𝐿²‖∇𝑞‖_𝐿²‖‖∇𝑞²‖‖_𝐿2≤𝑑𝑡_𝑡0_Ω−||∇𝑢||²−12||∇𝑇||²−𝐿^𝑒+2||∇𝑞||²𝑑𝑥𝑑𝑡_𝑡0‖∇𝑢‖^2𝐿2+3𝑃^2𝑟‖𝑢‖^2𝐿₂‖‖∇𝑢²‖‖^2𝐿₂+3‖𝑢‖^2𝐿₂‖‖∇𝑇²‖‖^2𝐿₂+3‖𝑢‖^2𝐿₂‖‖∇𝑞²‖‖^2𝐿₂+12‖∇𝑇‖^2𝐿2+‖‖∇𝑇²‖‖^2𝐿₂‖𝑇‖^2𝐿₂+𝐿^𝑒2‖∇𝑞‖^2𝐿2+2𝐿_𝑒‖𝑞‖^2𝐿₂||∇𝑞²||^2𝐿₂≤𝑑𝑡_𝑡0𝑃3^2𝑟‖𝑢‖^2𝐿₂‖‖∇𝑢²‖‖^2𝐿₂+3‖𝑢‖^2𝐿₂‖‖∇𝑇²‖‖^2𝐿₂+3‖𝑢‖^2𝐿₂‖‖∇𝑞²‖‖^2𝐿₂+‖‖∇𝑇²‖‖^2𝐿₂‖𝑇‖^2𝐿₂+2𝐿_𝑒‖𝑞‖^2𝐿₂‖‖∇𝑞²‖‖^2𝐿₂𝑑𝑡.(3.3) Then, ‖𝑢‖^2𝐿₂+‖𝑇‖^2𝐿₂+‖𝑞‖^2𝐿₂≤≤𝐶𝐶^𝑡0_𝑡0‖𝑢‖^2𝐿2+‖𝑇‖^2𝐿₂+‖𝑞‖^2𝐿₂‖‖∇𝑢²‖‖^2𝐿₂+‖‖∇𝑇²‖‖^2𝐿₂+‖‖∇𝑞²‖‖^2𝐿₂𝑑𝑡‖𝑢‖^2𝐿2+‖𝑇‖^2𝐿₂+‖𝑞‖^2𝐿₂𝑑𝑡.(3.4)
By using the Gronwall inequality, it follows that ‖𝑢‖^2𝐿₂+‖𝑇‖^2𝐿₂+‖𝑞‖^2𝐿₂≤0,(3.5) which imply (𝑢,𝑇,𝑞)≡0. Thus, the weak solution to (1.1)–(1.7) is unique.

4. Existence of Global Attractor

Theorem 4.1. If 𝜎𝛽₁≥max{(𝑅+1)²,(𝑅−1)²/𝐿_𝑒}, and 𝛽₁ is the first eigenvalue of elliptic equation (2.6), then (1.1)–(1.7) have a global attractor in 𝐿²(Ω,𝑅⁴).

Proof. According to Lemma 2.4, we prove Theorem 4.1 in the following two steps.
Step  1. Equations (1.1)–(1.7) have an absorbing set in 𝐻.
Multiply (1.1) by 𝑢 and integrate the product in Ω: 1𝑃_𝑟_Ω𝑑𝑢𝑑𝑡𝑢𝑑𝑥=_ΩΔ𝑢−∇𝑝−𝜎𝑢+𝑅𝑇−𝑅𝑞𝜅−𝑃1_𝑟(𝑢⋅∇)𝑢𝑢𝑑𝑥.(4.1) Then, 12𝑃_𝑟𝑑𝑑𝑡_Ω𝑢²𝑑𝑥=_Ω−||∇𝑢||²−𝜎𝑢⋅𝑢+𝑅𝑇−𝑅𝑞𝑢₂𝑑𝑥.(4.2)
Multiply (1.2) by 𝑇 and integrate the product in Ω: _Ω𝑑𝑇𝑑𝑡𝑇𝑑𝑥=_ΩΔ𝑇+𝑢₂−(𝑢⋅∇)𝑇+𝑄𝑇𝑑𝑥.(4.3) Then, 12𝑑𝑑𝑡_Ω𝑇²𝑑𝑥=_Ω−||∇𝑇||²+𝑢₂𝑇𝑑𝑥+𝑄𝑇𝑑𝑥.(4.4)
Multiply (1.3) by 𝑞 and integrate the product in Ω: _Ω𝑑𝑞𝑑𝑡𝑞𝑑𝑥=_Ω𝐿_𝑒Δ𝑞+𝑢₂−(𝑢⋅∇)𝑞+𝑄𝑞𝑑𝑥.(4.5) Then, 12𝑑𝑑𝑡_Ω𝑞²𝑑𝑥=_Ω−𝐿^𝑒||∇𝑞||²+𝑢₂𝑞𝑑𝑥+𝐺𝑞𝑑𝑥.(4.6)
We deduce from (4.2)–(4.6) the following: 12𝑑𝑑𝑡_Ω𝑃1_𝑟𝑢²+𝑇²+𝑞²𝑑𝑥=_Ω−||∇𝑢||²+(𝑅+1−)𝑇||∇𝑢₂𝑇||²−𝐿_𝑒||∇𝑞||²−𝜎𝑢⋅𝑢≤−𝑅−1𝑞𝑢₂+𝑄𝑇+𝐺𝑞𝑑𝑥_Ω−||∇𝑢||²−||∇𝑇||²−𝐿_𝑒||∇𝑞||²−𝜎|𝑢|²+𝜎||𝑢₂||²+(𝑅+1)²2𝜎||𝑇||²+𝑅−1²2𝜎||𝑞||²+𝜀||𝑇||²+𝜀||𝑞||²+1𝜀||𝑄||²+||𝐺||²𝑑𝑥.(4.7) Let 𝜀>0 be appropriate small such that 𝑑𝑑𝑡_Ω𝑢²≤𝐶₁+𝑇²+𝑞²𝑑𝑥_Ω−||∇𝑢||²−||∇𝑇||²−||∇𝑞||²𝑑𝑥+𝐶²_Ω||𝑄||²+||𝐺||²𝑑𝑥.(4.8)
Then, 𝑑𝑑𝑡_Ω𝑢²+𝑇²+𝑞²𝑑𝑥≤−𝐶₃_Ω|𝑢|²+||𝑇||²+||𝑞||²𝑑𝑥+𝐶⁴.(4.9)
Applying the Gronwall inequality, it follows that ‖(𝑢,𝑇,𝑞)(𝑡)‖^2𝐿₂≤‖(𝑢,𝑇,𝑞)(0)‖^2𝐿₂𝑒^−𝐶3𝑡+𝐶⁴𝐶₃1−𝑒^−𝐶3𝑡.(4.10)
Then, when 𝑀²>𝐶₄/𝐶₃, for any (𝑢₀,𝑇₀,𝑞₀)∈𝐵, here 𝐵 is a bounded in 𝐻, there exists 𝑡_∗>0 such that 𝑆(𝑡)𝑢₀,𝑇₀,𝑞₀=(𝑢(𝑡),𝑇(𝑡),𝑞(𝑡))∈𝐵_𝑀,𝑡>𝑡_∗,(4.11) where 𝐵_𝑀 is a ball in 𝐻, at 0 of radius 𝑀. Thus, (1.1)–(1.7) have an absorbing 𝐵_𝑀 in 𝐻.
Step  2. 𝐶-condition is satisfied.
The eigenvalue equation: 𝑢𝑥Δ𝑢=𝜆𝑢,₁𝑢0,,0𝑥₂=𝑢𝑥₂,0=0,div=𝑢𝑢=20𝜋,𝑥₂,(4.12) has eigenvalues 𝜆₁,𝜆₂,…,𝜆_𝑘,… and eigenvector {𝑒_𝑘∣𝑘=1,2,3,…}, and 𝜆₁≥𝜆₂≥⋯≥𝜆_𝑘≥⋯. If 𝑘→∞, then 𝜆_𝑘→−∞. {𝑒_𝑘∣𝑘=1,2,3,…} constitutes an orthogonal base of 𝐿²(Ω).
For all (𝑢,𝑇,𝑞)∈𝐻, we have 𝑢=^∞_𝑘=1𝑢_𝑘𝑒_𝑘,‖𝑢‖^2𝐿₂=^∞_𝑘=1𝑇=_∞𝑢^2𝑘,_𝑘=1𝑇_𝑘𝑒_𝑘,‖𝑇‖^2𝐿₂=^∞_𝑘=1𝑞=_∞𝑇^2𝑘,_𝑘=1𝑞_𝑘𝑒_𝑘,‖𝑞‖^2𝐿₂=^∞_𝑘=1𝑞_𝑘.(4.13)
When 𝑘→∞, 𝜆_𝑘→−∞. Let 𝛿 be small positive constant, and 𝑁=1/𝛿. There exists positive integer 𝑘 such that −𝑁≥𝜆_𝑗,𝑗≥𝑘+1.(4.14)
Introduce subspace 𝐸₁=span{𝑒₁,𝑒₂,…,𝑒_𝑘}⊂𝐿²(Ω). Let 𝐸₂ be an orthogonal subspace of 𝐸₁ in 𝐿²(Ω).
For all (𝑢,𝑇,𝑞)∈𝐻, we find that 𝑣₁𝑢=𝑣₁+𝑣₂,𝑇=𝑇₁+𝑇₂,𝑞=𝑞₁+𝑞₂,=^𝑘_𝑖=1𝑥_𝑖𝑒_𝑖∈𝐸₁×𝐸₁,𝑣₂=^∞_𝑗=_𝑘+1𝑇₁𝑥_𝑗𝑒_𝑗∈𝐸₂×𝐸₂,=^𝑘_𝑖=1𝑇_𝑖𝑒_𝑖∈𝐸₁,𝑇₂=^∞_𝑗=_𝑘+1𝑞₁𝑇_𝑗𝑒_𝑗∈𝐸₂,=^𝑘_𝑖=1𝑞_𝑖𝑒_𝑖∈𝐸₁,𝑞₂=^∞_𝑗=_𝑘+1𝑞_𝑗𝑒_𝑗∈𝐸₂.(4.15)
Let 𝑃_𝑖∶𝐿²(Ω)→𝐸_𝑖 be the orthogonal projection. Thanks to Definition 2.3, we will prove that for any bounded set 𝐵⊂𝐻 and 𝜀>0, there exists 𝑡₀>0 such that ‖‖‖‖𝑃𝑃₁𝑆(𝑡)𝐵‖‖_𝐻≤𝑀,∀𝑡>𝑡₀,𝑀isaconstant,(4.16)₂𝑆(𝑡)𝐵‖‖_𝐻≤𝜀,∀𝑡>𝑡₀,𝑢₀,𝑇₀,𝑞₀∈𝐵.(4.17)
From Step  1, 𝑆(𝑡) has an absorbing set 𝐵_𝑀. Then for any bounded set 𝐵⊂𝐻, there exists 𝑡_∗>0 such that 𝑆(𝑡)𝐵⊂𝐵_𝑀, for all 𝑡>𝑡_∗, which imply (4.16).
Multiply (1.1) by 𝑢 and integrate over (Ω). We obtain 𝑑𝑑𝑢𝑡,𝑢=𝑃^𝑟(Δ𝑢,𝑢)−𝑃_𝑟(𝜎𝑢,𝑢)+𝑃_𝑟𝑅𝑇+𝑅𝑞^→𝜅,𝑢−((𝑢⋅∇)𝑢,𝑢).(4.18) Then, ‖𝑢‖^2𝐿₂=𝑃_𝑟^𝑡0(Δ𝑢,𝑢)𝑑𝑡−𝑃_𝑟^𝑡0(𝜎𝑢,𝑢)𝑑𝑡+𝑃_𝑟^𝑡0𝑅𝑇+𝑅𝑞^→𝜅,𝑢𝑑𝑡+‖‖𝑢₀‖‖^2𝐿₂=𝜀₁𝑃_𝑟^𝑡0(Δ𝑢,𝑢)𝑑𝑡+1−𝜀₁𝑃_𝑟^𝑡0(Δ𝑢,𝑢)𝑑𝑡−𝑃_𝑟^𝑡0+𝑃_𝑟(𝜎𝑢,𝑢)𝑑𝑡^𝑡0𝑅𝑇+𝑅𝑞^→𝜅,𝑢𝑑𝑡+‖‖𝑢₀‖‖^2𝐿₂,(4.19) where 𝜀₁ is a constant which needs to be determined.
From (4.14), we find that (Δ𝑢,𝑢)=^∞_𝑖=1𝜆_𝑖𝑢^2𝑖=^𝑘_𝑖=1𝜆_𝑖𝑢^2𝑖+^∞_𝑗=_𝑘+1𝜆_𝑗𝑢^2𝑗≤𝜆^𝑘_𝑖=1𝑢^2𝑖−𝑁^∞_𝑖=_𝑘+1𝑢^2𝑗≤𝜆‖𝑢‖^2𝐿₂−𝑁‖‖𝜈₂‖‖^2𝐿₂,(4.20) where 𝜆=max{𝜆₁,𝜆₂,…,𝜆_𝑘}.
Thanks to (Δ𝑢,𝑢)=−∫_Ω|∇𝑢|²𝑑𝑥=−‖∇𝑢‖^2𝐿₂ and ‖𝑢‖_𝐿²≤𝐶‖∇𝑢‖_𝐿², it follows that (Δ𝑢,𝑢)=−‖∇𝑢‖^2𝐿₂≤−1𝐶²‖𝑢‖^2𝐿₂.(4.21)
We deduce from (4.11) the following: ‖𝑢‖^2𝐿₂+‖𝑇‖^2𝐿₂+‖𝑞‖^2𝐿₂≤𝑀²,𝑡≥𝑡_∗.(4.22)
Using (4.19)–(4.22), we find that ‖‖𝜈₂‖‖^2𝐿₂≤‖𝑢‖^2𝐿₂=𝜀₁𝑃_𝑟^𝑡0(Δ𝑢,𝑢)𝑑𝑡+1−𝜀₁𝑃_𝑟^𝑡0(Δ𝑢,𝑢)𝑑𝑡−𝑃_𝑟^𝑡0+𝑃_𝑟(𝜎𝑢,𝑢)𝑑𝑡^𝑡0𝑅𝑇+𝑅𝑞^→𝜅,𝑢𝑑𝑡+‖‖𝑢₀‖‖^2𝐿₂≤𝜀₁𝜆𝑃_𝑟^𝑡0‖𝑢‖^2𝐿₂𝑑𝑡−𝜀₁𝑁𝑃_𝑟^𝑡0‖‖𝜈₂‖‖^2𝐿₂𝑑𝑡−1−𝜀¹𝑃_𝑟𝐶²^𝑡0+𝑃‖𝑢‖^2𝐿₂𝑑𝑡_𝑟𝑅2^𝑡0‖𝑢‖^2𝐿₂𝑑𝑡+^𝑡0‖𝑇‖^2𝐿₂𝑑𝑡+𝑃^𝑟𝑅2^𝑡0‖𝑢‖^2𝐿₂𝑑𝑡+^𝑡0‖𝑞‖^2𝐿₂𝑑𝑡+‖‖𝑢₀‖‖^2𝐿₂.(4.23) Let 𝜀₁ satisfy 𝜀₁𝜆≤(1−𝜀₁)/𝐶² and 𝐾₁=𝑃_𝑟𝑅𝑀+𝑃_𝑟𝑅𝑀. Then, ‖‖𝜈₂‖‖^2𝐿₂≤−𝜀₁𝑁𝑃_𝑟^𝑡0‖‖𝜈₂‖‖^2𝐿₂𝑑𝑡+𝐾₁𝑡+‖‖𝑢₀‖‖^2𝐿₂,𝑡>𝑡_∗.(4.24) By the Gronwall inequality, we find that ‖‖𝜈₂‖‖^2𝐿₂≤𝑒^{−𝜀1𝑁𝑃𝑟𝑡}‖‖𝑢₀‖‖^2𝐿₂+𝐾¹𝜀₁𝑁𝑃_𝑟1−𝑒^{−𝜀1𝑁𝑃𝑟𝑡},𝑡>𝑡_∗.(4.25) Then, there exists 𝑡₁>𝑡_∗ satisfying 𝑒^{−𝜀1𝑁𝑃𝑟𝑡1}‖‖𝑢₀‖‖^2𝐿₂≤𝐾¹2𝜀₁𝑁𝑃_𝑟,𝐾¹𝜀₁𝑁𝑃_𝑟1−𝑒^{−𝜀1𝑁𝑃𝑟𝑡1}≤2𝜀₁𝐾𝑁¹𝑃_𝑟.(4.26) Since 𝛿=1/𝑁, for 𝑡>𝑡₁ it follows that ‖‖𝜈₂‖‖^2𝐿₂≤𝐾¹𝜀₁𝑁𝑃_𝑟=𝐾¹𝜀₁𝑃_𝑟𝛿.(4.27)
Multiply (1.2) by 𝑇 and integrate over (Ω). We obtain 𝑑𝑑𝑇𝑡,𝑇=(Δ𝑇,𝑇)+𝑢²,𝑇−((𝑢⋅∇)𝑇,𝑇)+(𝑄,𝑇).(4.28) Then, ‖𝑇‖^2𝐿₂=^𝑡0(Δ𝑇,𝑇)𝑑𝑡+^𝑡0𝑢₂,𝑇𝑑𝑡+‖‖𝑇₀‖‖^2𝐿₂=𝜀₂^𝑡0(Δ𝑇,𝑇)𝑑𝑡+1−𝜀₂^𝑡0(Δ𝑇,𝑇)𝑑𝑡+^𝑡0𝑢₂,𝑇𝑑𝑡+‖‖𝑇₀‖‖^2𝐿₂,(4.29) where 𝜀₂ is a constant which needs to be determined.
From (4.14), we find that (Δ𝑇,𝑇)=^∞_𝑖=1𝜆_𝑖𝑇^2𝑖=^𝑘_𝑖=1𝜆_𝑖𝑇^2𝑖+^∞_𝑗=_𝑘+1𝜆_𝑗𝑇^2𝑗≤𝜆^𝑘_𝑖=1𝑇^2𝑖−𝑁^∞_𝑗=_𝑘+1𝑇^2𝑗≤𝜆‖𝑇‖^2𝐿₂−𝑁‖‖𝑇₂‖‖^2𝐿₂,(4.30) where 𝜆=max{𝜆₁,𝜆₂,⋯,𝜆_𝑘}.
Since (Δ𝑇,𝑇)=−∫_Ω|∇𝑇|²𝑑𝑥=−‖∇𝑇‖^2𝐿₂ and ‖𝑇‖_𝐿²≤𝐶‖∇𝑇‖_𝐿², it follows that (Δ𝑇,𝑇)=−‖∇𝑇‖^2𝐿₂≤−1𝐶²‖𝑇‖^2𝐿₂.(4.31)
Using (4.22) and (4.29)–(4.31), we find that ‖‖𝑇₂‖‖^2𝐿₂≤‖𝑇‖^2𝐿₂=𝜀₂^𝑡0(Δ𝑇,𝑇)𝑑𝑡+1−𝜀₂^𝑡0(Δ𝑇,𝑇)𝑑𝑡+^𝑡0𝑢₂,𝑇𝑑𝑡+‖‖𝑇₀‖‖^2𝐿₂≤𝜀₂𝜆^𝑡0‖𝑇‖^2𝐿₂𝑑𝑡−𝜀₂𝑁^𝑡0‖‖𝑇₂‖‖^2𝐿₂𝑑𝑡−1−𝜀²𝐶²^𝑡0+1‖𝑇‖^2𝐿₂𝑑𝑡2^𝑡0‖𝑢‖^2𝐿₂𝑑𝑡+^𝑡0‖𝑇‖^2𝐿₂𝑑𝑡+‖‖𝑇₀‖‖^2𝐿₂.(4.32) Let 𝜀₂ satisfy 𝜀₂𝜆≤(1−𝜀₂)/𝐶². Then ‖‖𝑇₂‖‖^2𝐿₂≤−𝜀₂𝑁^𝑡0‖‖𝑇₂‖‖^2𝐿₂𝑑𝑡+𝑀𝑡+‖‖𝑇₀‖‖^2𝐿₂,𝑡>𝑡_∗.(4.33) By the Gronwall inequality, we find that ‖‖𝑇₂‖‖^2𝐿₂≤𝑒^{−𝜀2𝑁𝑡}‖‖𝑇₀‖‖^2𝐿₂+𝑀𝜀₂𝑁1−𝑒^{−𝜀2𝑁𝑡},𝑡>𝑡_∗.(4.34) Then, there exists 𝑡₂>𝑡_∗ satisfying 𝑒^{−𝜀2𝑁𝑡2}‖‖𝑇₀‖‖^2𝐿₂≤𝑀2𝜀₂𝑁,𝜀𝑀₂𝑁1−𝑒^{−𝜀2𝑁𝑡2}≤2𝜀𝑀₂𝑁.(4.35) Since 𝛿=1/𝑁, for 𝑡>𝑡₂, it follows that ‖‖𝑇₂‖‖^2𝐿₂≤𝑀𝜀₂𝑁=𝑀𝜀₂𝛿.(4.36)
Multiply (1.3) by 𝑞 and integrate over (Ω). We obtain 𝑑𝑑𝑞𝑡,𝑞=𝐿^𝑒(Δ𝑞,𝑞)+𝑢₂,𝑞−((𝑢⋅∇)𝑞,𝑞)+(𝐺,𝑞).(4.37) Then, ‖𝑞‖^2𝐿₂=𝐿_𝑒^𝑡0(Δ𝑞,𝑞)𝑑𝑡+^𝑡0𝑢₂,𝑞𝑑𝑡+‖‖𝑞₀‖‖^2𝐿₂=𝐿_𝑒𝜀₃^𝑡0(Δ𝑞,𝑞)𝑑𝑡+1−𝜀₃^𝑡0(Δ𝑞,𝑞)𝑑𝑡+^𝑡0𝑢₂,𝑞𝑑𝑡+‖‖𝑞₀‖‖^2𝐿₂,(4.38) where 𝜀₃ is a constant which needs to be determined.
From (4.14), we find that (Δ𝑞,𝑞)=^∞_𝑖=1𝜆_𝑖𝑞^2𝑖=^𝑘_𝑖=1𝜆_𝑖𝑞^2𝑖+^∞_𝑗=_𝑘+1𝜆_𝑗𝑞^2𝑗≤𝜆^𝑘_𝑖=1𝑞^2𝑖−𝑁^∞_𝑗=_𝑘+1𝑞^2𝑗≤𝜆‖𝑞‖^2𝐿₂−𝑁‖‖𝑞₂‖‖^2𝐿₂,(4.39) where 𝜆=max{𝜆₁,𝜆₂,…,𝜆_𝑘}.
Since (Δ𝑞,𝑞)=−∫_Ω|∇𝑞|²𝑑𝑥=−‖∇𝑞‖^2𝐿₂ and ‖𝑞‖_𝐿²≤𝐶‖∇𝑞‖_𝐿², we see that (Δ𝑞,𝑞)=−‖∇𝑞‖^2𝐿₂≤−1𝐶²‖𝑞‖^2𝐿₂.(4.40)
Using (4.22) and (4.38)–(4.40), we obtain ‖‖𝑞₂‖‖^2𝐿₂≤‖𝑞‖^2𝐿₂=𝜀₃𝐿_𝑒^𝑡0(Δ𝑞,𝑞)𝑑𝑡+1−𝜀₃𝐿_𝑒^𝑡0(Δ𝑞,𝑞)𝑑𝑡+^𝑡0𝑢₂,𝑞𝑑𝑡+‖‖𝑞₀‖‖^2𝐿₂≤𝜀₃𝜆𝐿_𝑒^𝑡0‖𝑞‖^2𝐿₂𝑑𝑡−𝜀₃𝑁𝐿_𝑒^𝑡0‖‖𝑞₂‖‖^2𝐿₂𝑑𝑡−1−𝜀³𝐶²𝐿_𝑒^𝑡0+1‖𝑞‖^2𝐿₂𝑑𝑡2^𝑡0‖𝑢‖^2𝐿₂𝑑𝑡+^𝑡0‖𝑞‖^2𝐿₂𝑑𝑡+‖‖𝑞₀‖‖^2𝐿₂.(4.41) Let 𝜀₃ satisfy 𝜀₃𝜆≤(1−𝜀₃)/𝐶². Then, ‖‖𝑞₂‖‖^2𝐿₂≤−𝜀₃𝑁𝐿_𝑒^𝑡0‖‖𝑞₂‖‖^2𝐿₂𝑑𝑡+𝑀𝑡+‖‖𝑞₀‖‖^2𝐿₂,𝑡>𝑡_∗.(4.42) By the Gronwall inequality, we find that ‖‖𝑞₂‖‖^2𝐿₂≤𝑒^{−𝜀3𝐿𝑒𝑁𝑡}‖‖𝑞₀‖‖^2𝐿₂+𝑀𝜀₃𝐿_𝑒𝑁1−𝑒^{−𝜀3𝐿𝑒𝑁𝑡},𝑡>𝑡_∗.(4.43) Then, there exists 𝑡₃>𝑡_∗ satisfying 𝑒^{−𝜀3𝐿𝑒𝑁𝑡3}‖‖𝑞₀‖‖^2𝐿₂≤𝑀2𝜀₃𝐿_𝑒𝑁,𝜀₃𝐿𝑀_𝑒𝑁1−𝑒^{−𝜀3𝐿𝑒𝑁𝑡3}≤2𝜀₃𝑀𝐿_𝑒𝑁.(4.44) Since 𝛿=1/𝑁, for 𝑡>𝑡₃, it follows that ‖‖𝑞₂‖‖^2𝐿₂≤𝑀𝜀₃𝐿_𝑒𝑁=𝜀𝑀₃𝐿_𝑒𝛿.(4.45)
From (4.27); (4.36) and (4.45) for all 𝛿>0 there exists 𝑡₀=max{𝑡₁,𝑡₂,𝑡₃} such that when 𝑡>𝑡₀, it follows that ‖‖𝑃₂𝑆(𝑡)𝑢₀,𝑇₀,𝑞₀‖‖^2𝐻=‖‖𝑣₂‖‖^2𝐿₂+‖‖𝑇₂‖‖^2𝐿₂+‖‖𝑞₂‖‖^2𝐿₂≤𝐾¹𝜀₁𝑃_𝑟+𝑀𝜀₂+𝑀𝜀₃𝐿_𝑒𝛿,(4.46) which imply (4.17). From Lemma 2.4, (1.1)–(1.7) have a global attractor in 𝐿²(Ω,𝑅⁴).

Acknowledgments

This work was supported by national natural science foundation of China (no. 11271271) and the NSF of Sichuan Education Department of China (no. 11ZA102).