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A comprehensive bi-objective optimization model to design circular supply chain networks for sustainable electric vehicle batteries

  • As electric vehicles (EVs) continue to advance, there is a growing emphasis on sustainability, particularly in the area of effectively managing the lifecycle of EV batteries. In this study, an efficient and novel optimization model was proposed for designing a circular supply chain network for EV batteries. In doing so, a comprehensive, bi-objective, mixed-integer linear programming model was employed. It is worth noting that the current model outlined in this paper involved both forward and reverse flows, illustrating the process of converting used batteries into their constituent materials or repurposing them for various applications. In line with the circular economy concept, the current model also minimized the total costs and carbon emission to develop an inclusive optimization framework. The LP-metric method was applied to solve the presented bi-objective optimization model. We simulated six problems with different sizes using data and experts' knowledge of a lithium-ion battery manufacturing industry in Canada, and evaluated the performance of the proposed model by simulated data. The results of the sensitivity analysis process of the objective functions coefficients showed that there was a balance between the two objective functions, and the costs should be increased to achieve lower emissions. In addition, the demand sensitivity analysis revealed that the increase in demand directly affects the increase in costs and emissions.

    Citation: Afshin Meraj, Tina Shoa, Fereshteh Sadeghi Naieni Fard, Hassan Mina. A comprehensive bi-objective optimization model to design circular supply chain networks for sustainable electric vehicle batteries[J]. AIMS Environmental Science, 2024, 11(2): 279-303. doi: 10.3934/environsci.2024013

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  • As electric vehicles (EVs) continue to advance, there is a growing emphasis on sustainability, particularly in the area of effectively managing the lifecycle of EV batteries. In this study, an efficient and novel optimization model was proposed for designing a circular supply chain network for EV batteries. In doing so, a comprehensive, bi-objective, mixed-integer linear programming model was employed. It is worth noting that the current model outlined in this paper involved both forward and reverse flows, illustrating the process of converting used batteries into their constituent materials or repurposing them for various applications. In line with the circular economy concept, the current model also minimized the total costs and carbon emission to develop an inclusive optimization framework. The LP-metric method was applied to solve the presented bi-objective optimization model. We simulated six problems with different sizes using data and experts' knowledge of a lithium-ion battery manufacturing industry in Canada, and evaluated the performance of the proposed model by simulated data. The results of the sensitivity analysis process of the objective functions coefficients showed that there was a balance between the two objective functions, and the costs should be increased to achieve lower emissions. In addition, the demand sensitivity analysis revealed that the increase in demand directly affects the increase in costs and emissions.



    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:

    {ut(ν+νr)Δu2νrrotω+(u)u+p=f,ωt(ca+cd)Δω+4νrω+(u)ω(c0+cdca)divω2νrrotu=˜f,u=0, (1)

    where u=(u1,u2,u3) is the velocity, ω=(ω1,ω2,ω3) is the angular velocity field of rotation of particles, p represents the pressure, f=(f1,f2,f3) and ˜f=(˜f1,˜f2,˜f3) stand for the external force and moment, respectively. The positive parameters ν,νr,c0,ca and cd are viscous coefficients. Actually, ν represents the usual Newtonian viscosity and νr is called microrotation viscosity.

    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 L2-global attractors and its Hausdorff dimension and fractal dimension estimation. Chen, Chen and Dong proved the existence of H2-global attractor and uniform attractor in [1] and [2], respectively. Łukaszewicz and Tarasińska [9] investigated the existence of H1-pullback attractor. Zhao, Sun and Hsu [18] established the existence of L2-pullback attractor and H1-pullback attractor of solutions for a universe given by a tempered condition, respectively. For the case of 2D unbounded domains: Dong and Chen [3] investigated the existence and regularity of global attractors. Zhao, Zhou and Lian [19] established the existence of H1-uniform attractor and further gave the inclusion relation between L2-uniform attractor and the H1-uniform attractor. Sun and Li [15] verified the existence of pullback attractor and further investigated the tempered behavior and upper semicontinuity of the pullback attractor. More recently, Sun, Cheng and Han [14] investigated the existence of random attractors for 2D stochastic micropolar fluid flows.

    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 u3 in the x3-direction is zero and the axes of rotation of particles are parallel to the x3 axis, that is u=(u1,u2,0), ω=(0,0,ω3), f=(f1,f2,0), ˜f=(0,0,˜f3). Let ΩR2 be an open set with boundary Γ that is not necessarily bounded but satisfies the following Poincaré inequality:

    There exists λ1>0 such that λ1φ2L2(Ω)φ2L2(Ω),  φH10(Ω). (2)

    Then we discuss the following 2D non-autonomous incompressible micropolar fluid flows with finite delay:

    {ut(ν+νr)Δu2νr×ω+(u)u+p=f(t,x)+g(t,ut),in(τ,+)×Ω,ωtˉαΔω+4νrω2νr×u+(u)ω=˜f(t,x)+˜g(t,ωt),in(τ,+)×Ω,u=0,in(τ,+)×Ω, (3)

    where ˉα:=c0+2cd>0, x:=(x1,x2)ΩR2, u:=(u1,u2), g and ˜g stand for the external force containing some hereditary characteristics ut and ωt, which are defined on (h,0) as follows

    ut(s):=u(t+s), ωt(s):=ω(t+s), tτ, s(h,0).

    where h is a positive fixed number, and

    ×u:=u2x1u1x2,×ω:=(ωx2,ωx1).

    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 H2-boundedness of the pullback attractors of the 2D Navier-Stokes equations in bounded domains. Motivated by [5] and following its main idea, we generalize their results to the 2D micropolar fluid flows with finite delay in unbounded domains. Compared with the Navier-Stokes equations (ω=0,νr=0), the micropolar fluid flow consists of the angular velocity field ω, which leads to a different nonlinear term B(u,w) and an additional term N(u) in the abstract equations (13). In addition, the time-delay term considered in this work also increases the difficulty.

    The second purpose of this work is to investigate the upper semicontinuity of the pullback attractor with respect to the domain Ω. To this end, using the arguments in [15,17], we first let {Ωm}m=1 be an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that m=1Ωm=Ω. Then we consider the Cauchy problem (3)-(5) in Ωm. We will conclude that there exists a pullback attractor ˆAˆH(Ωm) for the problem (3)-(5) in each Ωm. Finally, we establish the upper semicontinuity by showing limmdistE2ˆH(AˆH(Ωm)(t),AˆH(t))=0, tR.

    Throughout this paper, we denote the usual Lebesgue space and Sobolev space by Lp(Ω) and Wm,p(Ω) endowed with norms p and m,p, respectively. Especially, we denote Hm(Ω):=Wm,2(Ω).

    V:=V(Ω):={φC0(Ω)×C0(Ω)|φ=(φ1,φ2),φ=0},ˆV:=ˆV(Ω):=V×C0(Ω),H:=H(Ω):=closureofVinL2(Ω)×L2(Ω),withnormH anddualspaceH,V:=V(Ω):=closureofVinH1(Ω)×H1(Ω),withnormV anddualspaceV,ˆH:=ˆH(Ω):=closureofˆVinL2(Ω)×L2(Ω)×L2(Ω),withnormˆH anddualspaceˆH,ˆV:=ˆV(Ω):=closureofˆVinH1(Ω)×H1(Ω)×H1(Ω),withnormˆV anddualspaceˆV.

    (,) the inner product in L2(Ω),H or ˆH, , the dual pairing between V and V or between ˆV and ˆV. Throughout this article, we simplify the notations 2, H and ˆH by the same notation if there is no confusion. Furthermore, we denote

    Lp(I;X):=space of strongly measurable functions on the closed interval I   with values in the Banach space X, endowed with normφLp(I;X):=(IφpXdt)1/p,  for 1p<,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 XMandYMdefinedbydistM(X,Y)=supxXinfyYdistM(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 X for X{E2ˆH,E2ˆV,E2ˆH×L2ˆV} is defined as

    (w,v)E2ˆH:=(w2ˆv+v2L2ˆH)1/2,(w,v)E2ˆV:=(w2+v2L2ˆV)1/2,(w,v)E2ˆH×L2ˆV:=(w2+v2L2ˆV)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 A,B(,) and N() by

    {Aw,φ:=(ν+νr)(u,Φ)+ˉα(ω,φ3), w=(u,ω),φ=(Φ,φ3)ˆV,B(u,w),φ:=((u)w,φ), uV,w=(u,ω)ˆV,φˆV,N(w):=(2νr×ω,2νr×u+4νrω), w=(u,ω)ˆV. (6)

    What follows are some useful estimates and properties for the operators A, B(,) and N(), which have been established in works [11,13].

    Lemma 2.1. (1) The operator A is linear continuous both from ˆV to ˆV and from D(A) to ˆH, and so is for the operator N() from ˆV to ˆH, where D(A):=ˆV(H2(Ω))3.

    (2) The operator B(,) is continuous from V׈V to ˆV. Moreover, for any uV and wˆV, there holds

    B(u,ψ),φ=B(u,φ),ψ, uV, ψˆV, φˆV. (7)

    Lemma 2.2. (1) There are two positive constants c1 and c2 such that

    c1Aw,ww2ˆVc2Aw,w, wˆV. (8)

    (2) There exists a positive constant α0 which depends only on Ω, such that for any (u,ψ,φ)V׈V׈V there holds

    |B(u,ψ),φ|{α0u12u12φ12φ12ψ,α0u12u12ψ12ψ12φ. (9)

    Moreover, if (u,ψ,φ)V×D(A)×D(A), then

    |B(u,ψ),Aφ|α0u12u12ψ12Aψ12Aφ. (10)

    (3) There exists a positive constant c(νr) such that

    N(ψ)c(νr)ψˆV, ψˆV. (11)

    In addition,

    δ1ψ2ˆVAψ,ψ+N(ψ),ψ, ψˆV, (12)

    where δ1:=min{ν,ˉα}.

    According to the definitions of operators A,B(,) and N(), equations (3)-(5) can be formulated into the following abstract form:

    {wt+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 F and G.

    Assumption 2.1. Assume that G:R×L2(h,0;ˆH)(L2(Ω))3 satisfies:

    (ⅰ) For any ξL2(h,0;ˆH), the mapping RtG(t,ξ)(L2(Ω))3 is measurable.

    (ⅱ) G(,0)=(0,0,0).

    (ⅲ) There exists a constant LG>0 such that for any tR and any ξ,ηL2(h,0;ˆH),

    G(t,ξ)G(t,η)LGξηL2(h,0;ˆH).

    (ⅳ) There exists CG(0,δ1) such that, for any tτ and any w,vL2(τh,t;ˆH),

    tτG(θ,wθ)G(θ,vθ)2dθC2Gtτhw(θ)v(θ)2dθ.

    Assumption 2.2. Assume that F(t,x)L2loc(R;ˆH), tτ,τR, and there exists a γ(0,2δ12CG) such that

    teγθF(θ,x)2ˆVdθ<+. (14)

    In order to facilitate the discussion, we denote by P(X) the family of all nonempty subsets of X. Let D be a nonempty class of families parameterized in time ˆD={D(t):tR}P(X), which will be called a universe in P(X). Based on these notations, we can construct the universe Dγ in the following.

    Definition 2.3. (Definition of universe Dγ) Set

    Rγ:={ρ(t):RR+ | limteγtρ2(t)=0}.

    We denote by Dγ the class of all families ˆD={D(t) | tR}P(E2ˆH) such that

    D(t)ˉBE2ˆH(0,ρˆD(t)), for some ρˆD(t)Rγ,

    where ˉBE2ˆH(0,ρˆD(t)) represents the closed ball in E2ˆH centered at zero with radius ρˆD(t).

    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 (win,ϕin(s))E2ˆH, there exists a unique weak solution w():=w(;τ,win,ϕin(s)) for system (13), which satisfies

    wC([τ,T];ˆH)L2(τ,T;ˆV)  and  wL2(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))),  tτ, (15)

    generates a continuous process in E2ˆH and E2ˆV, respectively, which satisfies the following properties:

    (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 ˆAˆH={AˆH(t)|tR} for the process {U(t,τ)}tτ that satisfies the following properties:

    Compactness: for any tR,AˆH(t) is a nonempty compact subset of E2ˆH;

    Invariance: U(t,τ)AˆH(τ)=AˆH(t),tτ;

    Pullback attracting: for any ˆB={B(θ)|θR}Dγ, it holds that

    limτdistE2ˆH(U(t,τ)B(τ),AˆH(t))=0, tR;

    Minimality: the family of sets ˆAˆH is the minimal in the sense that if

    ˆO={O(t)|tR}P(E2ˆH) is another family of closed sets such that

    limτdistE2ˆH(U(t,τ)B(τ),O(t))=0,  for anyˆB={B(θ)|θR}Dγ,

    then AˆH(t)O(t) for any tR.

    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 XY is continuous. Assume that {wn}n1 is a bounded sequence in L(τ,t;X) such that wnw weakly in Lq(τ,t;X) for some q[1,+) and wC([τ,t];Y). Then w(t)X and

    w(θ)Xlim 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 Dγ given by a tempered condition in space E2ˆH. To this end, we consider the Galerkin approximation of the solution w(t) of system (13), which is denoted by

    wn(t)=wn(t;τ,win,ϕin(s))=nj=1ξnj(t)ej,  wnt()=wn(t+), (16)

    where the sequence {ej}j=1 is an orthonormal basis of ˆH and formed by eigenvectors of the operator A, that is, for all j1,

    ejD(A)  and  Aej=λjej, (17)

    where the eigenvalues {λj}j1 of A are real number that we can order in such a way

    0<λ1λ2λj,  λj+ as j.

    It is not difficult to check that ξn,j(t) is the solution of the following ordinary differential equations:

    {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 τR,ϵ>0,t>τ+h+ϵ and (win,ϕin(s))AˆH(τ), we have

    (ⅰ) the set {wn(r;τ,win,ϕin(s))|r[τ+ϵ,t]}n1 is bounded in ˆV;

    (ⅱ) the set {wnr(;τ,win,ϕin(s))|r[τ+h+ϵ,t]}n1 is bounded in L2ˆV;

    (ⅲ) the set {wn(;τ,win,ϕin(s))}n1 is bounded in L2(τ+ϵ,t;D(A));

    (ⅳ) the set {wn(;τ,win,ϕin(s))}n1 is bounded in L2(τ+ϵ,t;ˆH).

    Proof. Multiplying (18) by βnj(t) and summing them for j=1 to n, then using (7), (12) and Young's inequality, we obtain

    12ddtwn(t)2+δ1wn(t)2ˆV12ddtwn(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 [τ,t],tτ, leads to

    wn(t)2+2δ1tτwn(θ)2ˆVdθwin2+(δ1CG)tτwn(θ)2ˆVdθ+1δ1CGtτF(θ)2ˆVdθ+CGtτwn(θ)2dθ+1CGtτG(θ,wnθ)2dθwin2+(δ1CG)tτwn(θ)2ˆVdθ+1δ1CGtτF(θ)2ˆVdθ+CGtτwn(θ)2dθ+CG(tτwn(θ)2dθ+0hϕin(s)2ds),

    which implies

    wn(t)2+(δ1CG)tτwn(θ)2ˆVdθmax{1,CG}(win,ϕin)2E2ˆH+1δ1CGtτF(θ)2ˆVdθ. (19)

    Thanks to (17), multiplying (18) by λjβnj(t) and summing the resultant equation for j=1 to n yields that

    12ddtAwn(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 un(t)wn(t), wn(t)wn(t)ˆV, using (10), (11) and Young's inequality, it is easy to see that

    |B(un,wn),Awn(t)|+|N(wn(t)),Awn(t)|α0un12un12wn12Awn12Awn+14Awn(t)2+c2(νr)wn(t)2ˆV12Awn(t)2+43α40wn(t)2wn(t)4ˆV+c2(νr)wn(t)2ˆV

    and

    (F(t),Awn(t))+(G(t,wnt),Awn(t))14Awn(t)2+2F(t)2+2G(t,wnt)2.

    Therefore

    ddtAwn(t),wn(t)+12Awn(t)24F(t)2+4G(t,wnt)2+128α40wn(t)2wn(t)4ˆV+2c2(νr)wn(t)2ˆV4F(t)2+4G(t,wnt)2+(128c2α40wn(t)2wn(t)2ˆV+2c2c2(νr))Awn(t),wn(t). (20)

    Set

    Hn(θ):=Awn(θ),wn(θ),In(θ):=4F(θ)2+4G(θ,wnθ)2,Jn(θ):=128c2α40wn(θ)2wn(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)+rrϵIn(θ)dθ)exp{rrϵJn(θ)dθ},  τrϵ˜rrt.

    Integrating the above inequality for ˜r from rϵ to r, we obtain

    ϵHn(r)(rrϵHn(˜r)d˜r+ϵrrϵIn(θ)dθ)exp{rrϵJn(θ)dθ}.

    Since

    rrϵHn(˜r)d˜r+ϵrrϵIn(θ)dθ=rrϵAwn(˜r),wn(˜r)d˜r+4ϵrrϵ(F(θ)2+G(θ,wnθ)2)dθ1c1tτwn(θ)2ˆVdθ+4ϵtτF(θ)2dθ+4ϵC2G(tτwn(θ)2dθ+0hϕin(s)2ds),
    rrϵJn(θ)dθ=128c2α40rrϵwn(θ)2wn(θ)2ˆVdθ+2ϵc2c2(νr)128c2α40maxθ[τ,t]wn(θ)2tτwn(θ)2ˆVdθ+2ϵc2c2(νr),

    we can conclude that

    wn(r)2ˆVc2Hn(r)[c2c1ϵtτwn(θ)2ˆVdθ+4c2tτF(θ)2dθ +4c2C2G(tτwn(θ)2dθ+0hϕin(s)2ds)]exp{128c2α40maxθ[τ,t]wn(θ)2tτwn(θ)2ˆVdθ+2ϵc2c2(νr)},

    which together with (19) and Assumption 2.2 implies the assertion (i).

    Now, integrating (20) over [τ+ϵ,t], we obtain

    tτ+ϵAwn(θ)2dθ2c1wn(τ+ϵ)2ˆV+8tτF(θ)2dθ+8C2G(tτwn(θ)2dθ+0hϕin(s)2ds)+(256α40maxθ[τ+ϵ,t](wn(θ)wn(θ)ˆV)2+4c2(νr))tτ+ϵwn(θ)2ˆVdθ,

    which together with (19), Assumption 2.2 and the assertion (ⅰ) gives the assertion (ⅲ).

    In addition,

    0hwnr(θ)2ˆVdθ=rrhwn(θ)2ˆVdθhmaxθ[τ+ϵ,t]wn(θ)2ˆV,τ+h+ϵrt, (22)

    which together with the assertion (ⅰ) yields the assertion (ⅱ).

    Finally, multiplying (18) by βnj(t) and summing the resultant equation for j=1 to n, we obtain

    wn(t)2+12ddtAwn(t),wn(t)+B(un,wn),wn(t)+N(wn(t)),wn(t)=(F(t),wn(t))+(G(t,wnt),wn(t)). (23)

    From Assumption 2.1, it follows that

    (F(t),wn(t))+(G(t,wnt),wn(t))(F(t)+G(t,wnt))wn(t)2F(t)2+2G(t,wnt)2+14wn(t)2. (24)

    By Lemma 2.2,

    |B(un,wn),wn(t)|α0un12un12wn12Awn12wn(t)α0wn12wnˆVAwn12wn(t)α20wnwn2ˆVAwn+14wn(t)2 (25)

    and

    |N(wn(t)),wn(t)|14wn(t)2+c2(νr)wn(t)2ˆV. (26)

    Taking (23)-(26) into account, we obtain

    wn(t)2+2ddtAwn(t),wn(t)8F(t)2+8G(t,wnt)2+4α20wn3ˆVAwn(t)+4c2(νr)wn(t)2ˆV.

    Integrating the above inequality, yields

    tτ+ϵwn(θ)dθ2c11wn(τ+ϵ)2ˆV+8tτ+ϵF(θ)2dθ+8tτ+ϵG(θ,wnθ)2dθ+4α20tτ+ϵwn(θ)3ˆVAwn(θ)dθ+4c2(νr)tτ+ϵwn(θ)2ˆVdθ2c11wn(τ+ϵ)2ˆV+8tτ+ϵF(θ)2dθ+8C2G(tτwn(θ)2ˆVdθ+0hϕin(s)2ds)+4c2(νr)tτ+ϵwn(θ)2ˆVdθ,+2α20maxθ[τ+ϵ,t]wn(θ)2ˆVtτ+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 H1-boundedness of the pullback attractor ˆAˆH for the universe Dγ.

    Theorem 3.2. Let assumptions 2.1-2.2 hold and ˆAˆH={AˆH(t)|tR} be the pullback attractor of system (13). Then for any τR,ϵ>0, t>τ+h+ϵ and (win,ϕin)E2ˆH, the set r[τ+h+ϵ,t]U(r,τ)AˆH(τ) is bounded in E2ˆV.

    Proof. Based on Lemma 3.1, following the standard diagonal procedure, there exist a subsequence (denoted still by) {wn}n1 and a function w()L(τ+ϵ,t;ˆV)L2(τ+ϵ,t;D(A)) with w()L2(τ+ϵ,t;ˆH) such that, as n,

    wn()w() weakly star in L(τ+ϵ,t;ˆV), (27)
    wn()w() weakly in L2(τ+ϵ,t;D(A)), (28)
    wn()w() weakly in L2(τ+ϵ,t;ˆH). (29)

    Furthermore, it follows from the uniqueness of the limit function that w() is a weak solution of system (13). According to compact embedding theorem, (28) and (29) implies w()C([τ+ϵ,t];ˆV). Then Theorem 3.1 follows from (27), Lemma 2.4 and Lemma 3.1.

    Remark 3.1. We here point out that the boundedness of pullback attractor ˆA in E2D(A) can be proved by using similar proof as that in E2ˆV if we improve the regularity of F(t) and G(t,wnt) with respect to t, where D(A):=ˆV(H2)3. Exactly, assume that

    (I) F(t,x)W1,2loc(R;ˆH), tτ,τR, and teγθF(θ,x)2ˆVdθ<+.

    (II) (G(t,ξ))=dGdt: R×L2(h,0;ˆH)(L2(Ω))3 satisfies:

    For any ξL2(h,0;ˆH), the mapping RtG(t,ξ)(L2(Ω))3 is measurable.

    (G(,0))=(0,0,0).

    There exists a constant ˜LG>0 such that for any tR and any ξ,ηL2(h,0;ˆH),

    (G(t,ξ))(G(t,η))˜LGξηL2(h,0;ˆH).

    There exists ˜CG(0,δ1) such that, for any tτ and any w,vL2(τh,t;ˆH),

    tτ(G(θ,wθ))(G(θ,vθ))2dθ˜C2Gtτhw(θ)v(θ)2dθ.

    Then we can deduce that the Galerkin approximate solutions {wn()}n1 is bounded in D(A)=ˆV(H2)3. Moreover, {wn()}n1 is bounded in ˆH. Further, we can conclude the H2-boundedness of the pullback attractor ˆA.

    In this section, we concentrate on verifying the upper semicontinuity of the pullback attractor ˆAˆH obtained in Propositon 2.2 with respect to the spatial domain. To this end, first we let {Ωm}m=1 be an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that m=1Ωm=Ω. Then we consider the system (3) in each Ωm and define the operators A,B(,) and N() as previous (in (6)) with the spatial domain Ω replaced by Ωm. Further we can formulate the weak version of problem (3)-(5) as follows:

    {wmt+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 Ωm, the well-posedness of solution can be established by Galerkin method and energy method, one can refer to [7].

    Lemma 4.1. Suppose Assumption 2.1 and Assumption 2.2 hold, then for any given (winm,ϕinm)E2ˆH(Ωm), system (30) has a unique weak solution wm satisfying

    wm()C([τ,T];ˆH(Ωm))L2(τ,T;ˆV(Ωm)),  wm()L2(τ,T;ˆV(Ωm)), T>τ.

    Moreover, the solution wm() depends continuously on the initial value winm with respect to ˆH(Ωm) norm.

    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))),  tτ, (31)

    generates a continuous process {U(t,τ)}tτ in ˆH(Ωm). Moreover, on any smooth bounded domain Ωm, with similar proof as those of Lemma 3.2, Lemma 3.3 and Lemma 3.6 in [16], we can obtain the existence of pullback DˆH(Ωm)γ-absorbing for the process {Um(t,τ)}tτ and the pullback DˆH(Ωm)γ-asymptotic compactness of the process in ˆH(Ωm). That is,

    Lemma 4.2. Under the assumptions 2.1 and 2.2, it holds that

    (1) for any (winm,ϕinm(s))E2ˆH(Ωm), the family ˆBˆH(Ωm):={BˆH(Ωm)(t)|tR} given by

    BˆH(Ωm)(t)={(w,ϕ)E2ˆH(Ωm)×L2ˆV(Ωm)|(w,ϕ)2EˆH(Ωm)×L2ˆV(Ωm)R1(t)}

    is pullback DˆH(Ωm)γ-absorbing for the process {Um(t,τ)}, where R1(t) is bounded for all tR.

    (2) for any ϵ>0, there exist rm:=rm(ϵ,t,ˆBˆH(Ωm))>0, τm:=τm(ϵ,t,ˆBˆH(Ωm))<t such that for any r[rm,m],ττm,

    wm(t,τ,winm,ϕinm(s))L2(ΩmΩr)ϵ,  (winm,ϕinm(s))BˆH(Ωm)(τ).

    (3) the process {Um(t,τ)}tτ is pullback DˆH(Ωm)γ-asymptotically compact in ˆH(Ωm).

    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 ˆAˆH(Ωm)={AˆH(Ωm)(t)|tR} for the system (30) in ˆH(Ωm).

    In the following, we investigate the relationship between the solutions of system (30) and (13). Indeed, we devoted to proving the solutions wm of system (30) converges to the solution of system (13) as m. To this end, for wmˆH(Ωm), we extend its domain from Ωm to Ω by setting

    ˜wm={wm,xΩm,0,xΩΩm, (32)

    then it holds that

    wmˆH(Ωm)=˜wmˆH(Ωm)=˜wmˆH(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 {(winm,ϕinm(s))}m1 be a sequence in E2ˆH(Ωm)×L2ˆV(Ωm) converging weakly to an element (win,ϕin(s))E2ˆH×L2ˆV as m. Then for any tτ,

    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(th,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 tR, any sequence {(wm(t),wmt(s))}m1 with (wm(τ),wmτ(s))=(winm,ϕinm(s))AˆH(Ωm)(τ),m=1,2,, there exists (w(t),wt(s))AˆH(t) such that

    (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))}m1 is bounded in E2ˆH. Hence, there exist a subsequence (denoted still by) {(winm,ϕinm(s))}m1 and a (win,ϕin(s))AˆH(τ) such that

    (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 tR, there exist a (wm(t),wmt(s))AˆH(Ωm)(t) with (wm(τ),wmτ(s))AˆH(Ωm)(τ) and a (w(t),wt(s))AˆH(t) with (w(τ),wτ(s))AˆH(τ) such that

    (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 tR, it holds that

    limmdistE2ˆH(AˆH(Ωm)(t),AˆH(t))=0, (38)

    where AˆH(t) and AˆH(Ωm)(t) are the pullback attractor of system (13) and system (30), respectively.

    Proof. Suppose the assertion (38) is false, then for any m=1,2,, there exist t0R,ϵ0>0 and a sequence (wm(t0),wmt0(s))AˆH(Ωm)(t0) such that

    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

    limkdistE2ˆH((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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