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Backwards Asymptotically Autonomous Dynamics for 2D MHD Equations.

1. Introduction

In this paper, we consider the existence and backwards compactness of pullback attractors for the nonautonomous MHD equations on a bounded domain O [subset] [R.sup.2]:

[mathematical expression not reproducible]. (1)

The unknown u = ([u.sub.1], [u.sub.2]) is the velocity vector, B = ([B.sub.1], [B.sub.2]) is the magnetic field, and p is the pressure. The positive constants [v.sub.1] = 1/Re, [v.sub.2] = 1/Rm, and [v.sub.3] = [M.sup.2] [v.sub.1][v.sub.2], where Re, Rm, and M are called the Reynolds number, magnetic Reynolds number, and Hartman number, respectively (see [1]). The external force f [member of] [L.sup.2.sub.loc](R, [L.sup.2](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 force f 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 about backwards compactness of a pullback attractor, which means that the union of a pullback attractor over the past time is precompact; i.e., [[universal].sub.s[much less than]t] A(s) is precompact for all t [member of] 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 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 H and V, respectively. In this case, we need only to assume that the nonautonomous external force f is backwards tempered and backwards limiting. The spectrum decomposition technique is used to give required backwards uniform estimates in V.

Finally, we consider the asymptotically autonomous dynamics of PDE. Let S be an evolution process with a pullback attractor A = {A(t) : t [member of] r} and T a semigroup with a global attractor [S.sub.[infinity]] on a Banach space X. We say that S is asymptotically autonomous to T if

[mathematical expression not reproducible], (2)

whenever [[parallel][x.sub.[tau]]-[x.sub.0][parallel].sub.X] [right arrow] 0 as [tau] [right arrow] -[infinity], while S is uniformly asymptotically autonomous to T if the convergence in (2) is uniform in t [much greater than] 0; i.e.,

[mathematical expression not reproducible]. (3)

There is not much research on this kind of problem. The representative literature is published by Kloeden [21] which proved that if S is uniformly asymptotically autonomous to T and the pullback attractor a is uniformly compact (i.e., [[universal].sub.s[member of]R] A(s) is precompact), then the pullback attractor converges to the global attractor in the Hausdorff semidistance sense:

[mathematical expression not reproducible]. (4)

where [tau] [right arrow] +[infinity] 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 [[universal].sub.s[less than or equal to]t] A(s) is precompact) and find that A is backwards compact if and only if the upper semicontinuity holds; i.e.,

[mathematical expression not reproducible], (5)

if S is weakly asymptotically autonomous ([tau] [right arrow] -[infinity]) 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, [[parallel]*[parallel].sub.X]) be a Banach space and D is the collection of all bounded nonempty subsets of X. A set-valued mapping D : R [right arrow] [2.sup.X] \ 0 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 [member of] R. We also say that a nonautonomous set D(t) is increasing if D(s) [subset] D(t) for s [less than or equal to] t.

Definition 1. A nonautonomous set D [subset] X is called backwards compact (resp., backwards bounded) if [[universal].sub.s[less than or equal to]t] D(s) is precompact (resp., bounded) in X with each t [member of] R.

Definition 2. An evolution process S in X is a family of mappings S(t, [tau]) : X [right arrow] X with t [much greater than] [tau], which satisfies

[mathematical expression not reproducible], (6)

and (t, [tau], x) [right arrow] S (t, [tau]) x

is continuous for t [greater than or equal to] t and x [member of] X.

Definition 3. A nonautonomous set A(*) 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 [greater than or equal to] s;

(3) A(*) is pullback attracting set, which means that it pullback attracts every bounded subset D [member of] D, i.e.,

[mathematical expression not reproducible], (7)

where and throughout this paper dist(*,*) is Hausdorff semidistance, i.e.,

[mathematical expression not reproducible]. (8)

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.sub.1](*) of closed sets that pullback attracts bounded subsets of X, then A(t) [subset] [A.sub.1](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 set k in X is a pullback absorbing set at time t [member of] R for an evolution process S if, for each bounded subset D in X, there is [[tau].sub.0] := [[tau].sub.0] (t, D) > 0 such that

S(t, t-r) D [subset] K (t), for all [tau] [greater than or equal to] [[tau].sub.0]. (9)

Definition 6. An evolution process S in X is said to possess the backwards pullback flattening condition if given a bounded set D [subset] X, t [member of] R and [epsilon] > 0; there exist [[tau].sub.0] := [[tau].sub.0]([epsilon], t, D) > 0 and a finite dimensional subspace [X.sub.1] of X such that, for a bounded projector P : X [right arrow] [X.sub.1],

[mathematical expression not reproducible], (10)

and

[mathematical expression not reproducible] (11)

Theorem 7 (see [18]). Let S 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.

Then S has a backwards compact attractor A given by

[mathematical expression not reproducible]. (12)

Let an evolution process S have a pullback attractor a and a semigroup T with a global attractor [A.sub.[infinity]].

Definition 8. An evolution process S is said to be weakly asymptotically autonomous to T if for each t [greater than or equal to] 0,

[mathematical expression not reproducible], (13)

whenever [x.sub.[tau]] [member of] A(t), [x.sub.0] [member of] [A.sub.[infinity]] and [x.sub.[tau]] [right arrow] [x.sub.0].

Theorem 9. Let S be weakly asymptotically autonomous to T. Then the upper semicontinuity holds; i.e.,

[mathematical expression not reproducible] (14)

if and only if A is backwards compact.

Proof.

Sufficiency. We argue by contradiction. Since A is backwards compact, then C := [bar.[[universal].sub.s[less than or equal to]0] A(s)] is compact. Suppose that the semicontinuity (14) is not true, then there are [delta] > 0 and 0 > [[tau].sub.n] [down arrow] -[infinity] such that [dist.sub.X] (A([[tau].sub.n]), [A.sub.[infinity]]) [greater than or equal to] 4[delta] for all n [member of] N. We choose [x.sub.n] [member of] A([[tau].sub.n]) such that

d ([x.sub.n], [A.sub.[infinity]]) [greater than or equal to] [dist.sub.X] (A ([[tau].sub.n]), [A.sub.[infinity]]) - [delta] [greater than or equal to] 3[delta]. (15)

By the attraction of [A.sub.[infinity]] under the semigroup, there is a [n.sub.0] [member of] N such that

[mathematical expression not reproducible]. (16)

By the invariance of the pullback attractor A, we see that, for any [x.sub.n] [member of] A([[tau].sub.n]), there exists [mathematical expression not reproducible] such that

[mathematical expression not reproducible]. (17)

Since {[y.sub.n]} is included into the compact set C, it follows that there exist a subsequence [mathematical expression not reproducible] and y [member of] C such that [mathematical expression not reproducible] in X as k [right arrow] [infinity].

Applying the (13) in the case that [mathematical expression not reproducible], we find

[mathematical expression not reproducible], (18)

if k is large enough. From (16) and (18), we obtain that

[mathematical expression not reproducible], (19)

which contradicts with (15). Therefore the semicontinuity (14) holds true.

Necessity. Suppose the semicontinuity (14) holds true. We need to prove the precompactness of [[universal].sub.s[less than or equal to]t] A(s) for each fixed t [member of] R. Taking a sequence {[x.sub.n]} from this set, we then choose [s.sub.n] [less than or equal to] t such that [x.sub.n] [member of] A([s.sub.n]). We will prove that the sequence {[x.sub.n]} has a convergent subsequence in two case.

Case 1. If [s.sub.0] = [inf.sub.n[member of]N][s.sub.n] [not equal to] -[infinity], then for [mathematical expression not reproducible]. Define a mapping Y : [[s.sub.0], +[infinity]) x X [right arrow] X, (s, x) [right arrow] S(s, [s.sub.0])x, then the continuity assumption implies that [??] is a continuous mapping. By the invariance of the pullback attractor A, it is easy to see that

[mathematical expression not reproducible]. (20)

Then [mathematical expression not reproducible] is a compact set since the range of a continuous mapping on a compact set is compact. Hence {[x.sub.n]} is precompact as required.

Case 2. [s.sub.0] = [inf.sub.n[member of]N][s.sub.n] = -[infinity]. In this case, passing to a subsequence, we may assume [s.sub.n] [down arrow] -[infinity]. By the upper semicontinuity assumption (14), we have

d ([x.sub.n], [A.sub.[infinity]]) [less than or equal to] [dist.sub.X] (A([s.sub.n]), [A.sub.[infinity]]) [right arrow] 0, as n [right arrow] [infinity]. (21)

For each n [member of] n we choose a [y.sub.n] [member of] [A.sub.[infinity]] such that d([x.sub.n], [y.sub.n]) < d([x.sub.n], [A.sub.[infinity]])+1/n. Since the global attractor [A.sub.[infinity]] is a compact set, it implies that the sequence {[y.sub.n]} has a convergent subsequence such that [mathematical expression not reproducible]. Therefor,

[mathematical expression not reproducible], (22)

which together with (21) implies that [mathematical expression not reproducible] 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 to holds true at every single time (e.g., [mathematical expression not reproducible]), not uniformly in t [greater than or equal to] uniformness condition in [21, Theorem 3.2] successfully.

3. Nonautonomous 2D MHD Equations

3.1. Functional Spaces and Operators. Let O [subset] [R.sup.2] be a bounded, open, and simply connected subset with regular boundary [GAMMA]. We consider the following MHD equations defined on O x [[tau], +[infinity]):

[u.sub.t] + (u x [nabla]) u - [v.sub.1][DELTA]u - [v.sub.3] (B x [nabla]) B + [nabla][P.sub.0] = f (x, t), (23)

[B.sub.t] + (u * [nabla]) B - (B x [nabla]) u - [v.sub.2][DELTA]B = 0, (24)

div u = 0, div B = 0, (25)

where [P.sub.0] = p + [v.sub.3] [[absolute value of B].sup.2]/2 is the total pressure and [v.sub.i] are positive constants.

We consider the initial problem of (23)-(25) with mixed boundary conditions:

[mathematical expression not reproducible]. (26)

where n is the unit outward normal on [GAMMA]. For the mathematical setting of this problem, we introduce some Hilbert spaces. We set H = [H.sub.1] x [H.sub.2] and V = [V.sub.1] x [V.sub.2] with

[mathematical expression not reproducible], (27)

where [L.sup.2](O) = [L.sup.2][(O).sup.2], [H.sup.1](O) = [H.sup.1][(O).sup.2], and so on. We use (*,*) to denote the usual scalar product in [L.sup.2](O) and equip H = [H.sub.1] x [H.sub.2] with the scalar product [(*,*).sub.H] and norm [[parallel]*[parallel].sub.H] by

[mathematical expression not reproducible]. (28)

We take the scalar product in [V.sub.1] and [V.sub.2] with the general forms denoted by ((*,*)) and since O [subset] [R.sup.2] is a bounded smooth domain, we take equivalent norms on [V.sub.1] and [V.sub.2] to be the same symbol [parallel][nabla]*[parallel]; that is,

[mathematical expression not reproducible]. (29)

We equip V = [V.sub.1] x [V.sub.2] with the scalar product [((*,*)).sub.V] and the norm [[parallel]*[parallel].sub.V] given by

[mathematical expression not reproducible]. (30)

The trilinear form b(u, v, w) and the bilinear operator B from [H.sup.1](O) x [H.sup.1](O) into [H.sup.-1](O) are defined by

[mathematical expression not reproducible]. (31)

Moreover, we have the following useful relations (see [11, 25]):

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible], (33)

[mathematical expression not reproducible], (34)

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:

du/dt - [v.sub.1][DELTA]u= (-B (u, u) + [v.sub.3]B (B, B)) + f (x, t), (35)

dB/dt - [v.sub.2][DELTA]B = (-B (u, B) + B (B, u)), (36)

div u = 0, div B = 0, (37)

with the initial-boundary condition (26).

3.2. Assumptions on the Nonautonomous Force. In order to obtain a backwards compact attractor in H for (35)-(37), a basic assumption for external force is f [member of] [L.sup.2.sub.loc](R, [L.sup.2](O)). Furthermore, one has the following.

Assumption F1. f is backwards tempered; i.e.,

[mathematical expression not reproducible], (38)

for all [gamma] > 0 and t [member of] 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.,

[mathematical expression not reproducible], (39)

for all t [member of] 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. Let f [member of] [L.sup.2.sub.loc] (R, [L.sup.2](O)). Then for each ([u.sub.0], [B.sub.0]) [member of] H and for each [tau] [member of] R, there exists a unique weak solution

(u, B) [member of] [L.sup.2.sub.loc] ([tau], [infinity]; V) [intersection] C ([[tau], [infinity]); H) (40)

satisfying (35)-(37) in distribution sense with (u, B)[|.sub.t=[tau]] = ([u.sub.0], [B.sub.0]). Moreover, the mapping ([u.sub.0], [B.sub.0]) [??] (u, B) is continuous in H.

For convenience, we rewrite the solution of (35)-(37) by [phi] := (u, B) and the initial data by [[phi].sub.0] := ([u.sub.0], [B.sub.0]).

By Lemma 11, we can use the unique weak solution to define an evolution process S(t, [tau]) : H [right arrow] H by

S (t, [tau]) [[phi].sub.0] := [phi](t, [tau], [[phi].sub.0]), for all t [greater than or equal to] t and [[phi].sub.0] [member of] H. (41)

4. Backwards Compact Attractors for 2D MHD Equations

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

Lemma 12. Let f be backwards tempered, then for each t [member of] R and R > 0, there exists [[tau].sub.0] := [[tau].sub.0](R) [greater than or equal to] 2 such that, for all [tau] [greater than or equal to] [[tau].sub.0] and [[parallel][[phi].sub.0][parallel].sub.H] [much less than] R,

[mathematical expression not reproducible], (42)

[mathematical expression not reproducible], (43)

where [lambda] is given by (48) and M(t) is a nonnegative increasing function defined by

[mathematical expression not reproducible]. (44)

Proof. Let t [member of] R be fixed. For each s [less than or equal to] t, we multiply equation in (35) by u and (36) by [v.sub.3]B respectively and integrate over O, then the sum of them is

[mathematical expression not reproducible]. (45)

Notice from (31) and (32) that

[mathematical expression not reproducible]. (46)

For the nonlinear term, we have

[mathematical expression not reproducible], (47)

where we have used the notation [phi] = (u, B), [[parallel][phi][parallel].sup.2] = [[parallel]u[parallel].sup.2] + [v.sub.3] [[parallel]B[parallel].sup.2], and [lambda] > 0 is given by

[mathematical expression not reproducible]. (48)

Substituting the above into (45), we have

[mathematical expression not reproducible]. (49)

Multiplying (49) by [e.sup.[lambda]s] and integrating it over [s-[tau], s], we obtain

[mathematical expression not reproducible], (50)

for all [tau] [greater than or equal to] [[tau].sub.0] with some [[tau].sub.0] := [[tau].sub.0](R) [greater than or equal to] 2.

On the other hand, we multiply (49) by [e.sup.[lambda]s] and integrating it over [s-[tau], r] with r [member of] [s-2, s], we obtain

[mathematical expression not reproducible], (51)

for all [tau] [greater than or equal to] [[tau].sub.0] with some [[tau].sub.0] := [[tau].sub.0](R) [greater than or equal to] 2.

Taking the supremum with respect to the past time s [less than or equal to] 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. Let f be backwards tempered, then for each t [member of] R and R > 0, there exists [[tau].sub.0] := [[tau].sub.0](R) [greater than or equal to] 2 such that, for all [tau] [greater than or equal to] [[tau].sub.0] and [[parallel][[phi].sub.0][parallel].sub.H] [less than or equal to] R,

[mathematical expression not reproducible], (52)

where M(t) is given by (44).

Proof. Let t [member of] R be fixed. For each s [less than or equal to] t, we multiply equation in (35) by -[DELTA]u and (35) by -[v.sub.3][DELTA]B, respectively, then integrate over O, and sum the results to find

[mathematical expression not reproducible] (53)

Notice from (33) that

[mathematical expression not reproducible], (54)

[mathematical expression not reproducible]. (55)

On the other hand, by (33) and the inequality that abc [less than or equal to] C([[epsilon].sub.1], [[epsilon].sub.2])[a.sup.4] + [[epsilon].sub.1][b.sup.4] + [[epsilon].sub.2][c.sup.2], we have

[mathematical expression not reproducible]. (56)

The nonlinear term in (53) is controlled by

[mathematical expression not reproducible], (57)

Substituting (54)-(57) into (53), we find

[mathematical expression not reproducible], (58)

where h(s) = c[[parallel][phi](s)[parallel].sup.2.sub.H] [[parallel][nabla][phi](s)[parallel].sup.2.sub.H]. Integrate (58) over ([xi], r) with [xi] [member of] [s-2, s-1] and r [member of] [s-1, s] to obtain

[mathematical expression not reproducible] (59)

We integrate (59) with respect to [xi] over [s - 2, s - 1] with s [less than or equal to] t; we have

[mathematical expression not reproducible] (60)

On the other hand, by Lemma 12, we have

[mathematical expression not reproducible] (61)

Therefor, we insert (61) into (60) to obtain that

[mathematical expression not reproducible], (62)

for all [tau] [greater than or equal to] [[tau].sub.0] with some [[tau].sub.0] := [[tau].sub.0](R) [greater than or equal to] 2. Hence, we get (52) by taking the supremum in (62) with respect to s [much less than] t.

We now state our result as follows.

Theorem 14. Assume f is backwards tempered, then the evolution process S generated by nonautonomous 2D MHD equations possesses a backwards compact attractor A = [{A(t)}.sub.t[member of]R] in H.

Proof. Define a nonautonomous set by

[mathematical expression not reproducible], (63)

where M(t) is given by (44). By the compactness of the Sobolev embedding and (52), [K.sub.1] (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 in V. In this subsection, we prove the existence of backwards compact attractors in V. To do this, we first give a decomposition of an element in V. To this end, we consider the eigenvalue problem:

-[DELTA]v(x) = [lambda]v(x), x [member of] O, [v|.sub.[partial derivative]O] = 0, (64)

Then it is known that the above problem shows a family of complete orthonormal basis [{[e.sub.j]}.sup.[infinity].sub.j=1] of [L.sup.2](O) consisting of eigenvectors of -[DELTA] who has countable spectrum [[lambda].sub.j], j = 1, 2, ..., such that

[mathematical expression not reproducible]. (65)

Let [V.sub.m] = span[[e.sub.1], [e.sub.2], ..., [e.sub.m]} [subset] V and [P.sub.m] : V [right arrow] [V.sub.m] be the canonical projector and I be the identity. Then for every v [member of] V there exists a unique decomposition

[mathematical expression not reproducible], (66)

where [V.sup.[perpendicular to].sub.m] is the orthogonal complement of [V.sub.m].

Lemma 15. Let f be backwards tempered, then for each t [member of] R and R > 0, there exists [[tau].sub.0] := [[tau].sub.0](R) [greater than or equal to] 2 such that, for all [tau] [greater than or equal to] [[tau].sub.0] and [[parallel][[phi].sub.0][parallel].sub.H] [less than or equal to] R,

[mathematical expression not reproducible], (67)

where M(t) is given by (44).

Proof. By (58), we have

[mathematical expression not reproducible], (68)

Integrating (68) over [s-1, s], we can obtain

[mathematical expression not reproducible]. (69)

Thus by Lemmas 12 and 13 we have

[mathematical expression not reproducible]. (70)

Therefor, we obtain (67) by taking the supremum in (70) over all the past time s [less than or equal to] t.

Lemma 16. Let f be backwards tempered and backwards limiting, then for each [epsilon] > 0, t [member of] R, and R > 0, there exist [[tau].sub.1] := [[tau].sub.1]([epsilon], R) [greater than or equal to] 2 and N := N([epsilon], R) > 0 such that, for all [tau] [greater than or equal to] [[tau].sub.1], m [greater than or equal to] N and [[parallel][[phi].sub.0][parallel].sub.H] [less than or equal to] R,

[mathematical expression not reproducible]. (71)

Proof. Let t [member of] R be fixed. For each s [less than or equal to] t, we multiply equation in (35) by -[DELTA][u.sub.2] and (36) by -[v.sub.3][DELTA][B.sub.2], respectively, and then integrate over O to find that

[mathematical expression not reproducible] (72)

[mathematical expression not reproducible] (73)

Notice from (34) that we have

[mathematical expression not reproducible], (74)

[mathematical expression not reproducible]. (75)

[mathematical expression not reproducible], (76)

[mathematical expression not reproducible]. (77)

The nonlinear term in (72) is controlled by

[mathematical expression not reproducible], (78)

Then from (72) to (78) and using [[parallel][DELTA][u.sub.2][parallel].sup.2] [greater than or equal to] [[lambda].sub.m] [[parallel][DELTA][u.sub.2][parallel].sup.2], we find

[mathematical expression not reproducible]. (79)

We multiply (79) by [mathematical expression not reproducible] with s [member of] [r, t]] integrating the result in s [member of] [r, t], and then integrating it once again in r [member of] [t-1, t], we obtain

[mathematical expression not reproducible] (80)

We 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 [tau] [greater than or equal to] [[tau].sub.0] with some [[tau].sub.0] := [[tau].sub.0](R) [much greater than] 2,

[mathematical expression not reproducible]. (81)

Similar, by (52) and (67), we obtain

[mathematical expression not reproducible]. (82)

Finally, f is backwards limiting by assumption (39); thus

[mathematical expression not reproducible]. (83)

Hence, from (80) to (83), for all [tau] [greater than or equal to] [[tau].sub.1],

[mathematical expression not reproducible]. (84)

This completes the proof.

Theorem 17. Assume f 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)}.sub.t[member of]R] in V.

Proof. Define a nonautonomous set by

[mathematical expression not reproducible], (85)

where M(t) is given by (44). It is obvious that [K.sub.2](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)}.sub.t[member of]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 semi-continuity to the autonomous global attractor [A.sub.[infinity]] of the problem

[mathematical expression not reproducible], (86)

[mathematical expression not reproducible], (87)

[mathematical expression not reproducible], (88)

with initial-boundary values

[mathematical expression not reproducible]. (89)

For convenience, we rewrite the solution of (86)-(88) by [mathematical expression not reproducible] and the initial data by [mathematical expression not reproducible].

To discuss the asymptotically autonomous problem, we need to give a further assumption about the forcing f. We assume that f(t, *) [right arrow] [f.sub.[infinity]] as S is asymptotically autonomous to T.

Assumption F3. There is a function [f.sub.[infinity]] [member of] [L.sup.2](O)) such that

[mathematical expression not reproducible]. (90)

Lemma 18. Suppose assumptions F1 and F3 are satisfied. Then the solution [phi] of (35)-(37) is asymptotically autonomous to the solution [??] of (86)-(88). More precisely,

[mathematical expression not reproducible], (91)

whenever [[parallel][[phi].sub.0]-[[??].sub.0][parallel].sub.H] [right arrow] 0 as [tau] [right arrow] -[infinity].

Proof. Let [mathematical expression not reproducible]. Then subtract (35) from (86) and we obtain

[mathematical expression not reproducible]. (92)

Similarly, subtracting (36) from (87) we find that

[mathematical expression not reproducible]. (93)

Taking the inner product of (92) with [Q.sub.1] in H, we have

[mathematical expression not reproducible]. (94)

Using the trilinearity of b and relation (31) and (32), we have

[mathematical expression not reproducible]. (95)

Analogously to (95), for the second term on the right hand said of (94), we obtain

[mathematical expression not reproducible] (96)

From (94) to (96), we get

[mathematical expression not reproducible]. (97)

Similarly, we take the inner product of (93) with [v.sub.3][Q.sub.2] in H to get

[mathematical expression not reproducible]. (98)

Now, both (97) and (98) imply that

[mathematical expression not reproducible] (99)

and then that

[mathematical expression not reproducible]. (100)

Let [chi] be a continuous and trilinear operator on V x V x V given by

[mathematical expression not reproducible], (101)

for [v.sub.i] = ([u.sub.i], [B.sub.i]) [member of] V, i = 1, 2, 3. Thanks to the discrete Holder inequality we have

[mathematical expression not reproducible]. (102)

Therefor (100) can be rewritten as follows:

[mathematical expression not reproducible] (103)

Then applying the Gronwall inequality to (103), we have

[mathematical expression not reproducible] (104)

By Lemma 13, analogous results also hold for [[parallel][nabla][??](r)[parallel].sup.2.sub.H]. Hence, we have

[mathematical expression not reproducible] (105)

Since [mathematical expression not reproducible], it follows from Assumption F3 and (104) that

[mathematical expression not reproducible], (106)

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.

Theorem 19. Suppose assumptions F1 and F3 are satisfied. Then the nonautonomous MHD equations have a backwards compact pullback attractor A = {A(t) : t [member of] R}, which converges to the global attractor [A.sub.[infinity]] in H; that is,

[mathematical expression not reproducible]. (107)

https://doi.org/10.1155/2018/4948301

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

Each of the authors contributed to each part of this study equally. All authors read and proved the final vision of the manuscript.

Acknowledgments

This work was supported by the Program for the Innovation Research Grant for the Postgraduates of Guangzhou University (no. 2017GDJC-D08).

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Jiali Yu (iD), (1) Wenhuo Su (iD), (2) and Dongmei Xu (iD) (3)

(1) School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

(2) Center of Applied Mathematics, Yichun University, Yichun 336000, China

(3) School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China

Correspondence should be addressed to Dongmei Xu; xudongmei@126.com

Received 2 July 2018; Accepted 26 August 2018; Published 23 September 2018

Academic Editor: Maria Alessandra Ragusa
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Title Annotation:Research Article
Author:Yu, Jiali; Su, Wenhuo; Xu, Dongmei
Publication:Discrete Dynamics in Nature and Society
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Date:Jan 1, 2018
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