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Research article

Time-adaptive Lagrangian variational integrators for accelerated optimization

  • Received: 02 June 2022 Revised: 05 October 2022 Accepted: 06 October 2022 Published: 15 February 2023
  • 37M15, 37N40, 34A26, 65K10, 70H15

  • A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in [1] and [2]. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in [3] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.

    Citation: Valentin Duruisseaux, Melvin Leok. Time-adaptive Lagrangian variational integrators for accelerated optimization[J]. Journal of Geometric Mechanics, 2023, 15(1): 224-255. doi: 10.3934/jgm.2023010

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  • A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in [1] and [2]. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in [3] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.



    The Navier-Stokes equations are a typical nonlinear system, which model the mechanics law for fluid flow and have been applied in many fields. There are many research findings on the Navier-Stokes system, involving well-posedness, long-time behavior, etc., [1,2,3,4,5,6,7,8,9,10]. Furthermore, to simulate the fluid movement modeled by the Navier-Stokes equations, some regularized systems are proposed, such as the Navier-Stokes-Voigt equations. The Navier-Stokes-Voigt equations were introduced by Oskolkov in 1973, which describe the motion of Kelvin-Voigt viscoelastic incompressible fluid. Based on the global well-posedness of 3D Navier-Stokes-Voigt equations in [11], many interesting results on long-time behavior of solutions have been obtained, such as the existence of global attractor and pullback attractors, determining modes and estimate on fractal dimension of attractor [12,13,14] and references therein for details.

    The influence of past history term on dynamical system is well known, we refer to [15,16,17,18] for interesting conclusions, such as the global well-posedness, the existence of attractors and so on. In 2013, Gal and Tachim-Medjo [17] studied the Navier-Stokes-Voigt system with instantaneous viscous term and memory-type viscous term, and obtained the well-posedness of solution and exponential attractors of finite dimension. In 2018, Plinio et al. [18] considered the Navier-Stokes-Voigt system in [18], in which the instantaneous viscous term was completely replaced by the memory-type viscous term and the Ekman damping βu was presented. The authors showed the existence of regular global and exponential attractors with finite dimension. The presence of Ekman damping was to eliminate the difficulties brought by the memory term in deriving the dissipation of system.

    Some convergence results of solutions or attractors as perturbation vanishes for the non-autonomous dynamical systems without memory can be seen in [19,20,21,22]. However, there are few convergence results on the system with memory. Therefore, our purpose is to study the tempered pullback dynamics and robustness of the following 3D incompressible Navier-Stokes-Voigt equations on the bounded domain Ω with memory and the Ekman damping:

    {t(uαΔu)0g(s)Δu(ts)ds+(u)u+βu+p=fε(t,x), (t,x)Ωτ,divu=0, (t,x)Ωτ,u(t,x)=0, (t,x)Ωτ,u(τ,x)=u(τ), xΩ,u(τs,x)=φ(s,x), (s,x)Ω0, (1.1)

    where Ωτ=(τ,+)×Ω, Ωτ=(τ,+)×Ω, Ω0=(0,)×Ω, τR+ is the initial time, α>0 is a length scale parameter characterizing the elasticity of fluid, β>0 is the Ekman dissipation constant, u=(u1(t,x),u2(t,x),u3(t,x)) is the unknown velocity field of fluid, and p is the unknown pressure. The non-autonomous external force is fε(t,x)=f1(x)+εf2(t,x) (0ε<ε0), where ε0 is a fixed constant small enough. In addition, u(τ) is the initial velocity, and φ(s,x) denotes the past history of velocity. The memory kernel g: [0,)[0,) is supposed to be convex, smooth on (0,) and satisfies that

    g()=0, 0g(s)ds=1.

    In general, we give the past history variable

    η=ηt(s)=s0u(tσ)dσ, s0,

    which satisfies

    tη=sη+u(t).

    Also, η has the explicit representation

    {ηt(s)=s0u(tσ)dσ, 0<st,ηt(s)=η0(st)+t0u(tσ)dσ, s>t, (1.2)

    and

    ητ(s)=s0φ(σ)dσ.

    Next, we give the main features of this paper as follows.

    1) Inspired by [18,23], we provide a detailed representation and Gronwall type estimates for the energy of (1.1) dependent on ε in Lemma 2.1, with a focus on the parameters ω, Λ and the increasing function J(). Using these parameters, we construct the universe D and derive the existence of Dpullback absorbing sets, see Lemma 4.9.

    2) Via the decomposition method, we show that the process of the system has the property of Dκpullback contraction in the space NV, and the Dpullback asymptotic compactness is obtained naturally. Based on the theory of attractor in [1,24], the D-pullback attractors for the process {Sε(t,τ)} in NV are derived, see Theorem 3.3.

    3) When the perturbation parameter ε0 with the non-autonomous external force, the robustness is obtained via the upper semi-continuity of pullback attractors of (1.1) by using the technique in [18,21,22], see Theorems 3.2 in Section 3.

    This paper is organized as follows. Some preliminaries are given in Section 2, and the main results are stated in Section 3, which contains the global well-posedness of solution, the existence of pullback attractors and robustness. Finally, the detailed proofs are provided in Sections 4 and 5.

    ● The Sobolev spaces

    Let E={u|u(C0(Ω))3,divu=0}, H is the closure of E in (L2(Ω))3 topology with the norm and inner product as

    |u|=uH=(u,v)1/2, (u,v)=3j=1Ωuj(x)vj(x)dx,  u,vH.

    V is the closure of E in (H1(Ω))3 topology with the norm and inner product as

    u=uV=((u,u))1/2V, ((u,v))V=3i,j=1Ωujxivjxidx,  u,vV.

    Also, we denote

    ((u,v))Vα=(u,v)+α((u,v))V, ||u||2Vα=|u|2+α||u||2.

    H and V are Hilbert spaces with their dual spaces H and V respectively, and , denote the norm in V and the dual product between V and V respectively, and also H to itself.

    ● The fractional power functional spaces

    Let PL be the Helmholz-Leray orthogonal projection in (L2(Ω))3 onto H [3,7], and

    PL: HHH,

    where

    H={u(L2(Ω))3;  χ(L2loc(Ω))3: u=χ}.

    A=PLΔ is the Stokes operator with eigenvalues {λj}j=1 and orthonormal eigenfunctions {ωj}j=1.

    Define the fractional operator As by

    Asu=jλsj(u,ωj)ωj,  sR,  jZ+

    for u=j(u,ωj)ωj with the domain D(As)={u|AsuH}, and we use the norm of D(As) as

    u22s=|Asu|2=jλ2sj|(u,ωj)|2.

    Especially, denote W=D(A), and V=D(A1/2) with norm u1=|A1/2u|=u for any uV.

    ● The memory spaces

    For any s(0,), we define μ(s)=g(s), which is nonnegative, absolutely continuous, decreasing (μ0 almost everywhere) and

    κ=0μ(s)ds>0. (2.1)

    Also, there exists δ>0 such that

    μ(s)+δμ(s)0, a.e. s(0,). (2.2)

    Let

    MX=L2μ(R+;X), X=Vor W,

    which is a Hilbert space on R+ with inner product and norm

    ((η,ζ))MX=0μ(s)((η(s),ζ(s))Xds, ηMX=(0μ(s)||η(s)||2Xds)1/2.

    Moreover, the extended memory space can be defined as

    NX=X×MX

    equipped with the norm

    (u,η)2NX=u2X+η2MX.

    ● The bilinear and trilinear operators

    The bilinear and trilinear operators are defined as follows [8]

    B(u,v):=PL((u)v),   u,vV, (2.3)
    b(u,v,w)=<B(u,v),w>=3i,j=1Ωuivjxiwjdx. (2.4)

    Denote B(u)=B(u,u), B(u,v) is a continuous operator from V×V to V, and there hold

    b(u,v,v)=0, b(u,v,w)=b(u,w,v),  u,v,wV. (2.5)

    ● Some useful lemmas

    Lemma 2.1. ([23]) Assume that

    1) A nonnegative function h is locally summable on R+, and for any ε(0,ε0] and any tτ0 there holds

    εtτeε(ts)h(s)ds85supt0t+1th(s)ds<.

    2) The nonnegative function yε(t) is absolutely continuous on [τ,), and satisfies for some constants R,C00 that

    yε(t)Reε(tτ)+εptτeε(ts)h(s)yε(s)qds+C0ε1+r,

    where p,q,r0, and p1>(q1)(1+r)0.

    3) Let z(t)0 be a continuous function on (0,) equivalent to yε(t), which means there exist some constants M1, L0 such that

    z(t)Myε(t)M(z(t)+L).

    Then, there exist ω, Λ>0 and an increasing function J(): R+R+ such that

    z(t)J(MR)eω(tτ)+Λ(MC0+L).

    Remark 2.1. Under the assumptions in Lemma 2.1, there exists a constant θ(0,1) satisfying

    p1=pθθ+1q>0, p2=1θrθ>0.

    Denote

    p3=max{ε1/θ0,(2supt0t+1th(s)ds)1/p1,C1/p20}, p4=2max{6Rp13,1},

    then

    ω=ωθ,p,q,r,C0=12pθ3, Λ=Λθ,p,q,r,C0=5p1p23,
    J(R)=Jθ,p,q,r,C0(R)=2pq4p3exp(pθ412θln(6pq4)).

    Lemma 2.2. ([15]) Let η be the past history variable and (1.2) holds. Then

    ηt(s)2MXητ(s)2MX2tτ((ησ,u(σ)))MXdσ.

    We assume that f1(x) and f2(t,x) satisfy the following hypotheses:

    (C1) The function f1H.

    (C2) f2(t,x) is translation bounded in L2loc(R,H), which means there exists a constant K>0 such that

    suptRt+1t|f2(s)|2ds<K,

    and for any tR, there also holds

    teιsf2(s)2ds<, 0<ινε0, ν=min{ακδ72,1}, (3.1)

    where κ,δ are the same as parameters in (2.1) and (2.2) respectively.

    Construct the infinitesimal generator of right-translation semigroup on MX

    Tη=sη,

    whose domain is

    D(T)={ηMX: sηMX,  η(0)=0}.

    Given initial datum U(τ)=(u(τ),ητ)NV, then (1.1) can be transformed into the following abstract form

    {t(u+αAu)+0μ(s)Aη(s)ds+B(u,u)+βu=PLfε(t,x), (t,x)Ωτ,tη=Tη+u,divu=0, (t,x)Ωτ,u(t,x)=0, (t,x)Ωτ,u(τ,x)=u(τ), xΩ,ητ(s)=s0φ(σ)dσ. (3.2)

    ● Global well-posedness of solution

    Definition 3.1 A function U(t)=(u(t),ηt): [τ,+)NV is called the weak solution to (3.2), if for any fixed T>τ there hold

    (i) U(t)C([τ,T];NV), utL2(τ,T;V).

    (ii) U(τ)=(u(τ),ητ).

    (iii) for any wC1([τ,T];V) with w(T,x)=0, there holds

    Tτu+αAu,wtdt+Tτ0μ(s)((η(s),w))Vdsdt+Tτb(u,u,w)dt+Tτ(βu,w)dt=((u(τ),w(τ)))Vα+Tτ(PLfε,w)dt. (3.3)

    Theorem 3.2. Let U(τ)NV, and the hypotheses (C1)–(C2) hold. Then the global weak solution U(t,x) to system (3.2) uniquely exists on (τ,T), which generates a strongly continuous process

    Sε(t,τ): NVNV,  tτ,0ε<ε0

    and Sε(t,τ)U(τ)=U(t).

    Proof. The global well-posedness of solution can be obtained by the Galerkin approximation method, energy estimates and compact scheme. The detailed proof can be found in [15,18] and is omitted here.

    ● Existence of D-pullback attractors

    Theorem 3.3. Assume U(τ)NV and the hypotheses (C1)–(C2) hold. Then the process Sε(t,τ): NVNV generated by the system (3.2) possesses a minimal family of D-pullback attractors Aε={Aε(t)}tR in NV.

    Proof. See Section 4.2.

    When ε=0, the system (3.2) can be reduced to the following autonomous system

    {t(u+αAu)+0μ(s)Aη(s)ds+B(u,u)+βu=PLf1(x), (t,x)Ωτ,tη=Tη+u,divu=0, (t,x)Ωτ,u(t,x)=0, (t,x)Ωτ,u(τ,x)=u(τ), xΩ,ητ(s)=s0φ(σ)dσ. (3.4)

    Remark 3.1. The existence of global attractor A0 in NV can be achieved for the semigroup S0(tτ) generated by (3.4).

    ● Robustness: upper semi-continuity of D-pullback attractors

    Let be a metric space, and {Aλ}λ is a family of subsets in X. Then it is said that {Aλ} has the property of upper semi-continuity as λλ0 in X if

    limλλ0distX(Aλ,Aλ0)=0.

    The upper semi-continuity of attractors and related conclusions can be referred to [1,19,20,22] for more details.

    In the following way, we intend to establish some results on the convergence between D-pullback attractors Aε to system (3.2) and global attractor A0 to system (3.4) as ε0.

    Theorem 3.4. Let U(τ)NV, Aε is the family of D-pullback attractors of Sε(t,τ) in NV to system (3.2), and A0 is the global attractor of S0(tτ) in NV to system (3.4). Then the robustness of system is obtained by the following upper semi-continuity

    limε0distNV(Aε,A0)=0.

    Proof. See Section 5.

    In this section, we first give the fundamental theory of attractors for dissipative systems, and the related conclusions can be seen in [1,2,3,7].

    ● Some relevant definitions

    Definition 4.1. Assume that P(X) is the family of all nonempty subsets in a metric space X. If D is some nonempty class of families in the form ˆD={D(t):tR}P(X), where D(t)X is nonempty and bounded, then D is said to be a universe in P(X).

    Definition 4.2. The family ˆD0={D0(t):tR}P(X) is D-pullback absorbing for the process S(,) on X if for any tR and any ˆDD, there exists a τ0(t,ˆD)t such that

    S(t,τ)D(τ)D0(t),  ττ0(t,ˆD).

    Definition 4.3. A process S(,) on X is said to be D-pullback asymptotically compact if for any tR, any ˆDD, and any sequences {τn}(,t] and {xn}X satisfying τn and xnD(τn), the sequence {S(t,τn)xn} is relatively compact in X.

    The D-pullback asymptotic compactness can be characterized by the Kuratowski measure of noncompactness κ(B) (BX), relating definition and properties can be referred to [25,26], and the definition of Dκ-pullback contraction will be given as follows.

    Definition 4.4. For any tR and ε>0, a process S(t,τ) on X is said to be Dκ-pullback contracting if there exists a constant TD(t,ε)>0 such that

    κ(S(t,tτ)D(tτ))ε,  τTD(t,ε).

    Definition 4.5. A family A(t)={A(t)}tR is called the D-pullback attractors of process S(t,τ), if for any tR and any {D(t)}D, the following properties hold.

    (i) A(t) is compact in X.

    (ii) S(t,τ)A(τ)=A(t), tτ.

    (iii) limτdistX(S(t,τ)D(τ),A(t))=0.

    In addition, D-pullback attractor A is said to be minimal if whenever ˆC is another D-attracting family of closed sets, then A(t)C(t) for all tR.

    ● Some conclusions

    Theorem 4.6. ([1,27]) Let S(,):R2d×XX be a continuous process, where R2d={(t,τ)R2|tτ}, D is a universe in P(X), and a family ˆD0={D0(t):tR}P(X) is D-pullback absorbing for S(,), which is D-pullback asymptotically compact. Then, the family of D-pullback attractors AD={AD(t):tR} exists and

    AD(t)=s0¯τsS(t,tτ)D(tτ)X, tR.

    Remark 4.1. If ˆD0D, then AD is minimal family of closed subsets attracting pullback to D. It is said to be unique provided that ˆD0D, D0(t) is closed for any tR, and D is inclusion closed.

    Theorem 4.7. ([21]) Assume that ˜D={ˆ˜D(t)} is a family of sets in X, S(,) is continuous, and, for any tR, there exists a constant T(t,D,˜D) such that

    S(t,tτ)D(tτ)˜D(t),  τT(t,D,˜D).

    If S(,) is D-pullback absorbing and ˆDκ-pullback contracting, then the D-pullback attractors AD={AD(t):tR} exist for S(,).

    Lemma 4.8. ([28]) Assume that S(,)=S1(,)+S2(,), ˜D={ˆ˜D(t)} is a family of subsets in X, and for any tR and any τR+ there hold

    (i) For any u(tτ)˜D(tτ),

    S1(t,tτ)u(tτ)XΦ(t,τ)0 (τ+).

    (ii) For any Tτ, 0τTS2(t,tτ)˜D(tτ) is bounded, and S2(t,tτ)˜D(tτ) is relatively compact in X.

    Then S(,) is ˆDκ-pullback contracting in X.

    From Theorems 3.2, we know that the system (3.2) generates a continuous process Sε(t,τ) in NV. To obtain the D-pullback attractors, we need to establish the existence of D-pullback absorbing set and the D-pullback asymptotic compactness of Sε(t,τ).

    ● Existence of D-pullback absorbing set in NV

    Let D denote a family of all {D(t)}tRP(NV) satisfying

    limτeωτsupU(τ)D(τ)J(2|U(τ)|2)=0,

    where ω=ω3/4,1,4,3,fε>0 and J()=J3/4,1,4,3,fε(). Next, we establish the existence of D-pullback absorbing set.

    Lemma 4.9. Let (u(τ),ητ)NV, then the process {Sε(t,τ)} to system (3.2) possesses a D-pullback absorbing set ˆDε0(t)={Dε0(t)}tR in NV, where

    Dε0(t)=ˉBNV(0,ρεNV(t)),

    with radius

    ρεNV(t)=2Λ3/4,1,4,3,fε(2C(|f1|2+εK)+1). (4.1)

    Proof. Multiplying (3.2) by u, we have

    12ddtu2Vα+0μ(s)Aη(s)u(t)ds+β|u|2=(PLfε,u), (4.2)

    that is

    12ddtu2Vα+0μ(s)Aη(s)(tη(s)+sη(s))ds+β|u|2=12ddt(u2Vα+η2MV)+120μ(s)ddsη2ds+β|u|2|(fε,u)|. (4.3)

    Multiplying (3.2) by ut, we have

    ut2Vα+((η,ut))MV+12βddt|u|2+b(u,u,ut)=(PLfε,ut). (4.4)

    Then, the interpolation inequality and Young inequality lead to

    βddt|u|2+2ut2Vα2|((η,ut))MV|+2|(fε,ut)|+2|b(u,u,ut)|2|((η,ut))MV|+2|(fε,ut)|+CuL3uuL62|((η,ut))MV|+2|(fε,ut)|+C|u|1/2u1/2uuL6αut2+Cη2MV+C|u|u3+C|fε|2. (4.5)

    To estimate the term 0μ(s)ddsη2ds in (4.3) and avoid the possible singularity of μ at zero, we refer to [18] and construct the following new function

    ˜μ(s)={μ(˜s),0<s˜s,μ(s),s>˜s

    where ˜s is fixed such that ˜s0μ(s)dsκ/2. Also, if we set

    Φ(t)=4κ0˜μ(s)((η(s),u(t)))ds,

    then differentiating in t leads to

    ddtΦ(t)+u24μ(˜s)κ20μ(s)ddsη2ds+4ακεη2MV+αεut2. (4.6)

    We use the technique in [18] and set

    yε(t)=E(t)+νεΦ(t)+ε2Ψ(t),

    where

    E(t)=12(u2Vα+η2MV), Ψ(t)=2β|u|2.

    For sufficient small ε, it leads to

    E(t)2yε(t)2(E(t)+1),

    where we choose ε0 satisfying

    νε0supt[τ,T]Φ(t)+ε20supt[τ,T]Ψ(t)=1,

    and 0ε<ε0. Then, there holds

    ddtyε(t)+Cεyε(t)Cε4yε(t)3+C|fε|2, (4.7)

    and

    yε(t)yε(τ)eε(tτ)+Cε4tτeε(ts)1yε(s)3ds+CsuptRt+1t|fε|2ε1dsyε(τ)eε(tτ)+Cε4tτeε(ts)1yε(s)3ds+C(|f1|2+εK)ε1. (4.8)

    Then by Lemma 2.1, there exist

    ω=ω3/4,1,4,3,fε>0, Λ=Λ3/4,1,4,3,fε>0,

    and an increasing function

    J()=J3/4,1,4,3,fε(): R+R+

    such that

    E(t)J(2E(τ))eω(tτ)+Λ(2C(|f1|2+εK)+1),

    which implies the conclusion holds.

    Remark 4.2. For the semigroup S0(tτ), it has the global absorbing set D00 in NV, where

    D00={UNV; UNVρ0NV=2Λ3/4,1,4,3,f1(2C|f1|2+1)} (4.9)

    and

    lim supε0ρεNV(t)=ρ0NV. (4.10)

    Dκ-pullback contraction of Sε(t,τ) in NV

    To verify the pullback contraction of Sε(t,τ), we decompose Sε(t,τ) as follows

    Sε(t,τ)U(τ)=Sε1(tτ)U1(τ)+Sε2(t,τ)U2(τ)=:U1(t)+U2(t),

    which solve the following two problems respectively

    {t(u1+αAu1)+0μ(s)Aη1(s)ds+B(u,u1)=0, (t,x)Ωτ,tη1=Tη1+u1,divu1=0, (t,x)Ωτ,u1(t,x)=0, (t,x)Ωτ,u1(τ,x)=u(τ), xΩ,ητ1(s)=s0φ(σ)dσ, (4.11)

    and

    {t(u2+αAu2)+0μ(s)Aη2(s)ds+B(u,u2)+βu2=PLfεβu1, (t,x)Ωτ,tη2=Tη2+u2,divu2=0, (t,x)Ωτ,u2(t,x)=0, (t,x)Ωτ,u2(τ,x)=0, xΩ,ητ2(s)=0. (4.12)

    Lemma 4.10. Let U(τ)Dε0(τ), then the solution Sε1(tτ)U(τ) to the system (4.11) satisfies

    Sε1(tτ)U(τ)NVJ(2E(τ))eω(tτ)0 (τ).

    Proof. Multiplying (4.11) by u1 and tu1 respectively, and repeating the reasonings as shown as in Lemma 4.9, in which β=0 and fε=0, we can derive the conclusion finally. The parameter ω is dependent on ε and the increasing function J() is different from the one in Lemma 4.9. Despite all this, these parameters can be unified in same representation, and the concrete details are omitted here.

    Lemma 4.11. Let U(τ)Dε0(τ), then for any tR, there exists Cε(t)>0 such that the solution Sε2(t,τ)U(τ) to the system (4.12) satisfies

    Sε2(t,τ)U(τ)NWCε(t).

    Proof. Multiplying (4.12) by Au2, we have

    12ddt(u22+α|Au2|2)+0μ(s)Aη2(s)Au2(t)ds+βu22+b(u,u2,Au2)=(PLfεβu1,Au2), (4.13)

    from the existence of pullback absorbing set, Lemma 4.10, the interpolation inequality and Young inequality, we have

    12ddt(u22+α|Au2|2+η22MW)+120μ(s)dds|Aη2|2ds+βu22|(fε,Au2)|+|(βu1,Au2)|+uL6u2L3|Au2|νε4|Au2|2+Cuu21/2|Au2|1/2|Au2|+C|fε|2νε2|Au2|2+C|fε|2+C. (4.14)

    Multiplying (4.12) by Atu2, we have

    tu22+α|Atu2|2+((η2,tu2))MW+12βddtu22+b(u,u2,tu2)=(PLfεβu1,Atu2). (4.15)

    By the existence of pullback absorbing set, Lemma 4.10 and Young inequality, one has

    βddtu22+2tu22+2α|Atu2|22|((η2,tu2))MW|+2|(PLfεβu1,Atu2)|+2|b(u,u2,tu2)|2|((η2,tu2))MW|+2|(PLfεβu1,Atu2)|+C|Au2||Atu2|α|Atu2|2+Cη22MW+C|Au2|2+C|fε|2. (4.16)

    To estimate the term 0μ(s)dds|Aη2|2ds in (4.14), we set

    Φ2(t)=6κ0˜μ(s)(Aη2(s),Au2(t))ds,

    and differentiating in t leads to

    ddtΦ2(t)+6κ0˜μ(s)(Au2(t),Au2(t))ds6κ0˜μ(s)(Asη2,Au2(t))ds6κ0˜μ(s)(Aη2(s),Atu2(t))ds, (4.17)

    where

    6κ0˜μ(s)(Au2(t),Au2(t))ds6κ˜s˜μ(s)ds|Au2(t)|2, (4.18)

    and

    6κ0˜μ(s)(Asη2,Au2(t))ds=6κ˜sμ(s)(Aη2,Au2(t))ds6κ˜sμ(s)|Aη2||Au2(t)|ds6κ(˜sμ(s)|Aη2|2ds)1/2(˜sμ(s)|Au2(t)|2ds)1/26κ(˜sμ(s)dds|Aη2|2ds)1/2(2μ(˜s))1/2|Au2(t)||Au2(t)|2+18μ(˜s)κ2˜sμ(s)dds|Aη2|2ds, (4.19)

    and

    6κ0˜μ(s)(Aη2(s),Atu2(t))ds6κ0μ(s)|Aη2(s)||Atu2(t)|dsαε|Atu2(t)|2+9ακ2εη2MW. (4.20)

    Also, from the fact that μ(s)+δμ(s)0, we have

    0μ(s)dds|Aη2|2ds0δμ(s)|Aη2|2ds=δη22MW. (4.21)

    Thus

    ddtΦ2(t)+2|Au2|218μ(˜s)κ20μ(s)dds|Aη2|2ds+9ακεη22MW+αε|Atu2|2. (4.22)

    We use the technique in [18] and set

    zε(t)=E2(t)+νεΦ2(t)+ε2Ψ2(t),

    where

    E2(t)=12(u22+α|Au2|2+η22MW), Ψ2(t)=βu2.

    For sufficient small enough ε, it leads to

    E2(t)2zε(t)2(E2(t)+1),

    and there holds

    ddtzε(t)+νεzε(t)C+C|fε|2, (4.23)

    it follows from the Gronwall lemma that

    E2(t)Ceνε(tτ)E2(τ)+Cεtτeνε(st)|f2(s,x)|2ds+C|f1|2+CCεeνεttτeνε0s|f2(s,x)|2ds+C|f1|2+C, (4.24)

    which means the conclusion holds.

    Above all, Lemmas 4.10, 4.11 and 4.8 lead to

    Lemma 4.12. Let U(τ)NV, then the process Sε(t,τ): NVNV generated by the system (3.2) is Dκ-pullback contracting in NV.

    Consequently, from Theorem 4.7, we can finish the proof of Theorem 3.3.

    By the definition of upper semi-continuity, the following lemmas can be used to obtain the robustness of pullback attractors for evolutionary systems.

    Lemma 5.1. ([20]) Let ε(0,ε0], {Sε(t,τ)} is the process of evolutionary system with non-autonomous term (depending on ε), which is obtained by perturbing the semigroup S0(τ) of system without ε, and, for any tR, there also hold that

    (i) Sε(t,τ) has the pullback attractors Aε(t), and A0 is the global attractor for S0(τ).

    (ii) For any τR+ and any uX, there holds uniformly that

    limε0dX(Sε(t,tτ)u,S0(τ)u)=0.

    (iii) There exists a compact subset GX such that

    limε0distX(Aε(t),G)=0.

    Then, for any tR, there holds

    limε0distX(Aε,A0)=0.

    Lemma 5.2. ([21]) For any tR, τR+, and ε(0,ε0], ˆDε0(t)={Dε0(t):tR} is the pullback absorbing set for Sε(t,τ), and ˆCε0(t)={Cε0(t):tR} is a family of compact subsets in X. Assume that Sε(,)=Sε1(,)+Sε2(,), and there hold

    (i) For any utτDε0(tτ),

    Sε1(t,tτ)utτXΦ(t,τ)0 (τ).

    (ii) For any Tτ, 0τTSε2(t,tτ)Dε0(tτ) is bounded, and there exists a constant TDε0(t), independent of ε, such that

    Sε2(t,tτ)Dε0(tτ)Cε0(t),  τ>TDε0(t).

    (iii) There is a compact subset GX such that

    limε0distX(Cε0(t),G)=0.

    Then, the process Sε(t,τ) has the pullback attractors Aε(t), and

    limε0distX(Aε,G)=0.

    We give the following procedure to verify Theorem 3.4.

    Lemma 5.3. Let (uε,ηε)=Sε(t,τ)U(τ) be the solution to system (3.2), and (u,η)=S0(tτ)U(τ) is the solution to system (3.4), then, for any bounded subset BNV, there holds

    limε0supU(τ)BdNV(Sε(t,τ)U(τ),S0(tτ)U(τ))=0.

    Proof. We know

    {t(uε+αAuε)+0μ(s)Aηε(s)ds+B(uε,uε)+βuε=PLfε(t,x), (t,x)Ωτ,tηε=Tηε+uε, (5.1)

    and

    {t(u+αAu)+0μ(s)Aη(s)ds+B(u,u)+βu=PLf1(x), (t,x)Ωτ,tη=Tη+u. (5.2)

    Let wε=uεu and ξε=ηεη, we can derive

    t(wε+αAwε)+0μ(s)Aξε(s)ds+B(uε,wε)+B(wε,u)+βwε=εPLf2(t,x), (5.3)

    and multiplying it by wε leads to

    12ddtwε2α+0μ(s)(Aξε(s),wε(t))ds+b(wε,u,wε)+β|wε|2=ε(PLf2(t,x),wε), (5.4)

    it follows that

    12ddtwε2α+0μ(s)((ξε(s),wε))ds|b(wε,u,wε)|+ε|(f2(t,x),wε)|. (5.5)

    Integrating (5.5) over [τ,t], from Lemma 2.2 we derive that

    wε(t)2α+ξε(ts)2MVwε(τ)2α+ξε(τs)2MV+2tτ|b(wε,u,wε)|ds+2εtτ|(f2(t,x),wε)|ds(wε,ξε)|τ2NV+Ctτuwε2ds+2εtτ|(f2(t,x),wε)|ds(wε,ξε)|τ2NV+ε2tτ|f2|2ds+Ctτwε2ds, (5.6)

    that is

    (wε,ξε)|t2NV(wε,ξε)|τ2NV+ε2tτ|f2|2ds+Ctτ(wε,ξε)|s2NVds, (5.7)

    and the Gronwall inequality leads to

    (wε,ξε)|t2NVC((wε,ξε)|τ2NV+ε2tτ|f2|2ds)0 (ε0), (5.8)

    which means that the conclusion is finished.

    Proof of Theorem 3.4. From (4.24) and the fact that WV is compact, we know that there exists a compact subset GNV such that

    limε0distX(Cε0(t),G)=0. (5.9)

    Combining Lemma 4.10, Lemma 5.2 and (5.9), we have

    limε0distX(Aε,G)=0.

    In addition, the confirmation of condition (ii) in Lemma 5.1 is finished from Lemma 5.3, and we have

    limε0distX(Aε,A0)=0.

    Rong Yang was partially supported by the Science and Technology Project of Beijing Municipal Education Commission (No. KM202210005011).

    The authors declare there is no conflict of interest.



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