# Backwards Asymptotically Autonomous Dynamics for 2D MHD Equations

Authors: Jiali Yu; Wenhuo Su; Dongmei Xu
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We consider the backwards topological property of pullback attractors for the nonautonomous MHD equations. Under some backwards assumptions of the nonautonomous force, it is shown that the theoretical existence result for such an attractor is derived from an increasing, bounded pullback absorbing and the backwards pullback flattening property. Meanwhile, some abstract results on the convergence of nonautonomous pullback attractors in asymptotically autonomous problems are established and applied to MHD equations.


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## Body

## 1. Introduction

In this paper, we consider the existence and backwards compactness of pullback attractors for the nonautonomous MHD equations on a bounded domainO⊂R2:(1)ut+u·∇u-ν1Δu-ν3B·∇B+∇p+ν3B22=fx,t,Bt+u·∇B-B·∇u-ν2ΔB=0,div⁡u=0,div⁡B=0.The unknown u=(u1,u2) is the velocity vector, B=(B1,B2) is the magnetic field, and p is the pressure. The positive constants ν1=1/Re, ν2=1/Rm, and ν3=M2ν1ν2, where Re, Rm, and M are called the Reynolds number, magnetic Reynolds number, and Hartman number, respectively (see [1]). The external force f∈Lloc2(R,L2(O)).The system of equations describes a magnetized plasma as a one-component fluid and the magnetic field polarizes the conductive fluid, which changes the magnetic field reciprocally. Because of the important physical applications and the mathematical properties, MHD equations have been widely investigated in the literatures (see [2–8]).When the body forcef is time-independent, i.e., the MHD equation is autonomous, both well-posedness and ergodicity of the stochastic MHD equation were discussed in some papers (see [9, 10]) and the reference therein, while the existence of attractors was proved by many authors (see [5, 11]).Since the force is time-dependent, the dynamics is nonautonomous which is described by an important concept of pullback attractors. It is well-known that a pullback attractor is a time-dependent family of compact, invariant, and pullback attracting sets with the minimality, which was studied by many authors (see [12–16]).In this paper, we focus on a relatively new subject aboutbackwards compactness of a pullback attractor, which means that the union of a pullback attractor over the past time is precompact; i.e., ⋃s⩽tA(s) is precompact for all t∈R. To the best of our knowledge, there has been very little information on nonautonomous pullback attractors for evolution problems involving the backwards compactness (see [17–19]). To establish the theoretical results of a backwards compact attractor, we will introduce the flattening property presented by Kloeden [20] and promote this nature as abackwards pullback flattening property. We will prove that a nonautonomous system has a backwards compact attractor if it has an increasing, bounded, and pullback absorbing set and this system isbackwards pullback flattening. Similarly, we can introduce other relative concepts ofbackwards pullback asymptotic compactness. In fact, the two concepts mentioned above are equivalent in a uniform convex Banach space.As the application of theoretical results, we obtain that 2D MHD equations have a backwards compact attractor inH and V, respectively. In this case, we need only to assume that the nonautonomous external force f isbackwards tempered andbackwards limiting. The spectrum decomposition technique is used to give required backwards uniform estimates in V.Finally, we consider the asymptotically autonomous dynamics of PDE. LetS be an evolution process with a pullback attractor A={A(t):t∈R} and T a semigroup with a global attractor A∞ on a Banach space X. We say that S isasymptotically autonomous to T if(2)limτ→-∞⁡Sτ+t,τxτ-Ttx0X=0,for  eacht⩾0,whenever xτ-x0X→0 as τ→-∞, while S isuniformly asymptotically autonomous to T if the convergence in (2) is uniform in t⩾0; i.e.,(3)limτ→-∞⁡supt⩾0⁡Sτ+t,τxτ-Ttx0X=0.There is not much research on this kind of problem. The representative literature is published by Kloeden [21] which proved that if S isuniformly asymptotically autonomous to T and the pullback attractor A isuniformly compact (i.e., ⋃s∈RA(s) is precompact), then the pullback attractor converges to the global attractor in the Hausdorff semidistance sense:(4)limτ→+∞⁡distX⁡Aτ,A∞=0. where τ→+∞ which is different in this paper. Other forms of results can be found in [22–24] but all known results involved uniform convergence and uniform compactness.However, the uniformness condition is hard to verify in realistic models. Motivated by this dilemma, we establish an abstract result to reduce the uniformness condition (only⋃s⩽tA(s) is precompact) and find that A is backwards compact if and only if the upper semicontinuity holds; i.e.,(5)limτ→-∞⁡distX⁡Aτ,A∞=0, if S is weakly asymptotically autonomous (τ→-∞) to T, in this paper.


## 2. Preliminaries and Abstract Results

First, we review some basic concepts related to pullback attractors for nonautonomous dynamical system (see [12, 13, 15, 16]) and introduce the concept of a backwards compact attractor and then investigate its existence.Let(X,·X) be a Banach space and D is the collection of all bounded nonempty subsets of X. A set-valued mapping D:R→2X∖∅ is called a nonautonomous set in X, and it is said to have a topological property (such as boundedness, compactness, or closedness) if D(t) has this property for each t∈R. We also say that a nonautonomous set D(t) is increasing if D(s)⊂D(t) for s⩽t.Definition 1.
A nonautonomous setD⊂X is called backwards compact (resp., backwards bounded) if ⋃s⩽tD(s) is precompact (resp., bounded) in X with each t∈R.Definition 2.
An evolution processS in X is a family of mappings S(t,τ):X→X with t⩾τ, which satisfies (6)Sτ,τ=idX,St,τ=St,sSs,τ,for  allt⩾s⩾τwitht∈R,andt,τ,x→St,τxis  continuous  fort⩾τandx∈X.Definition 3.
A nonautonomous setA(·) in X is called a backwards compact attractors for a process S(·,·) if (1)
A(·) is backwards compact;(2)
A(·) is invariant, i.e., S(t,s)A(s)=A(t) for all t⩾s;(3)
A(·) is pullback attracting set, which means that it pullback attracts every bounded subset D∈D, i.e.,(7)limτ→∞⁡distX⁡St,t-τD,At=0,∀t∈R,
where and throughout this paperdist(·,·) is Hausdorff semidistance, i.e.,(8)distXA,B=supa∈A⁡infb∈B⁡a-bX,∀A,B⊆X.Remark 4.
Through the above definitions, a backwards compact attractor must be the minimal family of closed sets with property (3). This term can be interpreted as if there is another familyA1(·) of closed sets that pullback attracts bounded subsets of X, then A(t)⊂A1(t). Meanwhile, in general this is required to guarantee the uniqueness of the backwards compact attractor and by the minimality, it is shown that a backwards compact attractor must be a pullback attractor in the sense of [14, p.12]. If a pullback attractor is backwards compact, then it is a backwards compact attractor.Definition 5.
A nonautonomous setK in X is a pullback absorbing set at time t∈R for an evolution process S if, for each bounded subset D in X, there is τ0≔τ0(t,D)>0 such that (9)St,t-τD⊂Kt,for  allτ⩾τ0.Definition 6.
An evolution processS in X is said to possess the backwards pullback flattening condition if given a bounded set D⊂X,t∈R and ε>0; there exist τ0≔τ0(ε,t,D)>0 and a finite dimensional subspace X1 of X such that, for a bounded projector P:X→X1,(10)P⋃τ⩾τ0⋃s⩽tSs,s-τDis  bounded  inX, and(11)I-P⋃τ⩾τ0⋃s⩽tSs,s-τDX<εTheorem 7 (see [18]).
LetS be an evolution process in a Banach space X; assume that (i)
S has an increasing and bounded absorbing set K(·),(ii)
S is backwards pullback flattening.
ThenS has a backwards compact attractor A given by(12)At=ωKt,t≔⋂τ0>0⋃τ⩾τ0St,t-τKt¯X,for  eacht∈R.Let an evolution processS have a pullback attractor A and a semigroup T with a global attractor A∞.Definition 8.
An evolution processS is said to be weakly asymptotically autonomous to T if for each t⩾0,(13)limτ→-∞distXSτ+t,τxτ,Ttx0=0,whenever xτ∈A(τ),x0∈A∞ and xτ→x0.Theorem 9.
LetS be weakly asymptotically autonomous to T. Then the upper semicontinuity holds; i.e.,(14)limτ→-∞distXAτ,A∞=0 if and only if A is backwards compact.Proof.
Sufficiency. We argue by contradiction. Since A is backwards compact, then C≔⋃s⩽0A(s)¯ is compact. Suppose that the semicontinuity (14) is not true, then there are δ>0 and 0>τn↓-∞ such that distX(A(τn),A∞)⩾4δ for all n∈N. We choose xn∈A(τn) such that(15)dxn,A∞⩾distXAτn,A∞-δ⩾3δ. By the attraction of A∞ under the semigroup, there is a n0∈N such that(16)distXTτn0C,A∞<δ. By the invariance of the pullback attractor A, we see that, for any xn∈A(τn), there exists yn∈A(τn-τn0) such that (17)xn=Sτn-τn0+τn0,τn-τn0yn. Since {yn} is included into the compact set C, it follows that there exist a subsequence {ynk} and y∈C such that ynk→y in X as k→∞.
Applying the (13) in the case that t=τn0 and τ=τnk-τn0→-∞ as k→∞, we find(18)dxnk,Tτn0y=dSτnk-τn0+τn0,τnk-τn0ynk,Tτn0y<δ, if k is large enough. From (16) and (18), we obtain that (19)dxnk,A∞⩽dxnk,Tτn0y+distXTτn0C,A∞<2δ, which contradicts with (15). Therefore the semicontinuity (14) holds true.
Necessity. Suppose the semicontinuity (14) holds true. We need to prove the precompactness of ⋃s⩽tA(s) for each fixed t∈R. Taking a sequence {xn} from this set, we then choose sn⩽t such that xn∈A(sn). We will prove that the sequence {xn} has a convergent subsequence in two case.
Case 1. If s0=infn∈Nsn≠-∞, then for n∈N, sn∈[s0,t], and so {xn}⊂⋃s0⩽s⩽tA(s). Define a mapping Υ:[s0,+∞)×X→X,(s,x)→S(s,s0)x, then the continuity assumption implies that Υ is a continuous mapping. By the invariance of the pullback attractor A, it is easy to see that (20)⋃s0⩽s⩽tAs=Υs0,t×As0. Then ⋃s0⩽s⩽tA(s) is a compact set since the range of a continuous mapping on a compact set is compact. Hence {xn} is precompact as required.
Case 2. s0=infn∈Nsn=-∞. In this case, passing to a subsequence, we may assume sn↓-∞. By the upper semicontinuity assumption (14), we have(21)dxn,A∞⩽distX⁡Asn,A∞→0,asn→∞.For each n∈N we choose a yn∈A∞ such that d(xn,yn)<d(xn,A∞)+1/n. Since the global attractor A∞ is a compact set, it implies that the sequence {yn} has a convergent subsequence such that ynk→y as k→∞. Therefor,(22)dxnk,y⩽dxnk,ynk+dynk,y⩽dynk,y+dxnk,A∞+1nk, which together with (21) implies that xnk→y as required.Remark 10.
This proof (sufficiency) is different from Kloeden given in [21, Theorem 3.2]. At this moment, we only need that the convergence from S to T holds true at every single time (e.g., t=τn0), not uniformly in t⩾0. So we reduce the uniformness condition in [21, Theorem 3.2] successfully.


## 3. Nonautonomous 2D MHD Equations

### 3.1. Functional Spaces and Operators

LetO⊂R2 be a bounded, open, and simply connected subset with regular boundary Γ. We consider the following MHD equations defined on O×τ,+∞:(23)ut+u·∇u-ν1Δu-ν3B·∇B+∇P0=fx,t,(24)Bt+u·∇B-B·∇u-ν2ΔB=0,(25)div⁡u=0,div⁡B=0,where P0=p+ν3B2/2 is the total pressure and νi are positive constants.We consider the initial problem of (23)-(25) with mixed boundary conditions:(26)ux,τ=u0x,Bx,τ=B0xonOux,t=0,B·n=0.onΓ,t⩾τ where n is the unit outward normal on Γ. For the mathematical setting of this problem, we introduce some Hilbert spaces. We set H=H1×H2 and V=V1×V2 with(27)H1=H2=φ∈L2O:div⁡φ=0,φ·nΓ=0,V1=φ∈H01O:div⁡φ=0,V1′=φ∈H-1O:div⁡φ=0,V2=φ∈H1O:div⁡φ=0,φ·nΓ=0,where L2(O)=L2(O)2, H1(O)=H1(O)2, and so on. We use (·,·) to denote the usual scalar product in L2(O) and equip H=H1×H2 with the scalar product (·,·)H and norm ·H by(28)v1,v2H=u1,u2+ν3B1,B2,uH=u,uH1/2,vi=ui,Bi∈Hi.We take the scalar product in V1 and V2 with the general forms denoted by ((·,·)) and since O⊂R2 is a bounded smooth domain, we take equivalent norms on V1 and V2 to be the same symbol ∇·; that is,(29)u,v=∑i=1,j=12∫O∂ui∂xj∂vi∂xjdx,∇u=u,u1/2,u,v∈V1. We equip V=V1×V2 with the scalar product ((·,·))V and the norm ·V given by(30)v1,v2V=u1,u2+ν3B1,B2,vV=v,vV1/2,vi=ui,Bi∈V. The trilinear form b(u,v,w) and the bilinear operator B from H1(O)×H1(O) into H-1(O) are defined by(31)Bu,v,w=bu,v,w=∑i=1,j=12∫Oui∂vj∂xiwjdx.Moreover, we have the following useful relations (see [11, 25]):(32)bu,v,v=0,bu,v,w=-bu,w,v,u∈V2,v,w∈V1,(33)bu,v,w⩽cu1/2Δu1/2∇vw,u∈H2O,v∈H1O,w∈L2O,(34)bu,v,w⩽c∇u∇v1/2Δv1/2w,u∈H1O,v∈H2O,w∈L2O,where c is an intrinsic positive constant.On the other hand, through the above terms, (23)-(25) can be rewritten in a weak form as follows:(35)dudt-ν1Δu=-Bu,u+ν3BB,B+fx,t,(36)dBdt-ν2ΔB=-Bu,B+BB,u,(37)div⁡u=0,div⁡B=0,with the initial-boundary condition (26).


### 3.2. Assumptions on the Nonautonomous Force

In order to obtain a backwards compact attractor inH for (35)-(37), a basic assumption for external force is f∈Lloc2(R,L2(O)). Furthermore, one has the following.Assumption F1. f is backwards tempered; i.e.,(38)sups⩽t∫-∞seγr-sfr,·2dr<∞,for  allγ>0andt∈R.To prove the existence of backwards compact attractor in V for (35)-(37), we assume further the following.Assumption F2. f is backwards limiting; i.e.,(39)limγ→+∞⁡sups⩽t⁡∫-∞seγr-sfr,·2dr=0,for  allt∈R.By employing Galerkin method, we have the following well-possessedness of problem (35)-(37), which is similar to the nonautonomous case as given in [26].Lemma 11.
Letf∈Lloc2(R,L2(O)). Then for each (u0,B0)∈H and for each τ∈R, there exists a unique weak solution(40)u,B∈Lloc2τ,∞;V∩Cτ,∞;Hsatisfying (35)-(37) in distribution sense with u,Bt=τ=(u0,B0). Moreover, the mapping (u0,B0)↦(u,B) is continuous in H.For convenience, we rewrite the solution of (35)-(37) by φ≔(u,B) and the initial data by φ0≔(u0,B0).By Lemma11, we can use the unique weak solution to define an evolution process S(t,τ):H→H by(41)St,τφ0≔φt,τ,φ0,for  allt⩾τandφ0∈H.


## 3.1. Functional Spaces and Operators

LetO⊂R2 be a bounded, open, and simply connected subset with regular boundary Γ. We consider the following MHD equations defined on O×τ,+∞:(23)ut+u·∇u-ν1Δu-ν3B·∇B+∇P0=fx,t,(24)Bt+u·∇B-B·∇u-ν2ΔB=0,(25)div⁡u=0,div⁡B=0,where P0=p+ν3B2/2 is the total pressure and νi are positive constants.We consider the initial problem of (23)-(25) with mixed boundary conditions:(26)ux,τ=u0x,Bx,τ=B0xonOux,t=0,B·n=0.onΓ,t⩾τ where n is the unit outward normal on Γ. For the mathematical setting of this problem, we introduce some Hilbert spaces. We set H=H1×H2 and V=V1×V2 with(27)H1=H2=φ∈L2O:div⁡φ=0,φ·nΓ=0,V1=φ∈H01O:div⁡φ=0,V1′=φ∈H-1O:div⁡φ=0,V2=φ∈H1O:div⁡φ=0,φ·nΓ=0,where L2(O)=L2(O)2, H1(O)=H1(O)2, and so on. We use (·,·) to denote the usual scalar product in L2(O) and equip H=H1×H2 with the scalar product (·,·)H and norm ·H by(28)v1,v2H=u1,u2+ν3B1,B2,uH=u,uH1/2,vi=ui,Bi∈Hi.We take the scalar product in V1 and V2 with the general forms denoted by ((·,·)) and since O⊂R2 is a bounded smooth domain, we take equivalent norms on V1 and V2 to be the same symbol ∇·; that is,(29)u,v=∑i=1,j=12∫O∂ui∂xj∂vi∂xjdx,∇u=u,u1/2,u,v∈V1. We equip V=V1×V2 with the scalar product ((·,·))V and the norm ·V given by(30)v1,v2V=u1,u2+ν3B1,B2,vV=v,vV1/2,vi=ui,Bi∈V. The trilinear form b(u,v,w) and the bilinear operator B from H1(O)×H1(O) into H-1(O) are defined by(31)Bu,v,w=bu,v,w=∑i=1,j=12∫Oui∂vj∂xiwjdx.Moreover, we have the following useful relations (see [11, 25]):(32)bu,v,v=0,bu,v,w=-bu,w,v,u∈V2,v,w∈V1,(33)bu,v,w⩽cu1/2Δu1/2∇vw,u∈H2O,v∈H1O,w∈L2O,(34)bu,v,w⩽c∇u∇v1/2Δv1/2w,u∈H1O,v∈H2O,w∈L2O,where c is an intrinsic positive constant.On the other hand, through the above terms, (23)-(25) can be rewritten in a weak form as follows:(35)dudt-ν1Δu=-Bu,u+ν3BB,B+fx,t,(36)dBdt-ν2ΔB=-Bu,B+BB,u,(37)div⁡u=0,div⁡B=0,with the initial-boundary condition (26).


## 3.2. Assumptions on the Nonautonomous Force

In order to obtain a backwards compact attractor inH for (35)-(37), a basic assumption for external force is f∈Lloc2(R,L2(O)). Furthermore, one has the following.Assumption F1. f is backwards tempered; i.e.,(38)sups⩽t∫-∞seγr-sfr,·2dr<∞,for  allγ>0andt∈R.To prove the existence of backwards compact attractor in V for (35)-(37), we assume further the following.Assumption F2. f is backwards limiting; i.e.,(39)limγ→+∞⁡sups⩽t⁡∫-∞seγr-sfr,·2dr=0,for  allt∈R.By employing Galerkin method, we have the following well-possessedness of problem (35)-(37), which is similar to the nonautonomous case as given in [26].Lemma 11.
Letf∈Lloc2(R,L2(O)). Then for each (u0,B0)∈H and for each τ∈R, there exists a unique weak solution(40)u,B∈Lloc2τ,∞;V∩Cτ,∞;Hsatisfying (35)-(37) in distribution sense with u,Bt=τ=(u0,B0). Moreover, the mapping (u0,B0)↦(u,B) is continuous in H.For convenience, we rewrite the solution of (35)-(37) by φ≔(u,B) and the initial data by φ0≔(u0,B0).By Lemma11, we can use the unique weak solution to define an evolution process S(t,τ):H→H by(41)St,τφ0≔φt,τ,φ0,for  allt⩾τandφ0∈H.


## 4. Backwards Compact Attractors for 2D MHD Equations

### 4.1. Backwards Compact Attractors inH

In this subsection, our main work is to prove that the evolution process has an increasing bounded pullback absorbing set inH. From now on, we assume without loss of generality that c will be a positive constant which may alter its values everywhere.Lemma 12.
Letf be backwards tempered, then for each t∈R and R>0, there exists τ0≔τ0(R)⩾2 such that, for all τ⩾τ0 and φ0H⩽R,(42)sups⩽t⁡supr∈s-2,s⁡φr,s-τ,φ0H2⩽c1+Mt,(43)sups⩽t⁡∫s-τseλr-s∇φr,s-τ,φ0H2dr⩽c1+Mt,where λ is given by (48) and M(t) is a nonnegative increasing function defined by(44)Mt≔sups⩽t⁡∫-∞seλr-sfr,·2dr<∞.Proof.
Lett∈R be fixed. For each s⩽t, we multiply equation in (35) by u and (36) by ν3B respectively and integrate over O, then the sum of them is(45)ddsu2+ν3B2+2ν1∇u2+2ν2ν3∇B2=2-Bu,u+ν3BB,B,u+-Bu,B+BB,u,ν3B+2∫Of·udx.Notice from (31) and (32) that (46)Bu,u,u=0;Bu,B,ν3B=0;ν3BB,B,u=ν3bB,B,u;BB,u,ν3B=-ν3bB,B,u. For the nonlinear term, we have (47)2∫Of·udx⩽λu2+cfs,·2⩽λφ2+cfs,·2, where we have used the notation φ=(u,B), φ2=u2+ν3B2, and λ>0 is given by(48)2λu2⩽ν∇u2,2λB2⩽ν∇B2,∀u∈V1,B∈V2,ν=minν1,ν2. Substituting the above into (45), we have(49)ddsφH2+λφH2+ν∇φH2⩽cfs,·2. Multiplying (49) by eλs and integrating it over [s-τ,s], we obtain(50)φsH2+∫s-τseλr-s∇φrH2dr⩽e-λτφ0H2+c∫s-τseλr-sfr,·2dr⩽c1+∫-∞seλr-sfr,·2dr,for all τ⩾τ0 with some τ0≔τ0(R)⩾2.
On the other hand, we multiply (49) by eλs and integrating it over [s-τ,r] with r∈[s-2,s], we obtain(51)φrH2⩽eλs-r-τφ0H2+c∫s-τreλr1-rfr1,·2dr1⩽c1+∫-∞seλr1-sfr1,·2dr1,for all τ⩾τ0 with some τ0≔τ0(R)⩾2.
Taking the supremum with respect to the past times⩽t in (51) and (50), we get (42) and (43). By the assumption (38), M(t) is finite and increasing. This completes the proof.Lemma 13.
Letf be backwards tempered, then for each t∈R and R>0, there exists τ0≔τ0(R)⩾2 such that, for all τ⩾τ0 and φ0H⩽R,(52)sups⩽t⁡supr∈s-1,s⁡∇φr,s-τ,φ0H2⩽ce1+Mt2,where M(t) is given by (44).Proof.
Lett∈R be fixed. For each s⩽t, we multiply equation in (35) by -Δu and (35) by -ν3ΔB, respectively, then integrate over O, and sum the results to find(53)dds∇u2+ν3∇B2+2ν1Δu2+2ν2ν3ΔB2=2-Bu,u+ν3BB,B,-Δu+-Bu,B+BB,u,-ν3ΔB-2∫Of·Δudx Notice from (33) that(54)-Bu,u,-Δu=bu,u,Δu⩽cu1/2∇uΔu3/2⩽cu2∇u4+ν18Δu2,(55)BB,u,-ν3ΔB⩽cB2∇u4+ν2ν38ΔB2. On the other hand, by (33) and the inequality that abc≤C(ε1,ε2)a4+ε1b4+ε2c2, we have(56)ν3BB,B,-Δu=ν3bB,B,Δu⩽cB1/2∇BΔB1/2Δu⩽cB2∇B4+ν2ν38ΔB2+ν18Δu2,Bu,B,ν3ΔB⩽cu2∇B4+ν18Δu2+ν2ν38ΔB2. The nonlinear term in (53) is controlled by(57)∫Of·Δudx⩽ν18Δu2+ν2ΔB2+cfs,·2,Substituting (54)-(57) into (53), we find(58)dds∇φsH2+νΔφH2⩽hs∇φH2+cfs,·2, where h(s)=cφ(s)H2∇φ(s)H2. Integrate (58) over (ξ,r) with ξ∈[s-2,s-1] and r∈[s-1,s] to obtain(59)∇φr,s-τ,φ0H2⩽e∫ξrchρdρ∇φξ,s-τ,φ0H2+c∫ξre∫ρrchδdδfρ,·2dρ⩽e∫s-2rchρdρ∇φξ,s-τ,φ0H2+c∫s-2re∫ρrchδdδfρ,·2dρ We integrate (59) with respect to ξ over [s-2,s-1] with s⩽t; we have(60)∇φr,s-τ,φ0H2⩽ce∫s-2schρdρ∫s-2s∇φξ,s-τ,φ0H2dξ+∫s-2sfr,·2dr On the other hand, by Lemma 12, we have(61)∫s-2shρdρ=c∫s-2sφρH2∇φρH2dρ⩽c1+Mt∫s-2s∇φρH2dρ⩽c1+Mt2 Therefor, we insert (61) into (60) to obtain that(62)supr∈s-1,s⁡∇φr,s-τ,φ0H2⩽ce1+Mt2,for all τ⩾τ0 with some τ0≔τ0(R)⩾2. Hence, we get (52) by taking the supremum in (62) with respect to s⩽t.We now state our result as follows.Theorem 14.
Assumef is backwards tempered, then the evolution process S generated by nonautonomous 2D MHD equations possesses a backwards compact attractor A={A(t)}t∈R in H.Proof.
Define a nonautonomous set by(63)K1t≔φ∈V;φV2⩽ce1+Mt2,∀t∈R,where M(t) is given by (44). By the compactness of the Sobolev embedding and (52), K1(t) is compact and pullback absorbing in H. It is readily to check that the process S is backwards pullback asymptotically compact in H and thus is backwards pullback flattening follows from [17, Theorem 2.7]. Then the conclusion can be proved by Theorem 7.


### 4.2. Backwards Compact Attractors inV

In this subsection, we prove the existence of backwards compact attractors inV. To do this, we first give a decomposition of an element in V. To this end, we consider the eigenvalue problem: (64)-Δvx=λvx,x∈O,v∂O=0,Then it is known that the above problem shows a family of complete orthonormal basis {ej}j=1∞ of L2(O) consisting of eigenvectors of -Δ who has countable spectrum λj,j=1,2,…, such that (65)0<λ1⩽λ2⩽λ3⩽⋯→∞and-Δej=λjej. Let Vm=span{e1,e2,…,em}⊂V and Pm:V→Vm be the canonical projector and I be the identity. Then for every v∈V there exists a unique decomposition (66)v=v1+v2,v1=Pmv∈Vm,v2=I-Pmv∈Vm⊥, where Vm⊥ is the orthogonal complement of Vm.Lemma 15.
Letf be backwards tempered, then for each t∈R and R>0, there exists τ0≔τ0(R)⩾2 such that, for all τ⩾τ0 and φ0H⩽R,(67)sups⩽t∫s-1sΔφr,s-τ,φ0H2dr⩽ce21+Mt2,where M(t) is given by (44).Proof.
By (58), we have(68)dds∇φsH2+νΔφH2⩽hs∇φH2+cfs,·2,Integrating (68) over [s-1,s], we can obtain(69)ν∫s-1sΔφr,s-τ,φ0H2dr⩽c∫s-1s∇φrH4φrH2dr+∇φs-1H2+c∫s-1sfr,·2dr.Thus by Lemmas 12 and 13 we have(70)∫s-1sΔφr,s-τ,φ0H2dr⩽ce21+Mt2.Therefor, we obtain (67) by taking the supremum in (70) over all the past time s⩽t.Lemma 16.
Letf be backwards tempered and backwards limiting, then for each ε>0,t∈R, and R>0, there exist τ1≔τ1(ε,R)⩾2 and N≔N(ε,R)>0 such that, for all τ⩾τ1,m⩾N and φ0H⩽R,(71)sups⩽tI-Pmφs,s-τ,φ0V2<ε.Proof.
Lett∈R be fixed. For each s⩽t, we multiply equation in (35) by -Δu2 and (36) by -ν3ΔB2, respectively, and then integrate over O to find that(72)ddt∇u22+2ν1Δu22=2-Bu,u+ν3BB,B,-Δu2-2∫Of·Δu2dx(73)ν3ddt∇B22+2ν2ν3ΔB22=2-Bu,B+BB,u,-ν3ΔB2 Notice from (34) that we have(74)-Bu,u,-Δu2=bu,u,Δu2⩽c∇u3/2Δu1/2Δu2⩽c∇u3Δu+ν14Δu22,(75)BB,u,-ν3ΔB2=bB,u,-ν3ΔB2⩽c∇B∇u1/2Δu1/2ΔB22⩽c∇B2∇uΔu+ν2ν34ΔB22.(76)ν3BB,B,-Δu2=ν3bB,B,Δu2⩽c∇B3/2ΔB1/2Δu2⩽c∇B3ΔB+ν14Δu22,(77)Bu,B,ν3ΔB2=bB,u,ν3ΔB2⩽c∇u∇B1/2ΔB1/2ΔB22⩽c∇u2∇BΔB+ν2ν34ΔB22. The nonlinear term in (72) is controlled by(78)∫Of·Δu2dx⩽ν14Δu22+ν2ν34ΔB22+cft,·2, Then from (72) to (78) and using Δu22⩾λm∇u22, we find(79)ddt∇φ2H2+ν2λm∇φ2H2⩽c∇φH3ΔφH+cft,·2.We multiply (79) by e(ν/2)λms with s∈[r,t], integrating the result in s∈[r,t], and then integrating it once again in r∈[t-1,t], we obtain(80)∇φ2tH2⩽∫t-1teν/2λmr-t∇φ2rH2dr+∫t-1teν/2λmr-tΔφrH∇φrH3dr+c∫t-1teν/2λmr-tfr,·2drWe now take into account the supremum of each term in (80) over the past time. From (52) and the increasing property of M(t), we can see that, for all τ⩾τ0 with some τ0≔τ0(R)⩾2,(81)sups⩽t∫s-1seν/2λmr-s∇φ2rH2dr⩽ce1+Mt2sups⩽t⁡∫s-1seν/2λmr-sdr⩽cνλme1+Mt2. Similar, by (52) and (67), we obtain(82)sups⩽t⁡∫s-1seν/2λmr-sΔφrH∇φrH3dr⩽ce3/21+Mt2sups⩽t⁡∫s-1seν/2λmr-sΔφrHdr⩽ce3/21+Mt2sups⩽t⁡∫s-1sΔφrH2dr1/2∫s-1seνλmr-sdr1/2⩽cνλme31+Mt2. Finally, f is backwards limiting by assumption (39); thus(83)limm→∞⁡sups⩽t⁡∫-∞seν/2λmr-sfr,·2dr=0. Hence, from (80) to (83), for all τ⩾τ1,(84)sups⩽t⁡I-Pmφs,s-τ,φ0V2→0,asm→∞. This completes the proof.Theorem 17.
Assumef be backwards tempered and backwards limiting, then the evolution process S generated by nonautonomous 2D MHD equations possesses a backwards compact attractor A={A(t)}t∈R in V.Proof.
Define a nonautonomous set by(85)K2t≔φ∈V;φV2⩽ce1+Mt2,∀t∈R,where M(t) is given by (44). It is obvious that K2(t) is bounded and increasing absorbing set in V. On the other hand, by Lemmas 13 and 16 the process S is backwards pullback flattening in V. Then the all conditions in Theorem 7 are fulfilled. Therefor there exists a backwards compact attractor A={A(t)}t∈R in V.


## 4.1. Backwards Compact Attractors inH

In this subsection, our main work is to prove that the evolution process has an increasing bounded pullback absorbing set inH. From now on, we assume without loss of generality that c will be a positive constant which may alter its values everywhere.Lemma 12.
Letf be backwards tempered, then for each t∈R and R>0, there exists τ0≔τ0(R)⩾2 such that, for all τ⩾τ0 and φ0H⩽R,(42)sups⩽t⁡supr∈s-2,s⁡φr,s-τ,φ0H2⩽c1+Mt,(43)sups⩽t⁡∫s-τseλr-s∇φr,s-τ,φ0H2dr⩽c1+Mt,where λ is given by (48) and M(t) is a nonnegative increasing function defined by(44)Mt≔sups⩽t⁡∫-∞seλr-sfr,·2dr<∞.Proof.
Lett∈R be fixed. For each s⩽t, we multiply equation in (35) by u and (36) by ν3B respectively and integrate over O, then the sum of them is(45)ddsu2+ν3B2+2ν1∇u2+2ν2ν3∇B2=2-Bu,u+ν3BB,B,u+-Bu,B+BB,u,ν3B+2∫Of·udx.Notice from (31) and (32) that (46)Bu,u,u=0;Bu,B,ν3B=0;ν3BB,B,u=ν3bB,B,u;BB,u,ν3B=-ν3bB,B,u. For the nonlinear term, we have (47)2∫Of·udx⩽λu2+cfs,·2⩽λφ2+cfs,·2, where we have used the notation φ=(u,B), φ2=u2+ν3B2, and λ>0 is given by(48)2λu2⩽ν∇u2,2λB2⩽ν∇B2,∀u∈V1,B∈V2,ν=minν1,ν2. Substituting the above into (45), we have(49)ddsφH2+λφH2+ν∇φH2⩽cfs,·2. Multiplying (49) by eλs and integrating it over [s-τ,s], we obtain(50)φsH2+∫s-τseλr-s∇φrH2dr⩽e-λτφ0H2+c∫s-τseλr-sfr,·2dr⩽c1+∫-∞seλr-sfr,·2dr,for all τ⩾τ0 with some τ0≔τ0(R)⩾2.
On the other hand, we multiply (49) by eλs and integrating it over [s-τ,r] with r∈[s-2,s], we obtain(51)φrH2⩽eλs-r-τφ0H2+c∫s-τreλr1-rfr1,·2dr1⩽c1+∫-∞seλr1-sfr1,·2dr1,for all τ⩾τ0 with some τ0≔τ0(R)⩾2.
Taking the supremum with respect to the past times⩽t in (51) and (50), we get (42) and (43). By the assumption (38), M(t) is finite and increasing. This completes the proof.Lemma 13.
Letf be backwards tempered, then for each t∈R and R>0, there exists τ0≔τ0(R)⩾2 such that, for all τ⩾τ0 and φ0H⩽R,(52)sups⩽t⁡supr∈s-1,s⁡∇φr,s-τ,φ0H2⩽ce1+Mt2,where M(t) is given by (44).Proof.
Lett∈R be fixed. For each s⩽t, we multiply equation in (35) by -Δu and (35) by -ν3ΔB, respectively, then integrate over O, and sum the results to find(53)dds∇u2+ν3∇B2+2ν1Δu2+2ν2ν3ΔB2=2-Bu,u+ν3BB,B,-Δu+-Bu,B+BB,u,-ν3ΔB-2∫Of·Δudx Notice from (33) that(54)-Bu,u,-Δu=bu,u,Δu⩽cu1/2∇uΔu3/2⩽cu2∇u4+ν18Δu2,(55)BB,u,-ν3ΔB⩽cB2∇u4+ν2ν38ΔB2. On the other hand, by (33) and the inequality that abc≤C(ε1,ε2)a4+ε1b4+ε2c2, we have(56)ν3BB,B,-Δu=ν3bB,B,Δu⩽cB1/2∇BΔB1/2Δu⩽cB2∇B4+ν2ν38ΔB2+ν18Δu2,Bu,B,ν3ΔB⩽cu2∇B4+ν18Δu2+ν2ν38ΔB2. The nonlinear term in (53) is controlled by(57)∫Of·Δudx⩽ν18Δu2+ν2ΔB2+cfs,·2,Substituting (54)-(57) into (53), we find(58)dds∇φsH2+νΔφH2⩽hs∇φH2+cfs,·2, where h(s)=cφ(s)H2∇φ(s)H2. Integrate (58) over (ξ,r) with ξ∈[s-2,s-1] and r∈[s-1,s] to obtain(59)∇φr,s-τ,φ0H2⩽e∫ξrchρdρ∇φξ,s-τ,φ0H2+c∫ξre∫ρrchδdδfρ,·2dρ⩽e∫s-2rchρdρ∇φξ,s-τ,φ0H2+c∫s-2re∫ρrchδdδfρ,·2dρ We integrate (59) with respect to ξ over [s-2,s-1] with s⩽t; we have(60)∇φr,s-τ,φ0H2⩽ce∫s-2schρdρ∫s-2s∇φξ,s-τ,φ0H2dξ+∫s-2sfr,·2dr On the other hand, by Lemma 12, we have(61)∫s-2shρdρ=c∫s-2sφρH2∇φρH2dρ⩽c1+Mt∫s-2s∇φρH2dρ⩽c1+Mt2 Therefor, we insert (61) into (60) to obtain that(62)supr∈s-1,s⁡∇φr,s-τ,φ0H2⩽ce1+Mt2,for all τ⩾τ0 with some τ0≔τ0(R)⩾2. Hence, we get (52) by taking the supremum in (62) with respect to s⩽t.We now state our result as follows.Theorem 14.
Assumef is backwards tempered, then the evolution process S generated by nonautonomous 2D MHD equations possesses a backwards compact attractor A={A(t)}t∈R in H.Proof.
Define a nonautonomous set by(63)K1t≔φ∈V;φV2⩽ce1+Mt2,∀t∈R,where M(t) is given by (44). By the compactness of the Sobolev embedding and (52), K1(t) is compact and pullback absorbing in H. It is readily to check that the process S is backwards pullback asymptotically compact in H and thus is backwards pullback flattening follows from [17, Theorem 2.7]. Then the conclusion can be proved by Theorem 7.


## 4.2. Backwards Compact Attractors inV

In this subsection, we prove the existence of backwards compact attractors inV. To do this, we first give a decomposition of an element in V. To this end, we consider the eigenvalue problem: (64)-Δvx=λvx,x∈O,v∂O=0,Then it is known that the above problem shows a family of complete orthonormal basis {ej}j=1∞ of L2(O) consisting of eigenvectors of -Δ who has countable spectrum λj,j=1,2,…, such that (65)0<λ1⩽λ2⩽λ3⩽⋯→∞and-Δej=λjej. Let Vm=span{e1,e2,…,em}⊂V and Pm:V→Vm be the canonical projector and I be the identity. Then for every v∈V there exists a unique decomposition (66)v=v1+v2,v1=Pmv∈Vm,v2=I-Pmv∈Vm⊥, where Vm⊥ is the orthogonal complement of Vm.Lemma 15.
Letf be backwards tempered, then for each t∈R and R>0, there exists τ0≔τ0(R)⩾2 such that, for all τ⩾τ0 and φ0H⩽R,(67)sups⩽t∫s-1sΔφr,s-τ,φ0H2dr⩽ce21+Mt2,where M(t) is given by (44).Proof.
By (58), we have(68)dds∇φsH2+νΔφH2⩽hs∇φH2+cfs,·2,Integrating (68) over [s-1,s], we can obtain(69)ν∫s-1sΔφr,s-τ,φ0H2dr⩽c∫s-1s∇φrH4φrH2dr+∇φs-1H2+c∫s-1sfr,·2dr.Thus by Lemmas 12 and 13 we have(70)∫s-1sΔφr,s-τ,φ0H2dr⩽ce21+Mt2.Therefor, we obtain (67) by taking the supremum in (70) over all the past time s⩽t.Lemma 16.
Letf be backwards tempered and backwards limiting, then for each ε>0,t∈R, and R>0, there exist τ1≔τ1(ε,R)⩾2 and N≔N(ε,R)>0 such that, for all τ⩾τ1,m⩾N and φ0H⩽R,(71)sups⩽tI-Pmφs,s-τ,φ0V2<ε.Proof.
Lett∈R be fixed. For each s⩽t, we multiply equation in (35) by -Δu2 and (36) by -ν3ΔB2, respectively, and then integrate over O to find that(72)ddt∇u22+2ν1Δu22=2-Bu,u+ν3BB,B,-Δu2-2∫Of·Δu2dx(73)ν3ddt∇B22+2ν2ν3ΔB22=2-Bu,B+BB,u,-ν3ΔB2 Notice from (34) that we have(74)-Bu,u,-Δu2=bu,u,Δu2⩽c∇u3/2Δu1/2Δu2⩽c∇u3Δu+ν14Δu22,(75)BB,u,-ν3ΔB2=bB,u,-ν3ΔB2⩽c∇B∇u1/2Δu1/2ΔB22⩽c∇B2∇uΔu+ν2ν34ΔB22.(76)ν3BB,B,-Δu2=ν3bB,B,Δu2⩽c∇B3/2ΔB1/2Δu2⩽c∇B3ΔB+ν14Δu22,(77)Bu,B,ν3ΔB2=bB,u,ν3ΔB2⩽c∇u∇B1/2ΔB1/2ΔB22⩽c∇u2∇BΔB+ν2ν34ΔB22. The nonlinear term in (72) is controlled by(78)∫Of·Δu2dx⩽ν14Δu22+ν2ν34ΔB22+cft,·2, Then from (72) to (78) and using Δu22⩾λm∇u22, we find(79)ddt∇φ2H2+ν2λm∇φ2H2⩽c∇φH3ΔφH+cft,·2.We multiply (79) by e(ν/2)λms with s∈[r,t], integrating the result in s∈[r,t], and then integrating it once again in r∈[t-1,t], we obtain(80)∇φ2tH2⩽∫t-1teν/2λmr-t∇φ2rH2dr+∫t-1teν/2λmr-tΔφrH∇φrH3dr+c∫t-1teν/2λmr-tfr,·2drWe now take into account the supremum of each term in (80) over the past time. From (52) and the increasing property of M(t), we can see that, for all τ⩾τ0 with some τ0≔τ0(R)⩾2,(81)sups⩽t∫s-1seν/2λmr-s∇φ2rH2dr⩽ce1+Mt2sups⩽t⁡∫s-1seν/2λmr-sdr⩽cνλme1+Mt2. Similar, by (52) and (67), we obtain(82)sups⩽t⁡∫s-1seν/2λmr-sΔφrH∇φrH3dr⩽ce3/21+Mt2sups⩽t⁡∫s-1seν/2λmr-sΔφrHdr⩽ce3/21+Mt2sups⩽t⁡∫s-1sΔφrH2dr1/2∫s-1seνλmr-sdr1/2⩽cνλme31+Mt2. Finally, f is backwards limiting by assumption (39); thus(83)limm→∞⁡sups⩽t⁡∫-∞seν/2λmr-sfr,·2dr=0. Hence, from (80) to (83), for all τ⩾τ1,(84)sups⩽t⁡I-Pmφs,s-τ,φ0V2→0,asm→∞. This completes the proof.Theorem 17.
Assumef be backwards tempered and backwards limiting, then the evolution process S generated by nonautonomous 2D MHD equations possesses a backwards compact attractor A={A(t)}t∈R in V.Proof.
Define a nonautonomous set by(85)K2t≔φ∈V;φV2⩽ce1+Mt2,∀t∈R,where M(t) is given by (44). It is obvious that K2(t) is bounded and increasing absorbing set in V. On the other hand, by Lemmas 13 and 16 the process S is backwards pullback flattening in V. Then the all conditions in Theorem 7 are fulfilled. Therefor there exists a backwards compact attractor A={A(t)}t∈R in V.


## 5. Asymptotically Autonomous Dynamics

In this section, we will show that the dynamics of the original nonautonomous MHD equations is asymptotically autonomous and its pullback attractor converges upper semicontinuity to the autonomous global attractorA∞ of the problem(86)du^dt-ν1Δu^=-Bu^,u^+ν3BB^,B^+f∞,(87)dB^dt-ν2ΔB^=-Bu^,B^+BB^,u^,(88)div⁡u^=0,div⁡B^=0,with initial-boundary values (89)u^x,0=u^0x,B^x,0=B^0xonOu^x,t=0,B^·n=0.onΓ,t⩾0For convenience, we rewrite the solution of (86)-(88) by φ^≔(u^,B^) and the initial data by φ^0≔(u^0,B^0).To discuss the asymptotically autonomous problem, we need to give a further assumption about the forcingf. We assume that f(t,·)→f∞ as S is asymptotically autonomous to T.Assumption F3. There is a function f∞∈L2(O)) such that(90)limτ→-∞⁡∫0∞fτ+s-f∞2ds=0.Lemma 18.
Suppose assumptions F1 and F3 are satisfied. Then the solutionφ of (35)-(37) is asymptotically autonomous to the solution φ^ of (86)-(88). More precisely,(91)φτ+t,τ,φ0-φ^t,φ^0H→0asτ→-∞for  eacht⩾0,whenever φ0-φ^0H→0 as τ→-∞.Proof.
LetQ1(t)=u(t+τ,τ,u0)-u^(t,u^0),Q2(t)=B(t+τ,τ,B0)-B^(t,B^0) and Q=(Q1,Q2)=(u-u^,B-B^)=φ-φ^. Then subtract (35) from (86) and we obtain(92)dQ1dt-ν1ΔQ1=-Bu,u+Bu^,u^+ν3BB,B-BB^,B^+fτ+t-f∞.Similarly, subtracting (36) from (87) we find that(93)dQ2dt-ν2ΔQ2=-Bu,B+Bu^,B^+BB,u-BB^,u^.Taking the inner product of (92) with Q1 in H, we have(94)12ddtQ12+ν1∇Q12=-Bu,u+Bu^,u^,Q1+ν3BB,B-BB^,B^,Q1+fτ+t-f∞,Q1. Using the trilinearity of b and relation (31) and (32), we have(95)-Bu,u+Bu^,u^,Q1=-bu,u,Q1+bu^,u^,Q1)=-bu,u,Q1+bu,u^,Q1-bu,u^,Q1+bu^,u^,Q1)=-bu,Q1,Q1-bu,u^,Q1+bu^,u^,Q1)=-bQ1,u^,Q1. Analogously to (95), for the second term on the right hand said of (94), we obtain(96)ν3BB,B-BB^,B^,Q1=ν3bB,Q2,Q1+ν3bQ2,B^,Q1 From (94) to (96), we get(97)12ddtQ12+ν1∇Q12=-bQ1,u^,Q1+ν3bB,Q2,Q1+ν3bQ2,B^,Q1+fτ+t-f∞,Q1. Similarly, we take the inner product of (93) with ν3Q2 in H to get(98)ν32ddtQ22+ν2ν3∇Q22=-ν3bQ1,B^,Q2+ν3bB,Q1,Q2+ν3bQ2,u^,Q2. Now, both (97) and (98) imply that(99)12ddtQ12+ν3Q22+ν1∇Q12+ν2ν3∇Q22=-bQ1,u^,Q1+ν3bQ2,B^,Q1-ν3bQ1,B^,Q2+ν3bQ2,u^,Q2+fτ+t-f∞,Q1 and then that(100)ddtQH2+2ν∇QH2⩽2-bQ1,u^,Q1+ν3bQ2,B^,Q1-ν3bQ1,B^,Q2+ν3bQ2,u^,Q2+2fτ+t-f∞,Q1. Let χ be a continuous and trilinear operator on V×V×V given by(101)χv1,v2,v3=bu1,u2,u3-ν3bB1,B2,u3+ν3bu1,B2,B3-ν3bB1,u2,B3, for vi=(ui,Bi)∈V,i=1,2,3. Thanks to the discrete Hölder inequality we have(102)χv1,v2,v3⩽cv1H1/2∇v1H1/2v2Hv3H1/2∇v3H1/2,vi∈V. Therefor (100) can be rewritten as follows:(103)ddtQH2+2ν∇QH2⩽2χQ,φ^,Q+2fτ+t-f∞,Q1⩽c∇φ^H2QH2+2ν∇QH2+QH2+fτ+t-f∞H2⩽c∇φ^H2+1QH2+2ν∇QH2+fτ+t-f∞H2 Then applying the Gronwall inequality to (103), we have(104)QtH2⩽ec∫0t∇φ^rH2+1drQ0H2+cec∫0t∇φ^rH2+1dr∫0tfτ+s-f∞H2ds⩽ec∫0t∇φ^rH2+1drQ0H2+cec∫0t∇φ^rH2+1dr∫0∞fτ+s-f∞H2ds
By Lemma13, analogous results also hold for ∇φ^(r)H2. Hence, we have (105)∫0t∇φ^rH2+1dr<∞ Since Q(0)H=φ0-φ^0H→0, it follows from Assumption F3 and (104) that(106)φτ+t,τ,φ0-φ^t,φ^0H→0asτ→-∞for  eacht⩾0, This completes the proof.Finally, by using the existence of a backwards compact attractor given in Theorem14 and the asymptotic convergence given in Lemma 18, the following result was established following from Theorem 9 immediately.Theorem 19.
Suppose assumptions F1 and F3 are satisfied. Then the nonautonomous MHD equations have a backwards compact pullback attractorA={A(t):t∈R}, which converges to the global attractor A∞ in H; that is, (107)limt→-∞⁡distH⁡At,A∞=0.



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