Research Article | Open Access
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.
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 ). 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  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 ifwhenever 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  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 ). 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 haveFor 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 withwhere , , and so on. We use to denote the usual scalar product in and equip with the scalar product and norm byWe 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 byMoreover, we have the following useful relations (see [11, 25]):where is an intrinsic positive constant.
3.2. Assumptions on the Nonautonomous Force
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 isNotice 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 obtainfor all with some .
On the other hand, we multiply (49) by and integrating it over with , we obtainfor 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 bySubstituting (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 thatfor 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 bywhere 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).
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 findWe multiply (79) by with , integrating the result in , and then integrating it once again in , we obtain