The tempered pullback dynamics and robustness of the 3D Navier-Stokes-Voigt equations with memory and perturbed external force are considered in this paper. Based on the global well-posedness results and energy estimates involving memory, a suitable tempered universe is constructed, the robustness is finally established via the upper semi-continuity of tempered pullback attractors when the perturbation parameter epsilon tends to zero.
Citation: Keqin Su, Rong Yang. Pullback dynamics and robustness for the 3D Navier-Stokes-Voigt equations with memory[J]. Electronic Research Archive, 2023, 31(2): 928-946. doi: 10.3934/era.2023046
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The tempered pullback dynamics and robustness of the 3D Navier-Stokes-Voigt equations with memory and perturbed external force are considered in this paper. Based on the global well-posedness results and energy estimates involving memory, a suitable tempered universe is constructed, the robustness is finally established via the upper semi-continuity of tempered pullback attractors when the perturbation parameter epsilon tends to zero.
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(u−αΔu)−∫∞0g(s)Δu(t−s)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), x∈Ω,u(τ−s,x)=φ(s,x), (s,x)∈Ω0, | (1.1) |
where Ωτ=(τ,+∞)×Ω, ∂Ωτ=(τ,+∞)×∂Ω, Ω0=(0,∞)×Ω, τ∈R+ 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) (0≤ε<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(t−σ)dσ, s≥0, |
which satisfies
∂∂tη=−∂∂sη+u(t). |
Also, η has the explicit representation
{ηt(s)=∫s0u(t−σ)dσ, 0<s≤t,ηt(s)=η0(s−t)+∫t0u(t−σ)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 D−pullback absorbing sets, see Lemma 4.9.
2) Via the decomposition method, we show that the process of the system has the property of D−κ−pullback contraction in the space NV, and the D−pullback 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 ε→0 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∈(C∞0(Ω))3,divu=0}, H is the closure of E in (L2(Ω))3 topology with the norm and inner product as
|u|=‖u‖H=(u,v)1/2, (u,v)=3∑j=1∫Ωuj(x)vj(x)dx, ∀ u,v∈H. |
V is the closure of E in (H1(Ω))3 topology with the norm and inner product as
‖u‖=‖u‖V=((u,u))1/2V, ((u,v))V=3∑i,j=1∫Ω∂uj∂xi∂vj∂xidx, ∀ u,v∈V. |
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: H⊕H⊥→H, |
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, s∈R, j∈Z+ |
for u=∑j(u,ωj)ωj with the domain D(As)={u|Asu∈H}, and we use the norm of D(As) as
‖u‖22s=|Asu|2=∑jλ2sj|(u,ωj)|2. |
Especially, denote W=D(A), and V=D(A1/2) with norm ‖u‖1=|A1/2u|=‖u‖ for any u∈V.
● The memory spaces
For any s∈(0,∞), we define μ(s)=−g′(s), which is nonnegative, absolutely continuous, decreasing (μ′≤0 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, ‖η‖MX=(∫∞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=‖u‖2X+‖η‖2MX. |
● The bilinear and trilinear operators
The bilinear and trilinear operators are defined as follows [8]
B(u,v):=PL((u⋅∇)v), ∀ u,v∈V, | (2.3) |
b(u,v,w)=<B(u,v),w>=3∑i,j=1∫Ωui∂vj∂xiwjdx. | (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,w∈V. | (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 t≥τ≥0 there holds
ε∫tτe−ε(t−s)h(s)ds≤85supt≥0∫t+1th(s)ds<∞. |
2) The nonnegative function yε(t) is absolutely continuous on [τ,∞), and satisfies for some constants R,C0≥0 that
yε(t)≤Re−ε(t−τ)+εp∫tτe−ε(t−s)h(s)yε(s)qds+C0ε1+r, |
where p,q,r≥0, and p−1>(q−1)(1+r)≥0.
3) Let z(t)≥0 be a continuous function on (0,∞) equivalent to yε(t), which means there exist some constants M≥1, L≥0 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)e−ω(t−τ)+Λ(MC0+L). |
Remark 2.1. Under the assumptions in Lemma 2.1, there exists a constant θ∈(0,1) satisfying
p1=pθ−θ+1−q>0, p2=1−θ−rθ>0. |
Denote
p3=max{ε−1/θ0,(2supt≥0∫t+1th(s)ds)1/p1,C1/p20}, p4=2max{6Rp−13,1}, |
then
ω=ωθ,p,q,r,C0=12p−θ3, Λ=Λθ,p,q,r,C0=5p1−p23, |
J(R)=Jθ,p,q,r,C0(R)=2pq4p3exp(pθ41−2−θln(6pq4)). |
Lemma 2.2. ([15]) Let η be the past history variable and (1.2) holds. Then
‖ηt(s)‖2MX−‖ητ(s)‖2MX≤2∫tτ((ησ,u(σ)))MXdσ. |
We assume that f1(x) and f2(t,x) satisfy the following hypotheses:
(C1) The function f1∈H.
(C2) f2(t,x) is translation bounded in L2loc(R,H), which means there exists a constant K>0 such that
supt∈R∫t+1t|f2(s)|2ds<K, |
and for any t∈R, there also holds
∫t−∞eιs‖f2(s)‖2ds<∞, 0<ι≤νε0, 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)={η∈MX: ∂∂sη∈MX, 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), x∈Ω,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), ∂u∂t∈L2(τ,T;V).
(ii) U(τ)=(u(τ),ητ).
(iii) for any w∈C1([τ,T];V) with w(T,x)=0, there holds
−∫Tτ⟨u+αAu,wt⟩dt+∫Tτ∫∞0μ(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): NV→NV, ∀ t≥τ,0≤ε<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): NV→NV generated by the system (3.2) possesses a minimal family of D-pullback attractors Aε={Aε(t)}t∈R 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), x∈Ω,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(t−τ) 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 λ→λ0 in X if
limλ→λ0distX(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 ε→0.
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(t−τ) in NV to system (3.4). Then the robustness of system is obtained by the following upper semi-continuity
limε→0distNV(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):t∈R}⊂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):t∈R}⊂P(X) is D-pullback absorbing for the process S(⋅,⋅) on X if for any t∈R and any ˆD∈D, there exists a τ0(t,ˆD)≤t such that
S(t,τ)D(τ)⊂D0(t), ∀ τ≤τ0(t,ˆD). |
Definition 4.3. A process S(⋅,⋅) on X is said to be D-pullback asymptotically compact if for any t∈R, any ˆD∈D, and any sequences {τn}⊂(−∞,t] and {xn}⊂X satisfying τn→−∞ and xn∈D(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) (B⊂X), relating definition and properties can be referred to [25,26], and the definition of D−κ-pullback contraction will be given as follows.
Definition 4.4. For any t∈R and ε>0, a process S(t,τ) on X is said to be D−κ-pullback contracting if there exists a constant TD(t,ε)>0 such that
κ(S(t,t−τ)D(t−τ))≤ε, ∀ τ≥TD(t,ε). |
Definition 4.5. A family A(t)={A(t)}t∈R is called the D-pullback attractors of process S(t,τ), if for any t∈R and any {D(t)}∈D, the following properties hold.
(i) A(t) is compact in X.
(ii) S(t,τ)A(τ)=A(t), t≥τ.
(iii) limτ→−∞distX(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 t∈R.
● Some conclusions
Theorem 4.6. ([1,27]) Let S(⋅,⋅):R2d×X→X be a continuous process, where R2d={(t,τ)∈R2|t≥τ}, D is a universe in P(X), and a family ˆD0={D0(t):t∈R}⊂P(X) is D-pullback absorbing for S(⋅,⋅), which is D-pullback asymptotically compact. Then, the family of D-pullback attractors AD={AD(t):t∈R} exists and
AD(t)=⋂s≥0¯⋃τ≥sS(t,t−τ)D(t−τ)X, t∈R. |
Remark 4.1. If ˆD0∈D, then AD is minimal family of closed subsets attracting pullback to D. It is said to be unique provided that ˆD0∈D, D0(t) is closed for any t∈R, 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 t∈R, there exists a constant T(t,D,˜D) such that
S(t,t−τ)D(t−τ)⊂˜D(t), ∀ τ≥T(t,D,˜D). |
If S(⋅,⋅) is D-pullback absorbing and ˆD−κ-pullback contracting, then the D-pullback attractors AD={AD(t):t∈R} 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 t∈R and any τ∈R+ there hold
(i) For any u(t−τ)∈˜D(t−τ),
‖S1(t,t−τ)u(t−τ)‖X≤Φ(t,τ)→0 (τ→+∞). |
(ii) For any T≥τ, ∪0≤τ≤TS2(t,t−τ)˜D(t−τ) is bounded, and S2(t,t−τ)˜D(t−τ) is relatively compact in X.
Then S(⋅,⋅) is ˆD−κ-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)}t∈R⊂P(NV) satisfying
limτ→−∞eωτsupU(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)}t∈R in NV, where
Dε0(t)=ˉBNV(0,ρεNV(t)), |
with radius
ρεNV(t)=√2√Λ3/4,1,4,3,fε(2C(|f1|2+εK)+1). | (4.1) |
Proof. Multiplying (3.2) by u, we have
12ddt‖u‖2Vα+∫∞0μ(s)Aη(s)u(t)ds+β|u|2=(PLfε,u), | (4.2) |
that is
12ddt‖u‖2Vα+∫∞0μ(s)Aη(s)(∂tη(s)+∂sη(s))ds+β|u|2=12ddt(‖u‖2Vα+‖η‖2MV)+12∫∞0μ(s)dds‖η‖2ds+β|u|2≤|(fε,u)|. | (4.3) |
Multiplying (3.2) by ut, we have
‖ut‖2Vα+((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+2‖ut‖2Vα≤2|((η,ut))MV|+2|(fε,ut)|+2|b(u,u,ut)|≤2|((η,ut))MV|+2|(fε,ut)|+C‖u‖L3‖u‖‖u‖L6≤2|((η,ut))MV|+2|(fε,ut)|+C|u|1/2‖u‖1/2‖u‖‖u‖L6≤α‖ut‖2+C‖η‖2MV+C|u|‖u‖3+C|fε|2. | (4.5) |
To estimate the term ∫∞0μ(s)dds‖η‖2ds 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)ds≤κ/2. Also, if we set
Φ(t)=−4κ∫∞0˜μ(s)((η(s),u(t)))ds, |
then differentiating in t leads to
ddtΦ(t)+‖u‖2≤4μ(˜s)κ2∫∞0μ(s)dds‖η‖2ds+4ακε‖η‖2MV+αε‖ut‖2. | (4.6) |
We use the technique in [18] and set
yε(t)=E(t)+νεΦ(t)+ε2Ψ(t), |
where
E(t)=12(‖u‖2Vα+‖η‖2MV), 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 0≤ε<ε0. Then, there holds
ddtyε(t)+Cεyε(t)≤Cε4yε(t)3+C|fε|2, | (4.7) |
and
yε(t)≤yε(τ)e−ε(t−τ)+Cε4∫tτe−ε(t−s)⋅1⋅yε(s)3ds+Csupt∈R∫t+1t|fε|2ε−1ds≤yε(τ)e−ε(t−τ)+Cε4∫tτe−ε(t−s)⋅1⋅yε(s)3ds+C(|f1|2+εK)ε−1. | (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))e−ω(t−τ)+Λ(2C(|f1|2+εK)+1), |
which implies the conclusion holds.
Remark 4.2. For the semigroup S0(t−τ), it has the global absorbing set D00 in NV, where
D00={U∈NV; ‖U‖NV≤ρ0NV=√2√Λ3/4,1,4,3,f1(2C|f1|2+1)} | (4.9) |
and
lim supε→0ρεNV(t)=ρ0NV. | (4.10) |
● D−κ-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(t−τ)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), x∈Ω,ητ1(s)=∫s0φ(σ)dσ, | (4.11) |
and
{∂∂t(u2+αAu2)+∫∞0μ(s)Aη2(s)ds+B(u,u2)+βu2=PLfε−βu1, (t,x)∈Ωτ,∂∂tη2=Tη2+u2,divu2=0, (t,x)∈Ωτ,u2(t,x)=0, (t,x)∈∂Ωτ,u2(τ,x)=0, x∈Ω,ητ2(s)=0. | (4.12) |
Lemma 4.10. Let U(τ)∈Dε0(τ), then the solution Sε1(t−τ)U(τ) to the system (4.11) satisfies
‖Sε1(t−τ)U(τ)‖NV≤J(2E(τ))e−ω(t−τ)→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 t∈R, 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)‖NW≤Cε(t). |
Proof. Multiplying (4.12) by Au2, we have
12ddt(‖u2‖2+α|Au2|2)+∫∞0μ(s)Aη2(s)Au2(t)ds+β‖u2‖2+b(u,u2,Au2)=(PLfε−βu1,Au2), | (4.13) |
from the existence of pullback absorbing set, Lemma 4.10, the interpolation inequality and Young inequality, we have
12ddt(‖u2‖2+α|Au2|2+‖η2‖2MW)+12∫∞0μ(s)dds|Aη2|2ds+β‖u2‖2≤|(fε,Au2)|+|(βu1,Au2)|+‖u‖L6‖∇u2‖L3|Au2|≤νε4|Au2|2+C‖u‖‖u2‖1/2|Au2|1/2|Au2|+C|fε|2≤νε2|Au2|2+C|fε|2+C. | (4.14) |
Multiplying (4.12) by A∂tu2, we have
‖∂tu2‖2+α|A∂tu2|2+((η2,∂tu2))MW+12βddt‖u2‖2+b(u,u2,∂tu2)=(PLfε−βu1,A∂tu2). | (4.15) |
By the existence of pullback absorbing set, Lemma 4.10 and Young inequality, one has
βddt‖u2‖2+2‖∂tu2‖2+2α|A∂tu2|2≤2|((η2,∂tu2))MW|+2|(PLfε−βu1,A∂tu2)|+2|b(u,u2,∂tu2)|≤2|((η2,∂tu2))MW|+2|(PLfε−βu1,A∂tu2)|+C|Au2||A∂tu2|≤α|A∂tu2|2+C‖η2‖2MW+C|Au2|2+C|fε|2. | (4.16) |
To estimate the term ∫∞0μ(s)dds|Aη2|2ds in (4.14), we set
Φ2(t)=−6κ∫∞0˜μ(s)(Aη2(s),Au2(t))ds, |
and differentiating in t leads to
ddtΦ2(t)+6κ∫∞0˜μ(s)(Au2(t),Au2(t))ds≤−6κ∫∞0˜μ(s)(−A∂sη2,Au2(t))ds−6κ∫∞0˜μ(s)(Aη2(s),A∂tu2(t))ds, | (4.17) |
where
6κ∫∞0˜μ(s)(Au2(t),Au2(t))ds≥6κ∫∞˜s˜μ(s)ds⋅|Au2(t)|2, | (4.18) |
and
−6κ∫∞0˜μ(s)(−A∂sη2,Au2(t))ds=−6κ∫∞˜sμ′(s)(Aη2,Au2(t))ds≤6κ∫∞˜s−μ′(s)|Aη2||Au2(t)|ds≤6κ(∫∞˜s−μ′(s)|Aη2|2ds)1/2(∫∞˜s−μ′(s)|Au2(t)|2ds)1/2≤6κ(∫∞˜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
−6κ∫∞0˜μ(s)(Aη2(s),A∂tu2(t))ds≤6κ∫∞0μ(s)|Aη2(s)||A∂tu2(t)|ds≤αε|A∂tu2(t)|2+9ακ2ε‖η‖2MW. | (4.20) |
Also, from the fact that μ′(s)+δμ(s)≤0, we have
∫∞0μ(s)dds|Aη2|2ds≥∫∞0δμ(s)|Aη2|2ds=δ‖η2‖2MW. | (4.21) |
Thus
ddtΦ2(t)+2|Au2|2≤18μ(˜s)κ2∫∞0μ(s)dds|Aη2|2ds+9ακε‖η2‖2MW+αε|A∂tu2|2. | (4.22) |
We use the technique in [18] and set
zε(t)=E2(t)+νεΦ2(t)+ε2Ψ2(t), |
where
E2(t)=12(‖u2‖2+α|Au2|2+‖η2‖2MW), Ψ2(t)=β‖u‖2. |
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)≤Ce−νε(t−τ)E2(τ)+Cε∫tτeνε(s−t)|f2(s,x)|2ds+C|f1|2+C≤Cεe−νεt∫tτeνε0s|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): NV→NV generated by the system (3.2) is D−κ-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 t∈R, 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 τ∈R+ and any u∈X, there holds uniformly that
limε→0dX(Sε(t,t−τ)u,S0(τ)u)=0. |
(iii) There exists a compact subset G⊂X such that
limε→0distX(Aε(t),G)=0. |
Then, for any t∈R, there holds
limε→0distX(Aε,A0)=0. |
Lemma 5.2. ([21]) For any t∈R, τ∈R+, and ε∈(0,ε0], ˆDε0(t)={Dε0(t):t∈R} is the pullback absorbing set for Sε(t,τ), and ˆCε0(t)={Cε0(t):t∈R} is a family of compact subsets in X. Assume that Sε(⋅,⋅)=Sε1(⋅,⋅)+Sε2(⋅,⋅), and there hold
(i) For any ut−τ∈Dε0(t−τ),
‖Sε1(t,t−τ)ut−τ‖X≤Φ(t,τ)→0 (τ→∞). |
(ii) For any T≥τ, ∪0≤τ≤TSε2(t,t−τ)Dε0(t−τ) is bounded, and there exists a constant TDε0(t), independent of ε, such that
Sε2(t,t−τ)Dε0(t−τ)⊂Cε0(t), ∀ τ>TDε0(t). |
(iii) There is a compact subset G⊂X such that
limε→0distX(Cε0(t),G)=0. |
Then, the process Sε(t,τ) has the pullback attractors Aε(t), and
limε→0distX(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(t−τ)U(τ) is the solution to system (3.4), then, for any bounded subset B⊂NV, there holds
limε→0supU(τ)∈BdNV(Sε(t,τ)U(τ),S0(t−τ)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=uε−u and ξε=ηε−η, 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
12ddt‖wε‖2α+∫∞0μ(s)(Aξε(s),wε(t))ds+b(wε,u,wε)+β|wε|2=ε(PLf2(t,x),wε), | (5.4) |
it follows that
12ddt‖wε‖2α+∫∞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(t−s)‖2MV≤‖wε(τ)‖2α+‖ξε(τ−s)‖2MV+2∫tτ|b(wε,u,wε)|ds+2ε∫tτ|(f2(t,x),wε)|ds≤‖(wε,ξε)|τ‖2NV+C∫tτ‖u‖‖wε‖2ds+2ε∫tτ|(f2(t,x),wε)|ds≤‖(wε,ξε)|τ‖2NV+ε2∫tτ|f2|2ds+C∫tτ‖wε‖2ds, | (5.6) |
that is
‖(wε,ξε)|t‖2NV≤‖(wε,ξε)|τ‖2NV+ε2∫tτ|f2|2ds+C∫tτ‖(wε,ξε)|s‖2NVds, | (5.7) |
and the Gronwall inequality leads to
‖(wε,ξε)|t‖2NV≤C(‖(wε,ξε)|τ‖2NV+ε2∫tτ|f2|2ds)→0 (ε→0), | (5.8) |
which means that the conclusion is finished.
Proof of Theorem 3.4. From (4.24) and the fact that W↪V is compact, we know that there exists a compact subset G⊂NV such that
limε→0distX(Cε0(t),G)=0. | (5.9) |
Combining Lemma 4.10, Lemma 5.2 and (5.9), we have
limε→0distX(Aε,G)=0. |
In addition, the confirmation of condition (ii) in Lemma 5.1 is finished from Lemma 5.3, and we have
limε→0distX(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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