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[1] | Wenlong Sun . The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28(3): 1343-1356. doi: 10.3934/era.2020071 |
[2] | Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding . The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28(4): 1395-1418. doi: 10.3934/era.2020074 |
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[4] | Xiaojie Yang, Hui Liu, Haiyun Deng, Chengfeng Sun . Pullback $ \mathcal{D} $-attractors of the three-dimensional non-autonomous micropolar equations with damping. Electronic Research Archive, 2022, 30(1): 314-334. doi: 10.3934/era.2022017 |
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[7] | Keqin Su, Rong Yang . Pullback dynamics and robustness for the 3D Navier-Stokes-Voigt equations with memory. Electronic Research Archive, 2023, 31(2): 928-946. doi: 10.3934/era.2023046 |
[8] | Dingshi Li, Xuemin Wang . Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains. Electronic Research Archive, 2021, 29(2): 1969-1990. doi: 10.3934/era.2020100 |
[9] | Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang . Pullback dynamics of a 3D modified Navier-Stokes equations with double delays. Electronic Research Archive, 2021, 29(6): 4137-4157. doi: 10.3934/era.2021076 |
[10] | Lingrui Zhang, Xue-zhi Li, Keqin Su . Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D. Electronic Research Archive, 2023, 31(11): 6881-6897. doi: 10.3934/era.2023348 |
The micropolar fluid model is a qualitative generalization of the well-known Navier-Stokes model in the sense that it takes into account the microstructure of fluid [7]. The model was first derived in 1966 by Eringen [4] to describe the motion of a class of non-Newtonian fluid with micro-rotational effects and inertia involved. It can be expressed by the following equations:
{∂u∂t−(ν+νr)Δu−2νrrotω+(u⋅∇)u+∇p=f,∂ω∂t−(ca+cd)Δω+4νrω+(u⋅∇)ω−(c0+cd−ca)∇divω−2νrrotu=˜f,∇⋅u=0, | (1) |
where
Micropolar fluid models play an important role in the fields of applied and computational mathematics. There is a rich literature on the mathematical theory of micropolar fluid model. Particularly, the existence, uniqueness and regularity of solutions for the micropolar fluid flows have been investigated in [6]. Extensive studies on long time behavior of solutions for the micropolar fluid flows have also been done. For example, in the case of 2D bounded domains: Łukaszewicz [7] established the existence of
As we know, in the real world, delay terms appear naturally, for instance as effects in wind tunnel experiments (see [10]). Also the delay situations may occur when we want to control the system via applying a force which considers not only the present state but also the history state of the system. The delay of partial differential equations (PDE) includes finite delays (constant, variable, distributed, etc) and infinite delays. Different types of delays need to be treated by different approaches.
In this paper, we consider the situation that the velocity component
There exists λ1>0 such that λ1‖φ‖2L2(Ω)≤‖∇φ‖2L2(Ω), ∀φ∈H10(Ω). | (2) |
Then we discuss the following 2D non-autonomous incompressible micropolar fluid flows with finite delay:
{∂u∂t−(ν+νr)Δu−2νr∇×ω+(u⋅∇)u+∇p=f(t,x)+g(t,ut),in(τ,+∞)×Ω,∂ω∂t−ˉαΔω+4νrω−2νr∇×u+(u⋅∇)ω=˜f(t,x)+˜g(t,ωt),in(τ,+∞)×Ω,∇⋅u=0,in(τ,+∞)×Ω, | (3) |
where
ut(s):=u(t+s), ωt(s):=ω(t+s), ∀t≥τ, s∈(−h,0). |
where
∇×u:=∂u2∂x1−∂u1∂x2,∇×ω:=(∂ω∂x2,−∂ω∂x1). |
To complete the formulation of the initial boundary value problem to system (3), we give the following initial boundary conditions:
(u(τ),ω(τ))=(uin,ωin), (uτ(s),ωτ(s))=(ϕin1(s),ϕin2(s)), s∈(−h,0), | (4) |
u=0, ω=0,on(τ,+∞)×Γ. | (5) |
For problem (3)-(5), Sun and Liu established the existence of pullback attractor in [16], recently.
The first purpose of this work is to investigate the boundedness of the pullback attractor obtained in [16]. We remark that García-Luengo, Marín-Rubio and Real [5] proved the
The second purpose of this work is to investigate the upper semicontinuity of the pullback attractor with respect to the domain
Throughout this paper, we denote the usual Lebesgue space and Sobolev space by
V:=V(Ω):={φ∈C∞0(Ω)×C∞0(Ω)|φ=(φ1,φ2),∇⋅φ=0},ˆV:=ˆV(Ω):=V×C∞0(Ω),H:=H(Ω):=closureofVinL2(Ω)×L2(Ω),withnorm‖⋅‖H anddualspaceH∗,V:=V(Ω):=closureofVinH1(Ω)×H1(Ω),withnorm‖⋅‖V anddualspaceV∗,ˆH:=ˆH(Ω):=closureofˆVinL2(Ω)×L2(Ω)×L2(Ω),withnorm‖⋅‖ˆH anddualspaceˆH∗,ˆV:=ˆV(Ω):=closureofˆVinH1(Ω)×H1(Ω)×H1(Ω),withnorm‖⋅‖ˆV anddualspaceˆV∗. |
Lp(I;X):=space of strongly measurable functions on the closed interval I, with values in the Banach space X, endowed with norm‖φ‖Lp(I;X):=(∫I‖φ‖pXdt)1/p, for 1≤p<∞,C(I;X):=space of continuous functions on the interval I, with values in the Banach space X, endowed with the usual norm, L2loc(I;X):=space of locally square integrable functions on the interval I, with values in the Banach space X, endowed with the usual norm,distM(X,Y)− the Hausdorff semidistance between X⊆MandY⊆MdefinedbydistM(X,Y)=supx∈Xinfy∈YdistM(x,y). |
Following the above notations, we additionally denote
L2ˆH:=L2(−h,0;ˆH),L2ˆV:=L2(−h,0;ˆV), E2ˆH:=ˆH×L2ˆH, E2ˆV:=ˆV×L2ˆV, E2ˆH×L2ˆV:=ˆH×L2ˆV. |
The norm
‖(w,v)‖E2ˆH:=(‖w‖2ˆv+‖v‖2L2ˆH)1/2,‖(w,v)‖E2ˆV:=(‖w‖2+‖v‖2L2ˆV)1/2,‖(w,v)‖E2ˆH×L2ˆV:=(‖w‖2+‖v‖2L2ˆV)1/2. |
The rest of this paper is organized as follows. In section 2, we make some preliminaries. In section 3, we investigate the boundedness of the pullback attractor. In section 4, we prove the upper semicontinuity of the pullback attractor with respect to the domains.
In this section, for the sake of discussion, we first introduce some useful operators and put problem (3)-(5) into an abstract form. Then we recall some important known results about the non-autonomous micropolar fluid flows.
To begin with, we define the operators
{⟨Aw,φ⟩:=(ν+νr)(∇u,∇Φ)+ˉα(∇ω,∇φ3), ∀w=(u,ω),φ=(Φ,φ3)∈ˆV,⟨B(u,w),φ⟩:=((u⋅∇)w,φ), ∀u∈V,w=(u,ω)∈ˆV,∀φ∈ˆV,N(w):=(−2νr∇×ω,−2νr∇×u+4νrω), ∀w=(u,ω)∈ˆV. | (6) |
What follows are some useful estimates and properties for the operators
Lemma 2.1.
⟨B(u,ψ),φ⟩=−⟨B(u,φ),ψ⟩, ∀u∈V, ∀ψ∈ˆV, ∀φ∈ˆV. | (7) |
Lemma 2.2.
c1⟨Aw,w⟩≤‖w‖2ˆV≤c2⟨Aw,w⟩, ∀w∈ˆV. | (8) |
|⟨B(u,ψ),φ⟩|≤{α0‖u‖12‖∇u‖12‖φ‖12‖∇φ‖12‖∇ψ‖,α0‖u‖12‖∇u‖12‖ψ‖12‖∇ψ‖12‖∇φ‖. | (9) |
Moreover, if
|⟨B(u,ψ),Aφ⟩|≤α0‖u‖12‖∇u‖12‖∇ψ‖12‖Aψ‖12‖Aφ‖. | (10) |
‖N(ψ)‖≤c(νr)‖ψ‖ˆV, ∀ψ∈ˆV. | (11) |
In addition,
δ1‖ψ‖2ˆV≤⟨Aψ,ψ⟩+⟨N(ψ),ψ⟩, ∀ψ∈ˆV, | (12) |
where
According to the definitions of operators
{∂w∂t+Aw+B(u,w)+N(w)=F(t,x)+G(t,wt),in (τ,+∞)×Ω,∇⋅u=0,in (τ,+∞)×Ω,w=(u,ω)=0, on (τ,+∞)×Γ,w(τ)=(uin,ωin)=:win, wτ(s)=(uτ(s),ωτ(s))=(ϕin1(s),ϕin2(s)) =:ϕin(s), s∈(−h,0), | (13) |
where
w:=(u(t,x),ω(t,x)),F(t,x):=(f(t,x),˜f(t,x)),G(t,wt):=(g(t,ut),˜g(t,ωt)). |
Before recalling the known results for problem (13), we need to make the following assumptions with respect to
Assumption 2.1. Assume that
(ⅰ) For any
(ⅱ)
(ⅲ) There exists a constant
‖G(t,ξ)−G(t,η)‖≤LG‖ξ−η‖L2(−h,0;ˆH). |
(ⅳ) There exists
∫tτ‖G(θ,wθ)−G(θ,vθ)‖2dθ≤C2G∫tτ−h‖w(θ)−v(θ)‖2dθ. |
Assumption 2.2. Assume that
∫t−∞eγθ‖F(θ,x)‖2ˆV∗dθ<+∞. | (14) |
In order to facilitate the discussion, we denote by
Definition 2.3. (Definition of universe
Rγ:={ρ(t):R↦R+ | limt→−∞eγtρ2(t)=0}. |
We denote by
D(t)⊆ˉBE2ˆH(0,ρˆD(t)), for some ρˆD(t)∈Rγ, |
where
Based on the above assumptions, we can recall the global well-posedness of solutions and the existence of pullback attractor of problem (13).
Proposition 2.1. (Existence and uniqueness of solution, see [13,16])
Let Assumption 2.1 and Assumption 2.2 hold. Then for any
w∈C([τ,T];ˆH)∩L2(τ,T;ˆV) and w′∈L2(τ,T;ˆV∗), ∀T>τ. |
Remark 2.1. According to Proposition 2.1, the biparametric mapping defined by
U(t,τ):(win,ϕin(s))↦(w(t;τ,win,ϕin(s)),wt(s;τ,win,ϕin(s))), ∀t≥τ, | (15) |
generates a continuous process in
(i)U(τ,τ)(win,ϕin(s))=(win,ϕin(s)), |
(ii)U(t,θ)U(θ,τ)(win,ϕin(s))=U(t,τ)(win,ϕin(s)). |
Proposition 2.2. (Existence of pullback attractor, see [16]) Under the Assumption 2.1 and Assumption 2.2, there exists a pullback attractor
limτ→−∞distE2ˆH(U(t,τ)B(τ),AˆH(t))=0, ∀t∈R; |
limτ→−∞distE2ˆH(U(t,τ)B(τ),O(t))=0, for anyˆB={B(θ)|θ∈R}∈Dγ, |
then
Finally, we introduce a useful lemma, which plays an important role in the proof of higher regularity of the pullback attractor.
Lemma 2.4. (see [12]) Let X, Y be Banach spaces such that X is reflexive, and the inclusion
‖w(θ)‖X≤lim infn→+∞‖wn(θ)‖L∞(τ,t;X), ∀θ∈[τ,t]. |
This section is devoted to investigating the boundedness of the pullback attractor for the universe
wn(t)=wn(t;τ,win,ϕin(s))=n∑j=1ξnj(t)ej, wnt(⋅)=wn(t+⋅), | (16) |
where the sequence
ej∈D(A) and Aej=λjej, | (17) |
where the eigenvalues
0<λ1≤λ2≤⋯≤λj≤⋯, λj→+∞ as j→∞. |
It is not difficult to check that
{ddt(wn(t),ej)+⟨Awn(t)+B(un,wn)+N(wn(t)),ej⟩ =⟨F(t),ej⟩+(G(t,wnt),ej),(wn(τ),ej)=(win,ej), (wnτ(s),ej)=(ϕin(s),ej),s∈(−h,0), j=1,2,⋯,n. | (18) |
Next we verify the following estimates of the Galerkin approximate solutions defined by (16).
Lemma 3.1. Let Assumption (2.1) and Assumption (2.2) hold. Then for any
(ⅰ) the set
(ⅱ) the set
(ⅲ) the set
(ⅳ) the set
Proof. Multiplying (18) by
12ddt‖wn(t)‖2+δ1‖wn(t)‖2ˆV≤12ddt‖wn(t)‖2+⟨Awn(t),wn(t)⟩+⟨N(wn(t)),wn(t)⟩+⟨B(un,wn),wn(t)⟩=⟨F(t),wn(t)⟩+(G(t,wnt),wn(t)). |
Then integrating the above inequality over
‖wn(t)‖2+2δ1∫tτ‖wn(θ)‖2ˆVdθ≤‖win‖2+(δ1−CG)∫tτ‖wn(θ)‖2ˆVdθ+1δ1−CG∫tτ‖F(θ)‖2ˆV∗dθ+CG∫tτ‖wn(θ)‖2dθ+1CG∫tτ‖G(θ,wnθ)‖2dθ≤‖win‖2+(δ1−CG)∫tτ‖wn(θ)‖2ˆVdθ+1δ1−CG∫tτ‖F(θ)‖2ˆV∗dθ+CG∫tτ‖wn(θ)‖2dθ+CG(∫tτ‖wn(θ)‖2dθ+∫0−h‖ϕin(s)‖2ds), |
which implies
‖wn(t)‖2+(δ1−CG)∫tτ‖wn(θ)‖2ˆVdθ≤max{1,CG}‖(win,ϕin)‖2E2ˆH+1δ1−CG∫tτ‖F(θ)‖2ˆV∗dθ. | (19) |
Thanks to (17), multiplying (18) by
12ddt⟨Awn(t),wn(t)⟩+‖Awn(t)‖2+⟨B(un,wn),Awn(t)⟩+⟨N(wn(t)),Awn(t)⟩=(F(t),Awn(t))+(G(t,wnt),Awn(t)). |
Observe that
|⟨B(un,wn),Awn(t)⟩|+|⟨N(wn(t)),Awn(t)⟩|≤α0‖un‖12‖∇un‖12‖∇wn‖12‖Awn‖12‖Awn‖+14‖Awn(t)‖2+c2(νr)‖wn(t)‖2ˆV≤12‖Awn(t)‖2+43α40‖wn(t)‖2‖wn(t)‖4ˆV+c2(νr)‖wn(t)‖2ˆV |
and
(F(t),Awn(t))+(G(t,wnt),Awn(t))≤14‖Awn(t)‖2+2‖F(t)‖2+2‖G(t,wnt)‖2. |
Therefore
ddt⟨Awn(t),wn(t)⟩+12‖Awn(t)‖2≤4‖F(t)‖2+4‖G(t,wnt)‖2+128α40‖wn(t)‖2‖wn(t)‖4ˆV+2c2(νr)‖wn(t)‖2ˆV≤4‖F(t)‖2+4‖G(t,wnt)‖2+(128c2α40‖wn(t)‖2‖wn(t)‖2ˆV+2c2c2(νr))⟨Awn(t),wn(t)⟩. | (20) |
Set
Hn(θ):=⟨Awn(θ),wn(θ)⟩,In(θ):=4‖F(θ)‖2+4‖G(θ,wnθ)‖2,Jn(θ):=128c2α40‖wn(θ)‖2‖wn(θ)‖2ˆV+2c2c2(νr), |
then we get
ddθHn(θ)≤Jn(θ)Hn(θ)+In(θ). | (21) |
By Gronwall inequality, (21) yields
Hn(r)≤(Hn(˜r)+∫rr−ϵIn(θ)dθ)⋅exp{∫rr−ϵJn(θ)dθ}, ∀τ≤r−ϵ≤˜r≤r≤t. |
Integrating the above inequality for
ϵHn(r)≤(∫rr−ϵHn(˜r)d˜r+ϵ∫rr−ϵIn(θ)dθ)⋅exp{∫rr−ϵJn(θ)dθ}. |
Since
∫rr−ϵHn(˜r)d˜r+ϵ∫rr−ϵIn(θ)dθ=∫rr−ϵ⟨Awn(˜r),wn(˜r)⟩d˜r+4ϵ∫rr−ϵ(‖F(θ)‖2+‖G(θ,wnθ)‖2)dθ≤1c1∫tτ‖wn(θ)‖2ˆVdθ+4ϵ∫tτ‖F(θ)‖2dθ+4ϵC2G(∫tτ‖wn(θ)‖2dθ+∫0−h‖ϕin(s)‖2ds), |
∫rr−ϵJn(θ)dθ=128c2α40∫rr−ϵ‖wn(θ)‖2‖wn(θ)‖2ˆVdθ+2ϵc2c2(νr)≤128c2α40maxθ∈[τ,t]‖wn(θ)‖2∫tτ‖wn(θ)‖2ˆVdθ+2ϵc2c2(νr), |
we can conclude that
‖wn(r)‖2ˆV≤c2Hn(r)≤[c2c1ϵ∫tτ‖wn(θ)‖2ˆVdθ+4c2∫tτ‖F(θ)‖2dθ +4c2C2G(∫tτ‖wn(θ)‖2dθ+∫0−h‖ϕin(s)‖2ds)]⋅exp{128c2α40maxθ∈[τ,t]‖wn(θ)‖2∫tτ‖wn(θ)‖2ˆVdθ+2ϵc2c2(νr)}, |
which together with (19) and Assumption 2.2 implies the assertion (i).
Now, integrating (20) over
∫tτ+ϵ‖Awn(θ)‖2dθ≤2c1‖wn(τ+ϵ)‖2ˆV+8∫tτ‖F(θ)‖2dθ+8C2G(∫tτ‖wn(θ)‖2dθ+∫0−h‖ϕin(s)‖2ds)+(256α40maxθ∈[τ+ϵ,t](‖wn(θ)‖‖wn(θ)‖ˆV)2+4c2(νr))∫tτ+ϵ‖wn(θ)‖2ˆVdθ, |
which together with (19), Assumption 2.2 and the assertion (ⅰ) gives the assertion (ⅲ).
In addition,
∫0−h‖wnr(θ)‖2ˆVdθ=∫rr−h‖wn(θ)‖2ˆVdθ≤h⋅maxθ∈[τ+ϵ,t]‖wn(θ)‖2ˆV,τ+h+ϵ≤r≤t, | (22) |
which together with the assertion (ⅰ) yields the assertion (ⅱ).
Finally, multiplying (18) by
‖w′n(t)‖2+12ddt⟨Awn(t),wn(t)⟩+⟨B(un,wn),w′n(t)⟩+⟨N(wn(t)),w′n(t)⟩=(F(t),w′n(t))+(G(t,wnt),w′n(t)). | (23) |
From Assumption 2.1, it follows that
(F(t),w′n(t))+(G(t,wnt),w′n(t))≤(‖F(t)‖+‖G(t,wnt)‖)‖w′n(t)‖≤2‖F(t)‖2+2‖G(t,wnt)‖2+14‖w′n(t)‖2. | (24) |
By Lemma 2.2,
|⟨B(un,wn),w′n(t)⟩|≤α0‖un‖12‖∇un‖12‖∇wn‖12‖Awn‖12‖w′n(t)‖≤α0‖wn‖12‖wn‖ˆV‖Awn‖12‖w′n(t)‖≤α20‖wn‖‖wn‖2ˆV‖Awn‖+14‖w′n(t)‖2 | (25) |
and
|⟨N(wn(t)),w′n(t)|≤14‖w′n(t)‖2+c2(νr)‖wn(t)‖2ˆV. | (26) |
Taking (23)-(26) into account, we obtain
‖w′n(t)‖2+2ddt⟨Awn(t),wn(t)⟩≤8‖F(t)‖2+8‖G(t,wnt)‖2+4α20‖wn‖3ˆV‖Awn(t)‖+4c2(νr)‖wn(t)‖2ˆV. |
Integrating the above inequality, yields
∫tτ+ϵ‖w′n(θ)‖dθ≤2c−11‖wn(τ+ϵ)‖2ˆV+8∫tτ+ϵ‖F(θ)‖2dθ+8∫tτ+ϵ‖G(θ,wnθ)‖2dθ+4α20∫tτ+ϵ‖wn(θ)‖3ˆV‖Awn(θ)‖dθ+4c2(νr)∫tτ+ϵ‖wn(θ)‖2ˆVdθ≤2c−11‖wn(τ+ϵ)‖2ˆV+8∫tτ+ϵ‖F(θ)‖2dθ+8C2G(∫tτ‖wn(θ)‖2ˆVdθ+∫0−h‖ϕin(s)‖2ds)+4c2(νr)∫tτ+ϵ‖wn(θ)‖2ˆVdθ,+2α20maxθ∈[τ+ϵ,t]‖wn(θ)‖2ˆV∫tτ+ϵ(‖wn(θ)‖2ˆV+‖Awn(θ)‖2)dθ |
which together with (20), Assumption 2.2 and the assertions (ⅰ)-(ⅲ) gives the assertion (ⅳ). The proof is complete.
With the above lemma, we are ready to conclude this section with the following
Theorem 3.2. Let assumptions 2.1-2.2 hold and
Proof. Based on Lemma 3.1, following the standard diagonal procedure, there exist a subsequence (denoted still by)
wn(⋅)⇀∗w(⋅) weakly star in L∞(τ+ϵ,t;ˆV), | (27) |
wn(⋅)⇀w(⋅) weakly in L2(τ+ϵ,t;D(A)), | (28) |
w′n(⋅)⇀w′(⋅) weakly in L2(τ+ϵ,t;ˆH). | (29) |
Furthermore, it follows from the uniqueness of the limit function that
Remark 3.1. We here point out that the boundedness of pullback attractor
‖(G(t,ξ))′−(G(t,η))′‖≤˜LG‖ξ−η‖L2(−h,0;ˆH). |
∫tτ‖(G(θ,wθ))′−(G(θ,vθ))′‖2dθ≤˜C2G∫tτ−h‖w(θ)−v(θ)‖2dθ. |
Then we can deduce that the Galerkin approximate solutions
In this section, we concentrate on verifying the upper semicontinuity of the pullback attractor
{∂wm∂t+Awm+B(um,wm)+N(wm)=F(t,x)+G(t,wmt),in (τ,+∞)×Ωm,∇⋅um=0,in (τ,+∞)×Ωm,wm=(um,ωm)=0, on (τ,+∞)×Γ,wm(τ)=winm, wmτ(s)=ϕinm(s), s∈(−h,0). | (30) |
On each bounded domain
Lemma 4.1. Suppose Assumption 2.1 and Assumption 2.2 hold, then for any given
wm(⋅)∈C([τ,T];ˆH(Ωm))∩L2(τ,T;ˆV(Ωm)), w′m(⋅)∈L2(τ,T;ˆV∗(Ωm)), ∀T>τ. |
Moreover, the solution
According to Lemma 4.1, the map defined by
Um(t,τ):(winm,ϕinm(s))↦Um(t,τ,winm,ϕinm(s))=(wm(t),wmt(s;τ,win,ϕin(s))), ∀t≥τ, | (31) |
generates a continuous process
Lemma 4.2. Under the assumptions 2.1 and 2.2, it holds that
BˆH(Ωm)(t)={(w,ϕ)∈E2ˆH(Ωm)×L2ˆV(Ωm)|‖(w,ϕ)‖2EˆH(Ωm)×L2ˆV(Ωm)≤R1(t)} |
is pullback
‖wm(t,τ,winm,ϕinm(s))‖L2(Ωm∖Ωr)≤ϵ, ∀(winm,ϕinm(s))∈BˆH(Ωm)(τ). |
Then based on the Remark 2.1 in [16], we conclude that
Theorem 4.3. Let assumptions 2.1 and 2.2 hold. Then there exists a pullback attractor
In the following, we investigate the relationship between the solutions of system (30) and (13). Indeed, we devoted to proving the solutions
˜wm={wm,x∈Ωm,0,x∈Ω∖Ωm, | (32) |
then it holds that
‖wm‖ˆH(Ωm)=‖˜wm‖ˆH(Ωm)=‖˜wm‖ˆH(Ω):=‖wm‖. |
Next, using the same proof as that of Lemma 8.1 in [8], we have
Lemma 4.4. Let assumptions 2.1-2.2 hold and
wm(t;τ,winm,ϕinm(s))⇀w(t;τ,win,ϕin(s)) weakly in ˆH, | (33) |
wm(⋅;τ,winm,ϕinm(s))⇀w(⋅;τ,win,ϕin(s)) weakly in L2(t−h,t;ˆV). | (34) |
Based on Lemma 4.4, we set out to prove the following important lemma.
Lemma 4.5. Let assumptions 2.1-2.2 hold, then for any
(wm(t),wmt(s))→(w(t),wt(s)) strongly in E2ˆH. | (35) |
Proof. From the compactness of pullback attractor, it follows that the sequence
(winm,ϕinm(s))⇀(win,ϕin(s)) weakly in E2ˆH×L2ˆV as m→∞. | (36) |
Further, according to Lemma 4.4 and the invariance of the pullback attractor, we can conclude that for any
(wm(t),wmt(s))⇀(w(t),wt(s)) weakly in E2ˆH×L2ˆV as m→∞. | (37) |
Then, using the same way of proof as Lemma 3.6 in [16], we can obtain that the convergence relation of (37) is strong. The proof is complete. With the above lemma, we are ready to state the main result of this section.
Theorem 4.6. Let Assumption 2.1 and Assumption 2.2 hold, then for any
limm→∞distE2ˆH(AˆH(Ωm)(t),AˆH(t))=0, | (38) |
where
Proof. Suppose the assertion (38) is false, then for any
distE2ˆH((wm(t0),wmt0(s)),AˆH(t0))≥ϵ0. | (39) |
However, it follows from Lemma 4.5 that there exists a subsequence
{(wmk(t0),wmkt0(s))}⊆{(wm(t0),wmt0(s))} |
such that
limk→∞distE2ˆH((wmk(t0),wmkt0(s)),AˆH(t0))=0, |
which is in contradiction to (39). Therefore, (38) is true. The proof is complete.
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