# Global Attractor of Atmospheric Circulation Equations with Humidity Effect

Authors: Hong Luo
Journal: Abstract and Applied Analysis
 (2012)

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## 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 inL2(Ω,R4). Secondly, by using C-condition, it is obtained that atmospheric
circulation equations have a global attractor in L2(Ω,R4).


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## Body

## 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)):
(1.1)∂u∂t=Pr(Δu-∇p-σu)+Pr(RT-R~q)κ→-(u⋅∇)u,(1.2)∂T∂t=ΔT+u2-(u⋅∇)T+Q,(1.3)∂q∂t=LeΔq+u2-(u⋅∇)q+G,(1.4)div⁡u=0,
where Pr, R, R~, and Le 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:
(1.5)σ=(σ0ωωσ1).The problems (1.1)–(1.4) are supplemented with the following Dirichlet boundary condition at x2=0,1 and periodic condition for x1:
(1.6)(u,T,q)=0,x2=0,1,(u,T,q)(0,x2)=(u,T,q)(2π,x2),
and initial value conditions
(1.7)(u,T,q)=(u0,T0,q0),t=0.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 Section2, 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.∥
·
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X denote norm of the space X; C and Ci are variable constants. Let H={ϕ=(u,T,q)∈L2(Ω,R4)∣ϕ satisfy (1.4), (1.6)}, and H1={ϕ=(u,T,q)∈H1(Ω,R4)∣ϕ satisfy (1.4), (1.6)}.


## 2. Preliminaries

LetX and X1 be two Banach spaces, X1⊂X a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on X, given by
(2.1)dudt=Lu+G(u),u(x,0)=u0,
where u(t) is an unknown function, L:X1→X a linear operator, and G:X1→X a nonlinear operator.A family of operatorsS(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
(2.2)u(t,u0)=S(t)u0.Next, we introduce the concepts and definitions of invariant sets, global attractors, and ω-limit sets for the semigroup S(t).Definition 2.1.
LetS(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,
(2.3)inf⁡v∈Σ⁡‖S(t)u0-v‖X⟶0,ast⟶∞.
In this case, we say thatΣ attracts U. Particularly, if Σ attracts any bounded set of X, Σ is called a global attractor of S(t) in X.
For a setD⊂X, we define the ω-limit set of D as follows:
(2.4)ω(D)=⋂s≥0⋃t≥sS(t)D¯,
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.
LetS(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 tA≥0 such that S(t)u0∈B,  for  all  u0∈A and t>tA;(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 theω-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 thatS(t):X→X satisfies C-condition, if for any bounded set B⊂X and ε>0, there exist tB>0 and a finite dimensional subspace X1⊂X such that {PS(t)B} is bounded, and
(2.5)‖(I-P)S(t)u‖X<ε,∀t≥tB,u∈B,
where P:X→X1 is a projection.Lemma 2.4 (see [19]).
LetS(t):X→X (t≥0) be a dynamical systems. If the following conditions are satisfied:(1)
there exists a bounded absorbing setB⊂X;(2)
S(t) satisfies C-condition,
thenS(t) has a global attractor in X.From Linear elliptic equation theory, one has the following.Lemma 2.5.
The eigenvalue equation:(2.6)-ΔT(x1,x2)=βT(x1,x2),(x1,x2)∈(0,2π)×(0,1),T=0,x2=0,1,T(0,x2)=T(2π,x2)
has eigenvalue {βk}k=1∞, and
(2.7)0<β1≤β2≤⋯,βk⟶∞,ask⟶∞.


## 3. Uniqueness of Global Solution

Theorem 3.1.
Ifσ~β1≥max{(R+1)2,((R~-1)2/Le)}, 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
(3.1)1Pr∫Ω‍uvdx+∫Ω‍TSdx+∫Ω‍qzdx=∫0t‍∫Ω‍[-∇u∇v-σuv+(RT-R~q)v2-1Pr(u⋅∇)uv-∇T∇S+u2S-(u⋅∇)TS+QS-Le∇q∇z+u2z-(u⋅∇)qz+Gz-∇u∇v-σuv+(RT-R~q)v2]dxdt+1Pr∫Ω‍u0vdx+∫Ω‍T0Sdx+∫Ω‍q0zdx.
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,
(3.2)1Pr∫Ω‍uvdx+∫Ω‍TSdx+∫Ω‍qzdx=∫0t‍∫Ω‍[-∇u∇v-σuv+(RT-R~q)v2+1Pr(u2⋅∇)u2v-1Pr(u1⋅∇)u1v-∇T∇S+u2S+(u2⋅∇)T2S-(u1⋅∇)T1S-Le∇q∇z+u2z+(u2⋅∇)q2z-(u1⋅∇)q1z-∇u∇v-σuv+(RT-R~q)v2]dxdt.
Let(v,S,z)=(u,T,q). We obtain from (3.2) the following:
(3.3)1Pr∫Ω‍|u|2dx+∫Ω‍|T|2dx+∫Ω‍|q|2dx=∫0t‍∫Ω‍[-|∇u|2-σu⋅u+(RT-R~q)u2+1Pr(u2⋅∇)u2u-1Pr(u1⋅∇)u1u-|∇T|2+u2T+(u2⋅∇)T2T-(u1⋅∇)T1T-Le|∇q|2+u2q+(u2⋅∇)q2q-(u1⋅∇)q1q1Pr]dxdt≤∫0t‍∫Ω‍[-|∇u|2-|∇T|2-Le|∇q|2]dxdt+∫0t‍∫Ω‍[-σ~|u|2+(R+1)Tu2-(R~-1)qu2]dxdt+∫0t‍∫Ω‍[1Pr(u⋅∇)u2u+(u⋅∇)T2T+(u⋅∇)q2q]dxdt≤∫0t‍∫Ω‍[-|∇u|2-|∇T|2-Le|∇q|2]dxdt+∫0t‍∫Ω‍[-σ~|u|2+σ~|u2|2+(R+1)22σ~|T|2+(R~-1)22σ~|q|2]dxdt+∫0t‍[2Pr‖u‖L2‖∇u2‖L2‖∇u‖L2+2‖u‖L21/2‖∇u‖L21/2‖∇T2‖L2‖T‖L21/2‖∇T‖L21/2+2‖u‖L21/2‖∇u‖L21/2‖∇q2‖‖q‖L21/2‖∇q‖L21/22Pr]dt≤∫0t‍∫Ω‍[-|∇u|2-12|∇T|2-Le2|∇q|2]dxdt+∫0t‍[2Pr‖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‖L22Pr]dt≤∫0t‍∫Ω‍[-|∇u|2-12|∇T|2-Le2|∇q|2]dxdt+∫0t‍[‖∇u‖L22+3Pr2‖u‖L22‖∇u2‖L22+3‖u‖L22‖∇T2‖L22+3‖u‖L22‖∇q2‖L22+12‖∇T‖L22+‖∇T2‖L22‖T‖L22+Le2‖∇q‖L22+2Le‖q‖L22|∇q2|L22]dt≤∫0t‍[3Pr2‖u‖L22‖∇u2‖L22+3‖u‖L22‖∇T2‖L22+3‖u‖L22‖∇q2‖L22+‖∇T2‖L22‖T‖L22+2Le‖q‖L22‖∇q2‖L22]dt.
Then,
(3.4)‖u‖L22+‖T‖L22+‖q‖L22≤C∫0t‍[(‖u‖L22+‖T‖L22+‖q‖L22)(‖∇u2‖L22+‖∇T2‖L22+‖∇q2‖L22)]dt≤C∫0t‍(‖u‖L22+‖T‖L22+‖q‖L22)dt.
By using the Gronwall inequality, it follows that(3.5)‖u‖L22+‖T‖L22+‖q‖L22≤0,
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.
Ifσ~β1≥max{(R+1)2,(R~-1)2/Le}, 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 Lemma2.4, 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:
(4.1)1Pr∫Ω‍dudtudx=∫Ω‍[Δu-∇p-σu+(RT-R~q)κ~-1Pr(u⋅∇)u]udx.
Then,
(4.2)12Prddt∫Ω‍u2dx=∫Ω‍[-|∇u|2-σu⋅u+(RT-R~q)u2]dx.
Multiply (1.2) by T and integrate the product in Ω:
(4.3)∫Ω‍dTdtTdx=∫Ω‍[ΔT+u2-(u⋅∇)T+Q]Tdx.
Then,
(4.4)12ddt∫Ω‍T2dx=∫Ω‍(-|∇T|2+u2Tdx+QT)dx.
Multiply (1.3) by q and integrate the product in Ω:
(4.5)∫Ω‍dqdtqdx=∫Ω‍[LeΔq+u2-(u⋅∇)q+Q]qdx.
Then,
(4.6)12ddt∫Ω‍q2dx=∫Ω‍(-Le|∇q|2+u2qdx+Gq)dx.
We deduce from (4.2)–(4.6) the following:
(4.7)12ddt∫Ω‍(1Pru2+T2+q2)dx=∫Ω‍[-|∇u|2-|∇T|2-Le|∇q|2-σu⋅u+(R+1)Tu2-(R~-1)qu2+QT+Gq|∇u|2]dx≤∫Ω‍[1ε-|∇u|2-|∇T|2-Le|∇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.
Let ε>0 be appropriate small such that
(4.8)ddt∫Ω‍(u2+T2+q2)dx≤C1∫Ω‍[-|∇u|2-|∇T|2-|∇q|2]dx+C2∫Ω‍(|Q|2+|G|2)dx.
Then,(4.9)ddt∫Ω‍(u2+T2+q2)dx≤-C3∫Ω‍(|u|2+|T|2+|q|2)dx+C4.
Applying the Gronwall inequality, it follows that(4.10)‖(u,T,q)(t)‖L22≤‖(u,T,q)(0)‖L22e-C3t+C4C3(1-e-C3t).
Then, whenM2>C4/C3, for any (u0,T0,q0)∈B, here B is a bounded in H, there exists t*>0 such that
(4.11)S(t)(u0,T0,q0)=(u(t),T(t),q(t))∈BM,t>t*,
where BM is a ball in H, at 0 of radius M. Thus, (1.1)–(1.7) have an absorbing BM in H.
Step  2.
C-condition is satisfied.
The eigenvalue equation:(4.12)Δu=λu,u(x1,0)=u(x2,0)=0,u(0,x2)=u(2π,x2),div⁡u=0
has eigenvalues λ1,λ2,…,λk,… and eigenvector {ek∣k=1,2,3,…}, and λ1≥λ2≥⋯≥λk≥⋯. If k→∞, then λk→-∞. {ek∣k=1,2,3,…} constitutes an orthogonal base of L2(Ω).
For all(u,T,q)∈H, we have
(4.13)u=∑k=1∞‍ukek,‖u‖L22=∑k=1∞‍uk2,T=∑k=1∞‍Tkek,‖T‖L22=∑k=1∞‍Tk2,q=∑k=1∞‍qkek,‖q‖L22=∑k=1∞‍qk.
Whenk→∞, λk→-∞. Let δ be small positive constant, and N=1/δ. There exists positive integer k such that
(4.14)-N≥λj,j≥k+1.
Introduce subspaceE1=span{e1,e2,…,ek}⊂L2(Ω). Let E2 be an orthogonal subspace of E1 in L2(Ω).
For all(u,T,q)∈H, we find that
(4.15)u=v1+v2,T=T1+T2,q=q1+q2,v1=∑i=1k‍xiei∈E1×E1,v2=∑j=k+1∞‍xjej∈E2×E2,T1=∑i=1k‍Tiei∈E1,T2=∑j=k+1∞‍Tjej∈E2,q1=∑i=1k‍qiei∈E1,q2=∑j=k+1∞‍qjej∈E2.
LetPi:L2(Ω)→Ei be the orthogonal projection. Thanks to Definition 2.3, we will prove that for any bounded set B⊂H and ε>0, there exists t0>0 such that
(4.16)‖P1S(t)B‖H≤M,∀t>t0,Misaconstant,(4.17)‖P2S(t)B‖H≤ε,∀t>t0,(u0,T0,q0)∈B.
From Step  1,S(t) has an absorbing set BM. Then for any bounded set B⊂H, there exists t*>0 such that S(t)B⊂BM, for all t>t*, which imply (4.16).
Multiply (1.1) by u and integrate over (Ω). We obtain
(4.18)(dudt,u)=Pr(Δu,u)-Pr(σu,u)+Pr((RT+R~q)κ→,u)-((u⋅∇)u,u).
Then,
(4.19)‖u‖L22=Pr∫0t‍(Δu,u)dt-Pr∫0t‍(σu,u)dt+Pr∫0t‍((RT+R~q)κ→,u)dt+‖u0‖L22=ε1Pr∫0t‍(Δu,u)dt+(1-ε1)Pr∫0t‍(Δu,u)dt-Pr∫0t‍(σu,u)dt+Pr∫0t‍((RT+R~q)κ→,u)dt+‖u0‖L22,
where ε1 is a constant which needs to be determined.
From (4.14), we find that
(4.20)(Δu,u)=∑i=1∞‍λiui2=∑i=1k‍λiui2+∑j=k+1∞‍λjuj2≤λ∑i=1k‍ui2-N∑i=k+1∞‍uj2≤λ‖u‖L22-N‖ν2‖L22,
where λ=max{λ1,λ2,…,λk}.
Thanks to(Δu,u)=-∫Ω‍|∇u|2dx=-∥∇u∥L22 and ∥u∥L2≤C∥∇u∥L2, it follows that
(4.21)(Δu,u)=-‖∇u‖L22≤-1C2‖u‖L22.
We deduce from (4.11) the following:
(4.22)‖u‖L22+‖T‖L22+‖q‖L22≤M2,t≥t*.
Using (4.19)–(4.22), we find that
(4.23)‖ν2‖L22≤‖u‖L22=ε1Pr∫0t‍(Δu,u)dt+(1-ε1)Pr∫0t‍(Δu,u)dt-Pr∫0t‍(σu,u)dt+Pr∫0t‍((RT+R~q)κ→,u)dt+‖u0‖L22≤ε1λPr∫0t‍‖u‖L22dt-ε1NPr∫0t‍‖ν2‖L22dt-(1-ε1)PrC2∫0t‍‖u‖L22dt+PrR2(∫0t‍‖u‖L22dt+∫0t‍‖T‖L22dt)+PrR~2(∫0t‍‖u‖L22dt+∫0t‍‖q‖L22dt)+‖u0‖L22.
Let ε1 satisfy ε1λ≤(1-ε1)/C2 and K1=PrRM+PrR~M. Then,
(4.24)‖ν2‖L22≤-ε1NPr∫0t‍‖ν2‖L22dt+K1t+‖u0‖L22,t>t*.
By the Gronwall inequality, we find that
(4.25)‖ν2‖L22≤e-ε1NPrt‖u0‖L22+K1ε1NPr(1-e-ε1NPrt),t>t*.
Then, there exists t1>t* satisfying
(4.26)e-ε1NPrt1‖u0‖L22≤K12ε1NPr,K1ε1NPr(1-e-ε1NPrt1)≤K12ε1NPr.
Since δ=1/N, for t>t1 it follows that
(4.27)‖ν2‖L22≤K1ε1NPr=K1ε1Prδ.
Multiply (1.2) by T and integrate over (Ω). We obtain
(4.28)(dTdt,T)=(ΔT,T)+(u2,T)-((u⋅∇)T,T)+(Q,T).
Then,
(4.29)‖T‖L22=∫0t‍(ΔT,T)dt+∫0t‍(u2,T)dt+‖T0‖L22=ε2∫0t‍(ΔT,T)dt+(1-ε2)∫0t‍(ΔT,T)dt+∫0t‍(u2,T)dt+‖T0‖L22,
where ε2 is a constant which needs to be determined.
From (4.14), we find that
(4.30)(ΔT,T)=∑i=1∞‍λiTi2=∑i=1k‍λiTi2+∑j=k+1∞‍λjTj2≤λ∑i=1k‍Ti2-N∑j=k+1∞‍Tj2≤λ‖T‖L22-N‖T2‖L22,
where λ=max{λ1,λ2,⋯,λk}.
Since(ΔT,T)=-∫Ω‍|∇T|2dx=-∥∇T∥L22 and ∥T∥L2≤C∥∇T∥L2, it follows that
(4.31)(ΔT,T)=-‖∇T‖L22≤-1C2‖T‖L22.
Using (4.22) and (4.29)–(4.31), we find that
(4.32)‖T2‖L22≤‖T‖L22=ε2∫0t‍(ΔT,T)dt+(1-ε2)∫0t‍(ΔT,T)dt+∫0t‍(u2,T)dt+‖T0‖L22≤ε2λ∫0t‍‖T‖L22dt-ε2N∫0t‍‖T2‖L22dt-(1-ε2)C2∫0t‍‖T‖L22dt+12(∫0t‍‖u‖L22dt+∫0t‍‖T‖L22dt)+‖T0‖L22.
Let ε2 satisfy ε2λ≤(1-ε2)/C2. Then
(4.33)‖T2‖L22≤-ε2N∫0t‍‖T2‖L22dt+Mt+‖T0‖L22,t>t*.
By the Gronwall inequality, we find that
(4.34)‖T2‖L22≤e-ε2Nt‖T0‖L22+Mε2N(1-e-ε2Nt),t>t*.
Then, there exists t2>t* satisfying
(4.35)e-ε2Nt2‖T0‖L22≤M2ε2N,Mε2N(1-e-ε2Nt2)≤M2ε2N.
Since δ=1/N, for t>t2, it follows that
(4.36)‖T2‖L22≤Mε2N=Mε2δ.
Multiply (1.3) by q and integrate over (Ω). We obtain
(4.37)(dqdt,q)=Le(Δq,q)+(u2,q)-((u⋅∇)q,q)+(G,q).
Then,
(4.38)‖q‖L22=Le∫0t‍(Δq,q)dt+∫0t‍(u2,q)dt+‖q0‖L22=Leε3∫0t‍(Δq,q)dt+(1-ε3)∫0t‍(Δq,q)dt+∫0t‍(u2,q)dt+‖q0‖L22,
where ε3 is a constant which needs to be determined.
From (4.14), we find that
(4.39)(Δq,q)=∑i=1∞‍λiqi2=∑i=1k‍λiqi2+∑j=k+1∞‍λjqj2≤λ∑i=1k‍qi2-N∑j=k+1∞‍qj2≤λ‖q‖L22-N‖q2‖L22,
where λ=max{λ1,λ2,…,λk}.
Since(Δq,q)=-∫Ω‍|∇q|2dx=-∥∇q∥L22 and ∥q∥L2≤C∥∇q∥L2, we see that
(4.40)(Δq,q)=-‖∇q‖L22≤-1C2‖q‖L22.
Using (4.22) and (4.38)–(4.40), we obtain
(4.41)‖q2‖L22≤‖q‖L22=ε3Le∫0t‍(Δq,q)dt+(1-ε3)Le∫0t‍(Δq,q)dt+∫0t‍(u2,q)dt+‖q0‖L22≤ε3λLe∫0t‍‖q‖L22dt-ε3NLe∫0t‍‖q2‖L22dt-1-ε3C2Le∫0t‍‖q‖L22dt+12(∫0t‍‖u‖L22dt+∫0t‍‖q‖L22dt)+‖q0‖L22.
Let ε3 satisfy ε3λ≤(1-ε3)/C2. Then,
(4.42)‖q2‖L22≤-ε3NLe∫0t‍‖q2‖L22dt+Mt+‖q0‖L22,t>t*.
By the Gronwall inequality, we find that
(4.43)‖q2‖L22≤e-ε3LeNt‖q0‖L22+Mε3LeN(1-e-ε3LeNt),t>t*.
Then, there exists t3>t* satisfying
(4.44)e-ε3LeNt3‖q0‖L22≤M2ε3LeN,Mε3LeN(1-e-ε3LeNt3)≤M2ε3LeN.
Since δ=1/N, for t>t3, it follows that
(4.45)‖q2‖L22≤Mε3LeN=Mε3Leδ.
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
(4.46)‖P2S(t)(u0,T0,q0)‖H2=‖v2‖L22+‖T2‖L22+‖q2‖L22≤(K1ε1Pr+Mε2+Mε3Le)δ,
which imply (4.17). From Lemma 2.4, (1.1)–(1.7) have a global attractor in L2(Ω,R4).



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*Source: 172956-2012-08-30.xml*