Backwards Asymptotically Autonomous Dynamics for 2D MHD Equations

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.

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][3][4][5][6][7][8]).
When the body force 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][13][14][15][16]).
In this paper, we focus on a relatively new subject about backwards compactness of a pullback attractor, which means that the union of a pullback attractor over the past time is precompact; i.e., ⋃ ⩽ A( ) is precompact for all ∈ 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][18][19]). To establish the theoretical results of a backwards 2 Discrete Dynamics in Nature and Society compact attractor, we will introduce the flattening property presented by Kloeden [20] and promote this nature as a backwards 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 is backwards pullback flattening. Similarly, we can introduce other relative concepts of backwards 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 in and , respectively. In this case, we need only to assume that the nonautonomous external force is backwards tempered and backwards limiting. The spectrum decomposition technique is used to give required backwards uniform estimates in .
Finally, we consider the asymptotically autonomous dynamics of PDE. Let be an evolution process with a pullback attractor A = {A( ) : ∈ R} and a semigroup with a global attractor A ∞ on a Banach space . We say that is asymptotically autonomous to if whenever ‖ − 0 ‖ → 0 as → −∞, while is uniformly asymptotically autonomous to if the convergence in (2) is There is not much research on this kind of problem. The representative literature is published by Kloeden [21] which proved that if is uniformly asymptotically autonomous to and the pullback attractor A is uniformly compact (i.e., ⋃ ∈R A( ) is precompact), then the pullback attractor converges to the global attractor in the Hausdorff semidistance sense: where → +∞ which is different in this paper. Other forms of results can be found in [22][23][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 ⋃ ⩽ A( ) is precompact) and find that A is backwards compact if and only if the upper semicontinuity holds; i.e., if is weakly asymptotically autonomous ( → −∞) to , in this paper.

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 ( , ‖ ⋅ ‖ ) be a Banach space and D is the collection of all bounded nonempty subsets of . A set-valued mapping D : R → 2 \ 0 is called a nonautonomous set in , and it is said to have a topological property (such as boundedness, compactness, or closedness) if D( ) has this property for each ∈ R. We also say that a nonautonomous set D( ) is increasing if D( ) ⊂ D( ) for ⩽ .
where and throughout this paper dist(⋅, ⋅) is Hausdorff semidistance, i.e., 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 family A 1 (⋅) of closed sets that pullback attracts bounded subsets of , then A( ) ⊂ A 1 ( ). 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 set K in is a pullback absorbing set at time ∈ R for an evolution process if, for each bounded subset in , there is 0 fl 0 ( , ) > 0 such that Definition 6. An evolution process in is said to possess the backwards pullback flattening condition if given a bounded set ⊂ , ∈ R and > 0; there exist 0 fl 0 ( , , ) > 0 and a finite dimensional subspace 1 of such that, for a bounded projector : and Theorem 7 (see [18]). Let be an evolution process in a Banach space ; assume that (i) has an increasing and bounded absorbing set K(⋅), Then has a backwards compact attractor A given by Let an evolution process have a pullback attractor A and a semigroup with a global attractor A ∞ . Definition 8. An evolution process is said to be weakly asymptotically autonomous to if for each ⩾ 0, whenever ∈ A( ), 0 ∈ A ∞ and → 0 .
Theorem 9. Let be weakly asymptotically autonomous to . Then the upper semicontinuity holds; i.e., if and only if A is backwards compact.

Proof.
Sufficiency. We argue by contradiction. Since A is backwards compact, then fl ⋃ ⩽0 A( ) is compact. Suppose that the semicontinuity (14) is not true, then there are > 0 and 0 By the attraction of A ∞ under the semigroup, there is a 0 ∈ N such that By the invariance of the pullback attractor A, we see that, for Since { } is included into the compact set , it follows that there exist a subsequence { } and ∈ such that → in as → ∞.
Applying the (13) in the case that = | if is large enough. From (16) and (18), we obtain that which contradicts with (15). Therefore the semicontinuity (14) holds true.
Necessity. Suppose the semicontinuity (14) holds true. We need to prove the precompactness of ⋃ ⩽ A( ) for each fixed ∈ R. Taking a sequence { } from this set, we then choose ⩽ such that ∈ A( ). We will prove that the sequence { } has a convergent subsequence in two case.
Define a mapping Υ : [ 0 , +∞) × → , ( , ) → ( , 0 ) , 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 Then ⋃ 0 ⩽ ⩽ A( ) is a compact set since the range of a continuous mapping on a compact set is compact. Hence { } is precompact as required.
In this case, passing to a subsequence, we may assume ↓ −∞. By the upper semicontinuity assumption (14), we have For each ∈ N we choose a ∈ ∞ such that ( , ) < ( , ∞ )+1/ . Since the global attractor ∞ is a compact set, it implies that the sequence { } has a convergent subsequence such that → as → ∞. Therefor, which together with (21) implies that → as required.
We consider the initial problem of (23)-(25) with mixed boundary conditions: where n is the unit outward normal on Γ. For the mathematical setting of this problem, we introduce some Hilbert spaces. We set = 1 × 2 and = 1 × 2 with where , and so on. We use (⋅, ⋅) to denote the usual scalar product in L 2 (O) and equip = 1 × 2 with the scalar product (⋅, ⋅) and norm ‖ ⋅ ‖ by We take the scalar product in 1 and 2 with the general forms denoted by ((⋅, ⋅)) and since O ⊂ R 2 is a bounded smooth domain, we take equivalent norms on 1 and 2 to be the same symbol ‖∇ ⋅ ‖; that is, We equip = 1 × 2 with the scalar product ((⋅, ⋅)) and the norm ‖ ⋅ ‖ given by The trilinear form ( , V, ) and the bilinear operator B from Moreover, we have the following useful relations (see [11,25]): where is an intrinsic positive constant.
On the other hand, through the above terms, (23)-(25) can be rewritten in a weak form as follows: with the initial-boundary condition (26).

Assumptions on the Nonautonomous Force.
In order to obtain a backwards compact attractor in for (35)-(37), a basic assumption for external force is ∈ 2 (R, L 2 (O)). Furthermore, one has the following.
By employing Galerkin method, we have the following wellpossessedness of problem (35)-(37), which is similar to the nonautonomous case as given in [26].

Backwards Compact Attractors in .
In this subsection, our main work is to prove that the evolution process has an increasing bounded pullback absorbing set in . From now on, we assume without loss of generality that will be a positive constant which may alter its values everywhere.
We now state our result as follows.

Theorem 14. Assume is backwards tempered, then the evolution process generated by nonautonomous 2D MHD equations possesses a backwards compact attractor
Proof. Define a nonautonomous set by where ( ) is given by (44). By the compactness of the Sobolev embedding and (52), K 1 ( ) is compact and pullback absorbing in . It is readily to check that the process is backwards pullback asymptotically compact in and thus is backwards pullback flattening follows from [17, Theorem 2.7]. Then the conclusion can be proved by Theorem 7.
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Backwards Compact Attractors in .
In this subsection, we prove the existence of backwards compact attractors in . To do this, we first give a decomposition of an element in . To this end, we consider the eigenvalue problem: Then it is known that the above problem shows a family of complete orthonormal basis { } ∞ =1 of 2 (O) consisting of eigenvectors of −Δ who has countable spectrum , = 1, 2, . . ., such that Let = span{ 1 , 2 , . . . , } ⊂ and : → be the canonical projector and be the identity. Then for every V ∈ there exists a unique decomposition where ⊥ is the orthogonal complement of .
Proof. By (58), we have Integrating (68) over [ − 1, ], we can obtain Thus by Lemmas 12 and 13 we have Therefor, we obtain (67) by taking the supremum in (70) over all the past time ⩽ .

Lemma 16.
Let be backwards tempered and backwards limiting, then for each > 0, ∈ R, and > 0, there exist 1 fl 1 ( , ) ⩾ 2 and fl ( , ) > 0 such that, for all ⩾ 1 , ⩾ and ‖ 0 ‖ ⩽ , Proof. Let ∈ R be fixed. For each ⩽ , we multiply equation in (35) by −Δ 2 and (36) by −] 3 Δ 2 , respectively, and then integrate over O to find that Notice from (34) that we have The nonlinear term in (72) is controlled by Discrete Dynamics in Nature and Society Then from (72) to (78) and using ‖Δ 2 ‖ 2 ⩾ ‖∇ 2 ‖ 2 , we find We multiply (79) by (]/2) with ∈ [ , ], integrating the result in ∈ [ , ], and then integrating it once again in ∈ [ − 1, ], we obtain We now take into account the supremum of each term in (80) over the past time. From (52) and the increasing property of ( ), we can see that, for all ⩾ 0 with some 0 fl 0 ( ) ⩾ 2, Similar, by (52) and (67), we obtain Finally, is backwards limiting by assumption (39) Proof. Define a nonautonomous set by where ( ) is given by (44). It is obvious that K 2 ( ) is bounded and increasing absorbing set in . On the other hand, by Lemmas 13 and 16 the process is backwards pullback flattening in . Then the all conditions in Theorem 7 are fulfilled. Therefor there exists a backwards compact attractor A = {A( )} ∈R in .

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 attractor A ∞ of the problem̂− with initial-boundary valueŝ For convenience, we rewrite the solution of (86)-(88) bŷfl (̂,̂) and the initial data bŷ0 fl (̂0,̂0).
To discuss the asymptotically autonomous problem, we need to give a further assumption about the forcing . We assume that ( , ⋅) → ∞ as is asymptotically autonomous to .
Similarly, subtracting (36) from (87) we find that Taking the inner product of (92) with 1 in , we have Analogously to (95), for the second term on the right hand said of (94), we obtain From (94) to (96), we get Similarly, we take the inner product of (93) with ] 3 2 in to get Let be a continuous and trilinear operator on × × given by for V = ( , ) ∈ , = 1, 2, 3. Thanks to the discrete Hölder inequality we have This completes the proof.
Finally, by using the existence of a backwards compact attractor given in Theorem 14 and the asymptotic convergence given in Lemma 18, the following result was established following from Theorem 9 immediately. (107)

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.