Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 4948301, 11 pages
https://doi.org/10.1155/2018/4948301

Research Article

Backwards Asymptotically Autonomous Dynamics for 2D MHD Equations

Jiali Yu,¹ Wenhuo Su,² and Dongmei Xu³

¹School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
²Center of Applied Mathematics, Yichun University, Yichun 336000, China
³School 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

Copyright © 2018 Jiali Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In this paper, we consider the existence and backwards compactness of pullback attractors for the nonautonomous MHD equations on a bounded domain :

The unknown is the velocity vector, is the magnetic field, and is the pressure. The positive constants , , and , where , , and are called the Reynolds number, magnetic Reynolds number, and Hartman number, respectively (see [1]). The external force .

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 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., is precompact for all . 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 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 and a semigroup with a global attractor on a Banach space . We say that is asymptotically autonomous to if

whenever as , while is uniformly asymptotically autonomous to if the convergence in (2) is uniform in ; i.e.,

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 is uniformly compact (i.e., 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 [22–24] 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 is precompact) and find that is backwards compact if and only if the upper semicontinuity holds; i.e.,

if is weakly asymptotically autonomous () to , 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 be a Banach space and is the collection of all bounded nonempty subsets of . A set-valued mapping is called a nonautonomous set in , and it is said to have a topological property (such as boundedness, compactness, or closedness) if has this property for each . We also say that a nonautonomous set is increasing if for .

Definition 1. A nonautonomous set is called backwards compact (resp., backwards bounded) if is precompact (resp., bounded) in with each .

Definition 2. An evolution process in is a family of mappings with , which satisfies

Definition 3. A nonautonomous set in is called a backwards compact attractors for a process if

(1) is backwards compact;(2) is invariant, i.e., for all ;(3) is pullback attracting set, which means that it pullback attracts every bounded subset , i.e.,

where and throughout this paper 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 of closed sets that pullback attracts bounded subsets of , then . 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 in is a pullback absorbing set at time for an evolution process if, for each bounded subset in , there is such that

Definition 6. An evolution process in is said to possess the backwards pullback flattening condition if given a bounded set and ; there exist and a finite dimensional subspace 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 ,(ii) is backwards pullback flattening.

Then has a backwards compact attractor given by

Let an evolution process have a pullback attractor and a semigroup with a global attractor .

Definition 8. An evolution process is said to be weakly asymptotically autonomous to if for each ,

whenever and .

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

if and only if is backwards compact.

Proof. Sufficiency. We argue by contradiction. Since is backwards compact, then is compact. Suppose that the semicontinuity (14) is not true, then there are and such that for all . We choose such that

By the attraction of under the semigroup, there is a such that

By the invariance of the pullback attractor , we see that, for any , there exists such that

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 and as , we find

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 for each fixed . Taking a sequence from this set, we then choose such that . We will prove that the sequence has a convergent subsequence in two case.
Case 1. If , then for , , and so . Define a mapping , then the continuity assumption implies that is a continuous mapping. By the invariance of the pullback attractor , it is easy to see that

Then is a compact set since the range of a continuous mapping on a compact set is compact. Hence is precompact as required.
Case 2. . In this case, passing to a subsequence, we may assume . By the upper semicontinuity assumption (14), we have

For each we choose a such that . 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.

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., ), not uniformly in . So we reduce the uniformness condition in [21, Theorem 3.2] successfully.

3. Nonautonomous 2D MHD Equations

3.1. Functional Spaces and Operators

Let be a bounded, open, and simply connected subset with regular boundary . We consider the following MHD equations defined on :

where is the total pressure and are positive constants.

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

where is the unit outward normal on . For the mathematical setting of this problem, we introduce some Hilbert spaces. We set and with

where , , and so on. We use to denote the usual scalar product in and equip with the scalar product and norm by

We take the scalar product in and with the general forms denoted by and since is a bounded smooth domain, we take equivalent norms on and to be the same symbol ; that is,

We equip with the scalar product and the norm given by

The trilinear form and the bilinear operator from into are defined by

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).

3.2. Assumptions on the Nonautonomous Force

In order to obtain a backwards compact attractor in for (35)-(37), a basic assumption for external force is . Furthermore, one has the following.

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

To prove the existence of backwards compact attractor in for (35)-(37), we assume further the following.

Assumption F2. is backwards limiting; i.e.,

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 . Then for each and for each , there exists a unique weak solution

satisfying (35)-(37) in distribution sense with . Moreover, the mapping is continuous in .

For convenience, we rewrite the solution of (35)-(37) by and the initial data by .

By Lemma 11, we can use the unique weak solution to define an evolution process by

4. Backwards Compact Attractors for 2D MHD Equations

4.1. 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.

Lemma 12. Let be backwards tempered, then for each and , there exists such that, for all and ,

where is given by (48) and is a nonnegative increasing function defined by

Proof. Let be fixed. For each , we multiply equation in (35) by and (36) by respectively and integrate over , then the sum of them is

Notice from (31) and (32) that

For the nonlinear term, we have

where we have used the notation , , and is given by

Substituting the above into (45), we have

Multiplying (49) by and integrating it over , we obtain

for all with some .
On the other hand, we multiply (49) by and integrating it over with , we obtain

for all with some .
Taking the supremum with respect to the past time in (51) and (50), we get (42) and (43). By the assumption (38), is finite and increasing. This completes the proof.

Lemma 13. Let be backwards tempered, then for each and , there exists such that, for all and ,

where is given by (44).

Proof. Let be fixed. For each , we multiply equation in (35) by and (35) by , respectively, then integrate over , and sum the results to find

Notice from (33) that

On the other hand, by (33) and the inequality that , we have

The nonlinear term in (53) is controlled by

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

where . Integrate (58) over with and to obtain

We integrate (59) with respect to over with ; we have

On the other hand, by Lemma 12, we have

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

for all with some . Hence, we get (52) by taking the supremum in (62) with respect to .

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 in .

Proof. Define a nonautonomous set by

where is given by (44). By the compactness of the Sobolev embedding and (52), 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.

4.2. 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 of consisting of eigenvectors of who has countable spectrum , such that

Let and be the canonical projector and be the identity. Then for every there exists a unique decomposition

where is the orthogonal complement of .

Lemma 15. Let be backwards tempered, then for each and , there exists such that, for all and ,

where is given by (44).

Proof. By (58), we have

Integrating (68) over , 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 , and , there exist and such that, for all and ,

Proof. Let be fixed. For each , we multiply equation in (35) by and (36) by , respectively, and then integrate over to find that

Notice from (34) that we have

The nonlinear term in (72) is controlled by

Then from (72) to (78) and using , we find

We multiply (79) by with , integrating the result in , and then integrating it once again in , 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 with some ,

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

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

Hence, from (80) to (83), for all ,

This completes the proof.

Theorem 17. Assume be backwards tempered and backwards limiting, then the evolution process generated by nonautonomous 2D MHD equations possesses a backwards compact attractor in .

Proof. Define a nonautonomous set by

where is given by (44). It is obvious that 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 in .

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 attractor of the problem

with initial-boundary values

For convenience, we rewrite the solution of (86)-(88) by and the initial data by .

To discuss the asymptotically autonomous problem, we need to give a further assumption about the forcing . We assume that as is asymptotically autonomous to .

Assumption F3. There is a function such that

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

whenever as .

Proof. Let and . Then subtract (35) from (86) and we obtain

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

Taking the inner product of (92) with in , we have

Using the trilinearity of and relation (31) and (32), 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 in to get

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

and then that

Let be a continuous and trilinear operator on given by

for . Thanks to the discrete Hölder inequality we have

Therefor (100) can be rewritten as follows:

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

By Lemma 13, analogous results also hold for . Hence, we have

Since , it follows from Assumption F3 and (104) that

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 , which converges to the global attractor in ; that is,

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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