Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations

Gang Zhou; Rui Gao; Congyang Tian

doi:10.1515/math-2025-0174

Article Open Access

Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations

Gang Zhou , Rui Gao and Congyang Tian

Published/Copyright: July 16, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

The purpose of this article is to establish the existence of the pullback attractors for the non-autonomous magneto-micropolar fluid equations in 2D bounded domains. To this end, the asymptotic compactness of the processes generated by the solutions is required, which is proved by verifying the flattening property (also known as the “Condition C”) of the corresponding processes.

Keywords: magneto-micropolar fluid equations; pullback attractor; flattening property

MSC 2010: 35B40; 35B41; 76D07

1 Introduction

In this article, we consider the non-autonomous magneto-micropolar fluid model, which can be described by the following equations:

(1.1) ∂ u ∂ t − ( κ + χ ) Δ u + u ⋅ ∇ u + ∇ p + 1 2 ∣ h ∣ 2 = 2 χ ∇ × ω + γ h ⋅ ∇ h + f , k ∂ ω ∂ t − μ Δ ω + 4 χ ω + k u ⋅ ∇ ω + k u ⋅ ∇ ω = 2 χ ∇ × u + g , ∂ h ∂ t − α Δ h + u ⋅ ∇ h − h ⋅ ∇ u = 0 , div u = 0 , div h = 0 ,

where the functions u = ( u 1 ( x , t ) , u 2 ( x , t ) ) , ω = ω 3 ( x , t ) , and h = ( h 1 ( x , t ) , h 2 ( x , t ) ) denote the velocity, the micro-rotational velocity (angular velocity of rotation of particles), and the magnetic field, respectively. p is the pressure, and f = ( f 1 ( x , t ) , f 2 ( x , t ) , f 3 ) and g = g 3 ( x , t ) represent the external force and moment, respectively. κ , χ , γ , k , μ , and α are the positive constants related to the properties of the material, and for simplicity, let us take γ = k = 1 . In addition,

∇ × u = ∂ u 2 ∂ x 1 − ∂ u 1 ∂ x 2 , ∇ × ω = ∂ ω ∂ x 2 , − ∂ ω ∂ x 1 .

The magneto-micropolar fluid equations were first proposed by Galdi and Rionero [1], which can be used to describe the motion of electrically conducting micropolar fluid in the presence of a magnetic field. Physically, micropolar fluid is a fluid with microstructure, which represents a fluid consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, (see, e.g., [2–4]). If the magnetic field h = 0 , then equations (1.1) reduce to the micropolar fluid equations.

Magneto-micropolar fluid equations play an important role in the fields of applied and computational mathematics. There is a wide literature on the mathematical theory of magneto-micropolar fluid equations. In particular, the existence, uniqueness, and regularity of solutions for the magneto-micropolar fluid equations have been investigated in [5–7]. At the same time, extensive studies on the long-time behavior of solutions for the magneto-micropolar fluid equations have also been performed. For example, Łukaszewicz and Sadowski [8] obtained the existence of uniform attractor by using energy method in 2D unbounded domains. Matsuura [9] proved the existence of an exponential attractor for the magneto-micropolar fluid equations in 2D bounded domain. Zhao et al. [10] studied the existence of pullback attractor in ( L 2 ( Ω ) ) 3 . Later, by appropriately increasing the regularity of the initial value and the external force and moment, Li and Li [11] further investigated the regularity of the pullback attractor obtained in [10]. Recently, Yang et al. [12] proved the existence of trajectory attractor for the 3D magneto-micropolar fluid equations. For more results, see, e.g., [12–14] and references therein. In addition, we want to note that the asymptotic behavior of partial differential equations is one of the basic contents of the research, and there are many studies on this subject (see, e.g., [15–18]).

In this article, we consider equations (1.1) on a bounded domain Ω ⊆ R 2 with suitable smooth boundary ∂ Ω and satisfy the following initial and boundary conditions:

(1.2) w ( x , τ ) = ( u ( x , τ ) , ω ( x , τ ) , h ( x , τ ) ) = ( u τ ( x ) , ω τ ( x ) , h τ ( x ) ) , x ∈ Ω , τ ∈ R .

(1.3) u ( x , t ) = ω ( x , t ) = h ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ τ , + ∞ ) .

The main purpose of this article is to prove the existence of pullback attractor for systems (1.1)–(1.3). Indeed, the existence of pullback attractor has been established in [10,11] by showing the existence of the pullback absorbing family and verifying the asymptotically compactness of the generated evolution process. Similar problems have previously been considered in many studies (see, e.g., [19–23]). In all these cases, the existence of attractors has been proved by showing some sort of asymptotic compactness of the corresponding flow (or process in the non-autonomous case). However, here we prove the asymptotic compactness by verifying the flattening property of the generated evolution process. The arguments used here are essentially different from [10,11], which makes the analysis simpler.

We want to remark that to verify asymptotic compactness, one can either proceed directly, or make use of a splitting of the solutions into low and high components corresponding respectively to the finite-dimensional space and the residual space. Specifically, we decompose the infinite-dimensional Banach space X under consideration into two spaces X 1 and X 2 with dim X 1 < ∞ . Moreover, let P : X → X 1 , Q : X → X 2 be the canonical projectors, then consider splitting the solution u into P u and Q u parts to further study separately. Such a splitting is a very common technique in the study of the qualitative behavior of solutions for partial differential equation problems, such as in the construction of invariant manifolds [24] and inertial manifolds [25], the squeezing property [26], the notion of determining modes [27], and the theory of attractors [28]. In the context of proofs of the existence of attractors, it was formalized by Ma et al. [28] as their celebrated “Condition (C).” A more descriptive terminology, “flattening property,” was coined by Kloeden and Langa [29]. This method is then applied to prove the asymptotic compactness of other system (see, e.g., [30–32]). The idea of this article originates from [30,32]. The main advantage of this method is that one needs only to verify a necessary compactness condition with the same type of energy estimates as those for establishing the absorbing set. In other words, one does not need to obtain estimates in function spaces with stronger topology. This property is useful when higher regularity is not available, as demonstrated in the examples in [28] that proves the existence of the global attractors for the Navier-Stokes equations.

The rest of this article is organized as follows. In Section 2, we make some preliminaries. In Section 3, we concentrate on proving the existence of the pullback attractors for the universe of fixed bounded sets and for another universe with a tempered condition in spaces H ^ .

2 Preliminaries

In this section, we make some preliminaries. First, we introduce some notations and existing results about non-autonomous magneto-micropolar fluid equations, and then, we recall some concepts and important conclusions on infinite dynamical systems.

2.1 Notations and existing results

Throughout this article, we denote by L p ( Ω ) and W m , p ( Ω ) the usual Lebesgue space and Sobolev space endowed with norms ‖ ⋅ ‖ L p and ‖ ⋅ ‖ m , p , respectively. Particularly, H m ( Ω ) ≔ W m , 2 ( Ω ) . Moreover, let

V ≔ { φ ∈ C 0 ∞ ( Ω ) × C 0 ∞ ( Ω ) ∣ φ = ( φ 1 , φ 2 ) , ∇ ⋅ φ = 0 } , H ≔ closure of V in L 2 ( Ω ) × L 2 ( Ω ) , with norm ‖ ⋅ ‖ H and dual space H * , V ≔ closure of V in H 1 ( Ω ) × H 1 ( Ω ) , with norm ‖ ⋅ ‖ V and dual space V * , H ^ ≔ H × L 2 ( Ω ) × H , with norm ‖ ⋅ ‖ H ^ and dual space H ^ * , V ^ ≔ V × H 0 1 ( Ω ) × V , with norm ‖ ⋅ ‖ V ^ and dual space V ^ * .

Here,

‖ ( ϕ 1 , ϕ 2 ) ‖ H ≔ ( ‖ ϕ 1 ‖ L 2 2 + ‖ ϕ 2 ‖ L 2 2 ) 1 ⁄ 2 , ‖ ( ϕ 1 , ϕ 2 ) ‖ V ≔ ( ‖ ϕ 1 ‖ H 1 2 + ‖ ϕ 2 ‖ H 1 2 ) 1 ⁄ 2 , ‖ ( ϕ , ω , φ ) ‖ H ^ ≔ ( ‖ ϕ ‖ H 2 + ‖ ω ‖ L 2 2 + ‖ φ ‖ H 2 ) 1 ⁄ 2 , ‖ ( ϕ , ω , φ ) ‖ V ^ ≔ ( ‖ ϕ ‖ V 2 + ‖ ω ‖ H 1 2 + ‖ φ ‖ V 2 ) 1 ⁄ 2 .

In the subsequent instances, we simplify the notations ‖ ⋅ ‖ L 2 , ‖ ⋅ ‖ H and ‖ ⋅ ‖ H ^ by the same notation ‖ ⋅ ‖ if there is no confusion. In addition, we denote the compact embedding between spaces by ↪ ↪ . Clearly, V ^ ↪ ↪ H ^ ↪ V ^ * . We also denote by ( ⋅ , ⋅ ) the inner product in L 2 ( Ω ) , H or H ^ , and ⟨ ⋅ , ⋅ ⟩ the dual pairing between V and V * or between V ^ and V ^ * , and use dist M ( X , Y ) to represent the Hausdorff semidistance between X ⊆ M and Y ⊆ M with dist M ( X , Y ) = sup x ∈ X inf y ∈ Y dist M ( x , y ) . Furthermore, denote

L p ( I ; X ) ≔ space of strongly measurable functions on the closed interval I , with values in a Banach space X , endowed with norm ‖ φ ‖ L p ( I ; X ) ≔ ∫ I ‖ φ ‖ X p d t 1 ⁄ p , for 1 ⩽ p < ∞ , C ( I ; X ) ≔ space of continuous functions on the interval I , with values in the Banach space X , endowed with the usual norm, L loc 2 ( I ; X ) ≔ space of locally square integrable functions on the interval I , with values in the Banach space X , endowed with the usual norm .

For any w ≔ ( u , ω , h ) ∈ V ^ , we first define the operators A and L by

⟨ A ( w ) , Φ ⟩ ≔ ( κ + χ ) ( ∇ u , ∇ ξ ) + μ ( ∇ ω , ∇ η ) + α ( ∇ h , ∇ ζ ) , ∀ Φ ≔ ( ξ , η , ζ ) ∈ V ^ , ⟨ L ( w ) , Φ ⟩ ≔ − 2 χ ( ∇ × ω , ξ ) − 2 χ ( ∇ × u , η ) + 4 χ ( ω , η ) , ∀ Φ = ( ξ , η , ζ ) ∈ V ^ .

Remark 2.1

According to the definition of operator A and the classical spectral theory of elliptic operators [33], there exists a sequence { λ n } n = 1 ∞ (formed by the eigenvalues of A ) satisfying

0 < λ 1 ⩽ λ 2 ⩽ … ⩽ λ n ⩽ … , λ n → + ∞ , as n → ∞ ,

and a sequence of elements { e n } n = 1 ∞ ⊆ D ( A ) ≔ ( H 2 ( Ω ) ) 5 ∩ V ^ , which forms a orthonormal basis of H ^ , so that span { e 1 , e 2 , … , e n , … } is dense in V ^ , and A e n = λ n e n for ∀ n ∈ N .

In addition, for any w = ( u , ω , h ) , v = ( ϕ , φ , ψ ) , Φ = ( ξ , η , ζ ) ∈ V ^ , we introduce a bilinear continuous operator B : V ^ × V ^ → V ^ * defined by

⟨ B ( w , v ) , Φ ⟩ = b 1 ( u , ϕ , ξ ) + b 2 ( u , φ , η ) + b 1 ( u , ψ , ζ ) − b 1 ( h , ϕ , ζ ) − b 1 ( h , ψ , ξ ) ,

where

b 1 ( u , ϕ , ξ ) ≔ ∑ i , j = 1 2 ∫ Ω u i ∂ ϕ j ∂ x i ξ j d x , ∀ u , ϕ , ξ ∈ V , b 2 ( u , φ , η ) ≔ ∑ i = 1 2 ∫ Ω u i ∂ φ ∂ x i η d x , ∀ u ∈ V , φ , η ∈ H 0 1 ( Ω ) .

It is not difficult to check that

b 1 ( u , ϕ , ξ ) = − b 1 ( u , ξ , ϕ ) and b 1 ( u , ϕ , ϕ ) = 0 , ∀ u , ϕ , ξ ∈ V , b 2 ( u , φ , η ) = − b 2 ( u , η , φ ) and b 2 = ( u , φ , φ ) = 0 , ∀ u ∈ V , ∀ φ , η ∈ H 0 1 ( Ω ) .

Consequently,

(2.1) ⟨ B ( w , v ) , v ⟩ = 0 , ∀ w , v ∈ V ^ .

Some useful estimations for the operators A , B , and L have been established in [10,34,35].

Lemma 2.1

(1) There exists positive constant c 1 such that

∣ ⟨ B ( w , v ) , Φ ⟩ ∣ ⩽ c 1 ‖ A w ‖ ‖ v ‖ V ^ ‖ Φ ‖ , ∀ w ∈ D ( A ) , v ∈ V ^ , Φ ∈ H ^ , c 1 ‖ w ‖ 1 2 ‖ w ‖ V ^ 1 2 ‖ v ‖ 1 2 ‖ v ‖ V ^ 1 2 ‖ Φ ‖ V ^ , ∀ w , v , Φ ∈ V ^ .

(2) There are some positive constants δ 1 = min { κ , μ , α } , δ 2 ≔ δ 2 ( χ , Ω ) , c 2 , and c 3 such that

(2.2) δ 1 ‖ w ‖ V ^ 2 ⩽ ⟨ A w , w ⟩ + ⟨ L ( w ) , w ⟩ , ∀ w ∈ V ^ ,

(2.3) ‖ L ( w ) ‖ ⩽ δ 2 ‖ w ‖ V ^ , ∀ w ∈ V ^ ,

(2.4) c 2 ⟨ A w , w ⟩ ⩽ ‖ w ‖ V ^ ⩽ c 3 ⟨ A w , w ⟩ , ∀ w ∈ V ^ .

Based on the aforementioned operators A , B , and L , systems (1.1)–(1.3) can be formulated into the following abstract form:

(2.5) d d t w ( t ) + A w ( t ) + B ( w ( t ) , w ( t ) ) + L ( w ( t ) ) = F ( t ) , u ( x , t ) = ω ( x , t ) = h ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ τ , + ∞ ) , w ( τ ) = ( u ( τ ) , ω ( τ ) , h ( τ ) ) ≔ w τ , τ ∈ R ,

where w ( x , t ) ≔ ( u ( x , t ) , ω ( x , t ) , h ( x , t ) ) , and F ( t ) ≔ ( f ( t ) , g ( t ) , 0 ) .

Definition 2.1

Given τ ∈ R , for any T > τ , the function w is said to be a weak solution of (2.5) if, w = ( u , ω , h ) ∈ L ∞ ( τ , T ; H ^ ) ∩ L 2 ( τ , T ; V ^ ) such that, for any t ∈ [ τ , T ] , Ψ ∈ V ^ , the following equation

d d t w ( t ) , Ψ + ⟨ A w ( t ) , Ψ ⟩ + ⟨ B ( w ( t ) , w ( t ) ) , Ψ ⟩ + ⟨ L ( w ( t ) ) , Ψ ⟩ = ⟨ F ( t ) , Ψ ⟩

holds in the distribution sense of D ′ ( τ , + ∞ ) . Furthermore, if w is a weak solution of system (2.5) and w ∈ L 2 ( τ , T ; D ( A ) ) ∩ L ∞ ( τ , T ; V ^ ) for all T > τ , then w is called a strong solution of (2.5).

Now, we recall a known result with respect to the existence and uniqueness of solutions for system (2.5) as follows.

Lemma 2.2

(Existence and uniqueness of solution, see [6,7])

(i) Assume that F ( t ) ∈ L 2 ( τ , T ; V ^ * ) and w τ ∈ H ^ , then there exists a unique weak solution w ( t ; τ , w τ ) for system (2.5) satisfying

w ( t ; τ , w τ ) ∈ C ( [ τ , T ] ; H ^ ) ∩ L 2 ( τ , T ; V ^ ) ∩ L ∞ ( τ , T ; H ^ ) , ∀ t > τ .

(ii) Assume that F ( t ) ∈ L 2 ( τ , T ; H ^ ) and w τ ∈ V ^ , then there exists a unique strong solution w ( t ; τ , w τ ) for system (2.5) satisfying

w ( t ; τ , w τ ) ∈ C ( [ τ , T ] ; V ^ ) ∩ L 2 ( τ , T ; D ( A ) ) ∩ L ∞ ( τ , T ; V ^ ) , ∀ t > τ .

Indeed, it is not difficult to check that the solution of system (2.5) depends continuously on its initial value in H ^ . Then, based on Lemma 2.2, the biparametric mapping defined by

(2.6) U ( t , τ ) : w τ ↦ U ( t , τ ) w τ = w ( t ) , ∀ t ⩾ τ , w τ ∈ H ^ ,

generates a continuous process in H ^ , which satisfies the following properties:

( i ) U ( τ , τ ) w τ = w τ , ( i i ) U ( t , θ ) U ( θ , τ ) w τ = U ( t , τ ) w τ = w ( t ) .

2.2 Infinite dimensional dynamical systems

In order to facilitate the discussion, we denote by X the space H ^ or V ^ , and by P ( X ) the family of all nonempty subsets of X . Let D be a nonempty class of families parameterized in time D ^ = { D ( t ) : t ∈ R } ⊆ P ( X ) , which will be called a universe in P ( X ) . Based on these notations, we introduce the following definitions concerning the pullback attractors.

Definition 2.2

(1) A family of sets D ^ 0 = { D 0 ( t ) ∣ t ∈ R } ⊆ P ( X ) is called pullback D -absorbing for the process { U ( t , τ ) } t ⩾ τ in X if for any t ∈ R and any D ^ = { D ( t ) ∣ t ∈ R } ∈ D , there exists a τ 0 ( t , D ^ ) ⩽ t such that U ( t , τ ) D ( τ ) ⊆ D 0 ( t ) for all τ ⩽ τ 0 ( t , D ^ ) .

(2) The process { U ( t , τ ) } t ⩾ τ is said to be pullback D ^ 0 -asymptotically compact in X if for any t ∈ R , any sequences { τ n } ⊆ ( − ∞ , t ] and { x n } ⊆ X satisfying τ n → − ∞ as n → ∞ and x n ∈ D 0 ( τ n ) for all n , the sequence { U ( t , τ n ; x n ) } is relatively compact in X . { U ( t , τ ) } t ⩾ τ is called pullback D -asymptotically compact in X if it is pullback D ^ -asymptotically compact for any D ^ ∈ D .

(3) A family of sets A ^ D = { A D ( t ) ∣ t ∈ R } ⊆ P ( X ) is called a pullback D -attractor of the process { U ( t , τ ) } t ⩾ τ on X if it has the following properties:

Compactness: for any t ∈ R , A D ( t ) is a nonempty compact subset of X ;

Invariance: U ( t , τ ) A D ( τ ) = A D ( t ) , ∀ t ⩾ τ ;

Pullback attracting: A ^ D is pullback D -attracting in the following sense:

lim τ → − ∞ dist X ( U ( t , τ ) D ( τ ) , A D ( t ) ) = 0 , ∀ D ^ = { D ( s ) ∣ s ∈ R } ∈ D , t ∈ R ;

Minimality: the family of sets A ^ D is minimal in the sense that if O ^ = { O ( t ) ∣ t ∈ R } ⊆ P ( X ) is another family of closed sets satisfying

lim τ → − ∞ dist X ( U ( t , τ ) D ( τ ) , O ( t ) ) = 0 , ∀ D ^ = { D ( t ) ∣ t ∈ R } ∈ D ,

then A D ( t ) ⊆ O ( t ) for t ∈ R .

Furthermore, we recall a basic result (see Theorem 3.11 in [19]).

Proposition 2.1

Assume { U ( t , τ ) } t ⩾ τ is a closed process, D is a universe in P ( X ) , D ^ 0 = { D 0 ( t ) ∣ t ∈ R } ⊆ P ( X ) is pullback D -absorbing for the process, and { U ( t , τ ) } t ⩾ τ is pullback D ^ 0 -asymptotically compact. Then, the family A ^ D = { A D ( t ) ∣ t ∈ R } defined by

A D ( t ) = ⋃ D ^ ∈ D Λ ( D ^ , t ) ¯ X with Λ ( D ^ , t ) ≔ ⋂ s ⩽ t ⋃ τ ⩽ s U ( t , τ ) D ( τ ) ¯ X , ∀ t ∈ R ,

satisfies the following properties:

Compactness: for any t ∈ R , the set A D ( t ) is a nonempty compact subset of X, and A D ( t ) ⊆ Λ ( D ^ 0 , t ) ;
Invariance: A ^ D is invariant, i.e., U ( t , τ ) A D ( τ ) = A D ( t ) , for all τ ⩽ t ;
Pullback attracting: A ^ D is pullback D - attracting, i.e.,
lim τ → − ∞ dist X ( U ( t , τ ) D ( τ ) , A D ( t ) ) = 0 , f o r a l l D ^ ∈ D , t ∈ R ;
Minimality: the family A ^ D is minimal;
If D ^ 0 ∈ D , then A D ( t ) = Λ ( D ^ 0 , t ) ⊆ D 0 ( t ) ¯ X , for all t ∈ R .

Next, we introduce a notion called “flattening property” (see [29,30]), which is also known as “Condition (C)” in [28].

Definition 2.3

Assume that X is a Banach space with norm ‖ ⋅ ‖ X , and D ^ 0 = { D 0 ( t ) ∣ t ∈ R } is a given family. We say that the process { U ( t , τ ) } t ⩾ τ on X satisfies the pullback D ^ 0 -flattening property if, for any t ∈ R and ε > 0 , there exist τ ( ε , t , D ^ 0 ) < t , a finite-dimensional subspace X ( ε , t , D ^ 0 ) of X , and a map P ( ε , t , D ^ 0 ) : X ↦ X ( ε , t , D ^ 0 ) , such that

{ P U ( t , τ ) w τ ∣ τ ⩽ τ ( ε , t , D ^ 0 ) , w τ ∈ D 0 ( τ ) } is bounded in X ,

and

‖ ( I − P ) U ( t , τ ) w τ ‖ X < ε , for any τ ⩽ τ ( ε , t , D ^ 0 ) , w τ ∈ D 0 ( τ ) .

Remark 2.2

García-Luengo et al. [30, Proposition 9] pointed out that to ensure a process { U ( t , τ ) } t ⩾ τ is pullback D ^ 0 -asymptotically compact, it is sufficient to show that the process satisfies the pullback D ^ 0 -flattening property.

From now on, we denote by D H ^ the class of all families of nonempty subset D ^ = { D ( t ) ∣ t ∈ R } ⊆ P ( H ^ ) satisfying

lim τ → − ∞ ( e δ 1 τ sup w ∈ D ( τ ) ‖ w ‖ 2 ) = 0 .

And, we use D F H ^ to denote the class of families D ^ = { D ( t ) = D ∣ t ∈ R } with D a fixed nonempty bounded subset of H ^ . Evidently, it holds that D F H ^ ⊆ D H ^ .

3 Existence of pullback attractor

The purpose of this section is to prove the existence of pullback attractor in H ^ . To this end, we first show the existence of pullback absorbing family by establishing a priori estimates of the solutions of system (2.5). Then, we prove that the process defined by (2.6) satisfies the pullback flattening property.

To begin with, let us give an assumption as follows:

(A1) F ( t , x ) ∈ L loc 2 ( R ; V ^ * ) and ∫ − ∞ 0 e λ 1 δ 1 θ ‖ F ( θ ) ‖ V ^ * 2 d θ < + ∞ .

The following estimates have been established in [10, Lemma 3.2], so we omit the detailed proof here.

Lemma 3.1

Suppose that assumption (A1) holds and w ≔ ( u , ω , h ) is a weak solution of system (2.5), then for any t ∈ R and D ^ = { D ( t ) ∣ t ∈ R } ∈ D H ^ , there exists τ 0 ( D ^ , t ) < t − 2 such that, for any τ ⩽ τ 0 ( D ^ , t ) and w τ ∈ D ( τ ) ,

(3.1) ‖ w ( r ; τ , w τ ) ‖ 2 ⩽ ρ 1 ( t ) , ∀ r ∈ [ t − 2 , t ] ,

(3.2) ∫ r − 1 r ‖ w ( s ; τ , w τ ) ‖ V ^ 2 d s ⩽ ρ 2 ( t ) , ∀ r ∈ [ t − 1 , t ] ,

(3.3) ∫ r − 1 r ‖ w ′ ( s ; τ , w τ ) ‖ V ^ * 2 d s ⩽ ρ 3 ( t ) , ∀ r ∈ [ t − 1 , t ] ,

where

ρ 1 ( t ) = 1 + e − λ 1 δ 1 ( t − 2 ) δ 1 ∫ − ∞ t e λ 1 δ 1 s ‖ F ( s ) ‖ V ^ * 2 d s , ρ 2 ( t ) = 1 δ 1 max θ ∈ [ t − 2 , t ] ρ 1 ( θ ) + 1 δ 1 ∫ t − 2 t ‖ F ( s ) ‖ V ^ * 2 d s , ρ 3 ( t ) = ( 3 k 1 2 + 3 k 2 2 max θ ∈ [ t − 2 , t ] ρ 1 ( θ ) ) ρ 2 ( t ) + 3 ∫ t − 2 t ‖ F ( s ) ‖ V ^ * 2 d s ,

where k 1 and k 2 are the positive constants.

As a consequence of Lemma 3.1, we have

Lemma 3.2

Under the conditions of Lemma 3.1, the family D ^ 0 = { D 0 ( t ) ∣ t ∈ R } , with D 0 ( t ) = ℬ ¯ ( 0 , ℛ H ^ ( t ) ) being the pullback D H ^ -absorbing for the process { U ( t , τ ) } t ⩾ τ in H ^ , where

ℬ ¯ H ^ ( 0 , ℛ H ^ ( t ) ) = { w ∈ H ^ ∣ ‖ w ‖ 2 ⩽ ℛ H ^ ( t ) } , ℛ H ^ ( t ) ≔ 1 + e − δ 1 λ 1 ( t − 2 ) δ 1 ∫ − ∞ t e δ 1 λ 1 θ ‖ F ( θ ) ‖ V ^ * 2 d θ ,

being a closed ball in H ^ .

In order to prove the pullback D H ^ -flattening property of the process in H ^ , the following auxiliary conclusions are required.

Lemma 3.3

Assume that (A1) holds, then for any ε > 0 , t ∈ R , and D ^ ∈ D H ^ , there exists δ ≔ δ ( ε , t , D ^ ) ∈ ( 0 , 1 ) such that

(3.4) ∣ ‖ w ( t ; τ , w τ ) ‖ 2 − ‖ w ( t − s ; τ , w τ ) ‖ 2 ∣ < ε , ∀ s ∈ [ 0 , δ ] , τ ⩽ τ 0 ( D ^ , t ) , w τ ∈ D ( τ ) .

Moreover,

(3.5) ∫ t − δ t ‖ w ( θ ; τ , w τ ) ‖ 2 d θ < ε , ∀ τ ⩽ τ 0 ( D ^ , t ) , w τ ∈ D ( τ ) ,

where τ 0 ( D ^ , t ) comes from Lemma 3.1.

Proof

We will prove (3.4) by contradiction. To this end, suppose (3.4) is not true, then for any δ ∈ ( 0 , 1 ) , there exist an ε 0 > 0 , t ∈ R , D ^ ∈ D H ^ and three sequences { τ n } ⊆ ( − ∞ , t − 2 ] with τ n → − ∞ as n → ∞ , { w τ n } with w τ n ∈ D ( τ n ) , and { s n } with 0 ⩽ s n ⩽ 1 n such that

(3.6) ∣ ‖ w ( t ; τ n , w τ n ) ‖ 2 − ‖ w ( t − s n ; τ n , w τ n ) ‖ 2 ∣ ⩾ ε 0 , for all n ⩾ 1 .

Let w ( n ) ≔ w ( n ) ( ⋅ ) ≔ w ( ⋅ ; τ n , w τ n ) . Then, it follows from Lemma 3.1 that w ( n ) ( t , x ) and ( w ( n ) ) ′ ( t , x ) are uniformly bounded in L ∞ ( t − 2 , t ; H ^ ) ∩ L 2 ( t − 2 , t ; V ^ ) and L 2 ( t − 2 , t ; V ^ * ) , respectively, as τ n ⩽ τ 0 ( D ^ , t ) (where τ 0 ( D ^ , t ) comes from Lemma 3.1). Furthermore, from the diagonal procedure, we can conclude that there exist a subsequence (still denoted by) w ( n ) ( ⋅ ) and an element w ( ⋅ ) such that

(3.7) w ( n ) ⇀ * w , weakly star in L ∞ ( t − 2 , t ; H ^ ) , w ( n ) ⇀ w , weakly in L 2 ( t − 2 , t ; V ^ ) , ( w ( n ) ) ′ ⇀ w ′ , weakly in L 2 ( t − 2 , t ; V ^ * ) .

Based on the Aubin-Lions theorem (see, e.g., [36,37]) and the embedding V ^ ↪ ↪ H ^ ↪ V ^ * , we have

w ( n ) → w , strongly in L 2 ( t − 2 , t ; H ^ ) .

Hence,

w ( n ) → w , strongly in H ^ , a.e. on [ t − 2 , t ] .

In addition, (3.7) also implies

(3.8) w ( n ) ( ⋅ ) ∈ C ( [ t − 2 , t ] ; H ^ ) and w ( ⋅ ) ∈ C ( [ t − 2 , t ] ; H ^ ) .

Since { ( w ( n ) ) ′ } is bounded in L 2 ( t − 2 , t ; V ^ * ) and

w ( n ) ( s 2 ) − w ( n ) ( s 1 ) = ∫ s 1 s 2 ( w ( n ) ) ′ ( θ ) d θ in V ^ * , ∀ s 1 , s 2 ∈ [ t − 2 , t ] ,

we can conclude by the Ascoli-Arzelá theorem that

w ( n ) → w , strongly in C ( [ t − 2 , t ] ; V ^ * ) ,

which together with (3.8) gives

w ( n ) ( s n ) ⇀ w ( s * ) , weakly in H ^ , ∀ s n ⊆ [ t − 2 , t ] , with s n → s * as n → ∞ .

Furthermore, using the same arguments as the proof of (3.32) in [10], we can obtain

w ( n ) → w , strongly in C ( [ t − 2 , t ] ; H ^ ) .

Consequently,

‖ w ( t ; τ n , w τ n ) ‖ → ‖ w ( t ) ‖ and ‖ w ( t − s n ; τ n , w τ n ) ‖ → ‖ w ( t ) ‖ , as n → ∞ ,

which contradicts (3.6). Thus, (3.4) holds.

Next, testing (2.5) by w and using (2.1), we obtain

1 2 d d t ‖ w ( t ) ‖ 2 + ⟨ A w ( t ) , w ( t ) ⟩ + ⟨ L ( w ( t ) ) , w ( t ) ⟩ = ⟨ F ( t ) , w ( t ) ⟩ ,

which together with (2.2), Schwartz inequality, and Young’s inequality implies that

d d t ‖ w ( t ) ‖ 2 + 2 δ 1 ‖ w ( t ) ‖ 2 ⩽ d d t ‖ w ( t ) ‖ 2 + 2 δ 1 ‖ w ( t ) ‖ V ^ 2 ⩽ 2 ⟨ F ( t ) , w ( t ) ⟩ ⩽ δ 1 ‖ w ( t ) ‖ V ^ 2 + 1 δ 1 ‖ F ( t ) ‖ V ^ * 2 .

Integrating the aforementioned inequality with respect to time variable over [ t − δ , t ] , we obtain

δ 1 ∫ t − δ t ‖ w ( θ ) ‖ 2 d θ ⩽ ‖ w ( t − δ ) ‖ 2 − ‖ w ( t ) ‖ 2 + 1 δ 1 ∫ t − δ t ‖ F ( θ ) ‖ V ^ * 2 d θ .

Since F ( t , x ) ∈ L loc 2 ( R ; V ^ * ) , (3.5) follows from the aforementioned inequality and (3.4). The proof is complete.□

Now, we can concentrate on verifying the pullback D H ^ -flattening property of process. The proof is similar to that of Lemma 4.4 in [32]; for the reader’s convenience, we provide the full proof in the following.

Lemma 3.4

Assume that (A1) holds, then for any D ^ ∈ D H ^ , the process { U ( t , τ ) } t ⩾ τ satisfies the pullback D H ^ -flattening property in H ^ .

Proof

According to Definition 2.3, we need to show that for any ε > 0 , t ∈ R and D ^ ∈ D H ^ , there exists m = m ( ε , t , D ^ ) ∈ N such that the projection P m : H ^ ↦ H ^ m , with H ^ m = span { e 1 , e 2 , … , e m } ( { e n } n ⩾ 1 being the orthonormal basis of H ^ given in Remark 2.1), satisfies the following two properties:

{ P m U ( t , τ ) D ( τ ) : τ ⩽ τ 0 ( D ^ , t ) } is bounded in H ^ ,
‖ ( I − P m ) U ( t , τ ) w ( τ ) ‖ < ε , for any τ ⩽ τ 0 ( D ^ , t ) , w τ ∈ D ( τ ) ,

where τ 0 ( D ^ , t ) comes from Lemma 3.1.

Property (i) follows directly from the fact ‖ P m w ( t ) ‖ ⩽ ‖ w ( t ) ‖ and (3.1).

Next, we prove property (ii). First, according to Lemma 12 in [38], under the assumption (A1), it holds that

(3.9) lim c → ∞ e − c t ∫ − ∞ t e c θ ‖ F ( θ ) ‖ V ^ * 2 d θ = 0 , ∀ t ∈ R .

Then, for any τ ⩽ τ 0 ( D ^ , t ) and w τ ∈ D ( τ ) , let q m ( t ) ≔ w ( t ) − P m w ( t ) . Taking inner product of (2.5) with q m ( t ) , we have

(3.10) 1 2 d d t ‖ q m ( t ) ‖ 2 + ⟨ A q m ( t ) , q m ( t ) ⟩ + ⟨ B ( u , w ) , q m ( t ) ⟩ + ⟨ L ( w ) , q m ( t ) ⟩ = ⟨ F ( t ) , q m ( t ) ⟩ .

In what follows, we estimate the terms of (3.10) one by one. First, it follows from Lemma 2.1 that

∣ ⟨ B ( w ( t ) , w ( t ) ) , q m ( t ) ⟩ ∣ ⩽ c 1 ‖ w ( t ) ‖ ‖ w ( t ) ‖ V ^ ‖ q m ( t ) ‖ V ^ ⩽ c 1 2 δ 1 ‖ w ( t ) ‖ 2 ‖ w ( t ) ‖ V ^ 2 + δ 4 4 ‖ q m ( t ) ‖ V ^ 2

and

∣ ⟨ L ( w ( t ) ) , q m ( t ) ⟩ ∣ ⩽ ‖ L ( w ( t ) ) ‖ ‖ q m ( t ) ‖ ⩽ δ 2 ‖ w ( t ) ‖ V ^ ‖ q m ( t ) ‖ V ^ ⩽ 2 δ 2 2 δ 1 ‖ w ( t ) ‖ V ^ 2 + δ 1 8 ‖ q m ( t ) ‖ V ^ 2 .

In addition,

⟨ F ( t ) , q m ( t ) ⟩ ⩽ 2 δ 1 ‖ F ( t ) ‖ V ^ * 2 + δ 1 8 ‖ q m ‖ V ^ 2 and ‖ q m ( t ) ‖ V ^ ⩽ ‖ w ( t ) ‖ V ^ .

Taking the aforementioned three inequalities and (2.2) into account, we can deduce from (3.10) that

d d t ‖ q m ( t ) ‖ 2 + δ 1 ‖ q m ( t ) ‖ V ^ 2 ⩽ 4 δ 1 ‖ F ( t ) ‖ V ^ * 2 + 2 c 1 2 δ 1 ‖ w ( t ) ‖ 2 ‖ w ( t ) ‖ V ^ 2 + 4 δ 2 2 δ 1 ‖ w ( t ) ‖ V ^ 2 + 2 δ 2 ‖ w ( t ) ‖ V ^ 2 ,

which together with the following Poincaré inequality:

λ m + 1 ‖ q m ( t ) ‖ 2 ⩽ ‖ ∇ q m ( t ) ‖ 2 ⩽ ‖ q m ( t ) ‖ V ^ 2 ,

where λ m + 1 is given by Remark 2.1, yields that

d d t ‖ q m ( t ) ‖ 2 + δ 1 λ m + 1 ‖ q m ( t ) ‖ 2 ⩽ 4 δ 1 ‖ F ( t ) ‖ V ^ * 2 + 2 c 1 2 δ 1 ‖ w ( t ) ‖ 2 ‖ w ( t ) ‖ V ^ 2 + 4 δ 2 2 δ 1 + 2 δ 2 ‖ w ( t ) ‖ V ^ 2 .

Multiplying the aforementioned inequality by e δ 1 λ m + 1 t and changing the variable t to θ , then integrating the resultant inequality with respect to θ over [ t − 1 , t ] , we can obtain

e δ 1 λ m + 1 t ‖ q m ( t ) ‖ 2 − e δ 1 λ m + 1 ( t − 1 ) ‖ q m ( t − 1 ) ‖ 2 ⩽ 4 δ 1 ∫ t − 1 t e δ 1 λ m + 1 θ ‖ F ( θ ) ‖ V ^ * 2 d θ + 2 c 1 2 δ 1 ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ 2 ‖ w ( θ ) ‖ V ^ 2 d θ + 4 δ 2 2 δ 1 + 2 δ 2 ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ ,

which together with (3.1) gives

(3.11) ‖ q m ( t ) ‖ 2 ⩽ e − δ 1 λ m + 1 ‖ q m ( t − 1 ) ‖ 2 + 4 δ 2 2 δ 1 + 2 δ 2 e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ + 4 δ 1 e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ F ( θ ) ‖ V ^ * 2 d θ + 2 c 1 2 δ 1 e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ 2 ‖ w ( θ ) ‖ V ^ 2 d θ ⩽ e − δ 1 λ m + 1 ‖ q m ( t − 1 ) ‖ 2 + c 0 ( 1 + ρ 1 ( t ) ) e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ + 4 δ 1 e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ F ( θ ) ‖ V ^ * 2 d θ ,

where c 0 ≔ max 4 δ 2 2 δ 1 + 2 δ 2 , 2 c 1 2 δ 1 . Observe that

λ m + 1 → ∞ , as m → ∞ , and ‖ q m ( t − 1 ) ‖ ⩽ ‖ w ( t − 1 ) ‖ ⩽ ρ 1 ( t ) .

We can conclude that for any ε > 0 , there exists m 1 ≔ m 1 ( ε , t , D ^ ) such that, for any m ⩾ m 1 ,

(3.12) e − δ 1 λ m + 1 ‖ q m ( t − 1 ) ‖ 2 < ε 2 3 .

Moreover, from (3.9), we deduce that there exists m 2 ≔ m 2 ( ε , t ) such that, for any m ⩾ m 2 ,

(3.13) e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ F ( θ ) ‖ V ^ * 2 d θ < δ 1 ε 2 12 .

In addition, for any δ ∈ ( 0 , 1 ) , it follows from (3.2) that

e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ = e − δ 1 λ m + 1 t ∫ t − 1 t − δ e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ + e − δ 1 λ m + 1 t ∫ t − δ t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ ⩽ e − δ 1 λ m + 1 δ ∫ t − 1 t ‖ w ( θ ) ‖ V ^ 2 d θ + ∫ t − δ t ‖ w ( θ ) ‖ V ^ 2 d θ ⩽ e − δ 1 λ m + 1 δ ρ 2 ( t ) + ∫ t − δ t ‖ w ( θ ) ‖ V ^ 2 d θ ,

which together with (3.5) implies that for the aforementioned ε , there exists δ * ∈ ( 0 , 1 ) and m 3 ≔ m 3 ( ε , t , D ^ , δ * ) such that, for any m ⩾ m 3 ,

(3.14) e − δ 1 λ m + 1 t ∫ t − 1 t e δ 1 λ m + 1 θ ‖ w ( θ ) ‖ V ^ 2 d θ ⩽ e − δ 1 λ m + 1 δ * ρ 2 ( t ) + ∫ t − δ * t ‖ w ( θ ) ‖ V ^ 2 d θ

< ε 2 3 c 0 ( 1 + ρ 1 ( t ) ) , ∀ τ ⩽ τ 0 ( D ^ , t ) , w τ ∈ D ( τ ) .

Finally, substituting (3.12)–(3.14) into (3.11) and taking m ≔ max { m 1 , m 2 , m 3 } , we obtain

‖ q m ( t ) ‖ 2 < ε 2 for any τ ⩽ τ 0 ( D ^ , t ) , w τ ∈ D ( τ ) ,

which is property (ii). The proof is complete.□

At this stage, we can state the main results of this article.

Theorem 3.1

Under the assumption (A1), the process { U ( t , τ ) } t ⩾ τ defined by (2.6) possesses the minimal pullback D F H ^ - and D H ^ -attractors

A ^ D F H ^ = { A D F H ^ ( t ) ∣ t ∈ R } a n d A ^ D H ^ = { A D H ^ ( t ) ∣ t ∈ R } ∈ D H ^ ,

respectively. Furthermore,

A D F H ^ ( t ) ⊆ A D H ^ ( t ) , ∀ t ∈ R .

Proof

The result of Theorem 3.1 is a direct consequence of Proposition 2.1, Remark 2.2, Lemmas 3.2 and 3.4 and the fact D F V ^ ⊆ D H ^ .□

Acknowledgments

The authors thank warmly the anonymous referees for their pertinent comments and suggestions, which greatly improves the earlier manuscript.

Funding information: This work was supported by the Natural Science Foundation of Hubei Province, China (No. 2022CFB661) and the Young and Middle-aged Talent Fund of Hubei Education Department, China (No. Q20201307).
Author contributions: All authors contributed to the manuscript, read and approved the final manuscript.
Conflict of interest: The authors declare that there is no conflict of interests regarding this article.

References

[1] G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the microploar fluid equations, Int. J. Engrg. Sci. 15 (1977), 105–108. 10.1016/0020-7225(77)90025-8Search in Google Scholar

[2] S. C. Cowin, Polar fluids, Phys. Fluids 11 (1968), 1919–1927. 10.1063/1.1692219Search in Google Scholar

[3] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966), 1–8. 10.1512/iumj.1966.16.16001Search in Google Scholar

[4] G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser, Boston, 1999. 10.1007/978-1-4612-0641-5Search in Google Scholar

[5] C. Jia, Z. Tan, and J. Zhou, Global well-posedness of compressible magneto-micropolar fluid equations, J. Geom. Anal. 33 (2023), 358. 10.1007/s12220-023-01418-3Search in Google Scholar

[6] M. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solutions, Math. Nachr. 188 (1997), 301–319. 10.1002/mana.19971880116Search in Google Scholar

[7] M. Rojas-Medar and J. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut. 11 (1998), 443–460. 10.5209/rev_REMA.1998.v11.n2.17276Search in Google Scholar

[8] G. Łukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys. 55 (2004), 247–257. 10.1007/s00033-003-1127-7Search in Google Scholar

[9] K. Matsuura, Exponential attractors for 2D magneto-micropolar fluid flow in a bounded domain, Discrete Contin. Dyn. Syst. 2005 (2005), 634–641. Search in Google Scholar

[10] C. Zhao, Y. Li, and G. Łukaszewicz, Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids, Z. Angew. Math. Phys. 71 (2020), 141. 10.1007/s00033-020-01368-8Search in Google Scholar

[11] Y. Li and X. Li, Equivalence between invariant measures and statistical solutions for the 2D non-autonomous magneto-micropolar fluid equations, Math. Methods Appl. Sci. 45 (2022), 2638–2657. 10.1002/mma.7944Search in Google Scholar

[12] H. Yang, X. Han, X. Wang, and C. Zhao, Homogenization of trajectory statistical solutions for the 3D incompressible magneto-micropolar fluids, Discrete Contin. Dyn. Syst. Ser. S 16 (2023), 2672–2685. 10.3934/dcdss.2022202Search in Google Scholar

[13] C. J. Niche and C. F. Perusato, Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids, Z. Angew. Math. Phys. 73 (2022), 48. 10.1007/s00033-022-01683-2Search in Google Scholar

[14] Z. Tan, W. Wu, and J. Zhou, Global existence and decay estimate of solutions to magneto-micropolar fluid equations, J. Differential Equations 266 (2019), 4137–4169. 10.1016/j.jde.2018.09.027Search in Google Scholar

[15] Z. Jia and Z. Yang, Large time behavior to a chemotaxis-consumption model with singular sensitivity and logistic source, Math. Methods Appl. Sci. 44 (2021), 3630–3645. 10.1002/mma.6971Search in Google Scholar

[16] C. X. Lei, H. W. Li, and Y. J. Zhao, Dynamical behavior of a reaction-diffusion SEIR epidemic model with mass action infection mechanism in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B 29 (2024), 3163–3198. 10.3934/dcdsb.2023216Search in Google Scholar

[17] J. Wu, X. He, and X. Li, Finite-time stabilization of time-varying nonlinear systems based on a novel differential inequality approach, Appl. Math. Comput. 420 (2022), 126895. 10.1016/j.amc.2021.126895Search in Google Scholar

[18] D. F. Yan, G. C. Pang, J. L. Qiu, X. Y. Chen, A. C. Zhang, and Y. Liu, Finite-time stability analysis of switched systems with actuator saturation based on event-triggered mechanism, Discrete Contin. Dyn. Syst. Ser. S 16 (2023), 1929–1943. 10.3934/dcdss.2023058Search in Google Scholar

[19] J. García-Luengo, P. Marín-Rubio, and J. Real, Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations 252 (2012), 4333–4356. 10.1016/j.jde.2012.01.010Search in Google Scholar

[20] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998), 71–85. 10.1016/S0362-546X(97)00453-7Search in Google Scholar

[21] W. Sun, J. Cheng, and X. Han, Random attractors for 2D stochastic micropolar fluid flows on unbounded domains, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), 693–716. 10.3934/dcdsb.2020189Search in Google Scholar

[22] C. Zhao, W. Sun, and C. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ. 12 (2015), 265–288. 10.4310/DPDE.2015.v12.n3.a4Search in Google Scholar

[23] C. Zhao and W. Sun, Global well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays, Commun. Math. Sci. 15 (2017), 97–121. 10.4310/CMS.2017.v15.n1.a5Search in Google Scholar

[24] S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations 74 (1988), 285–317. 10.1016/0022-0396(88)90007-1Search in Google Scholar

[25] C. Foias, G. R. Sell, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), 309–353. 10.1016/0022-0396(88)90110-6Search in Google Scholar

[26] C. Foias, O. Manley, and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, ESAIM Math. Model. Numer. Anal. 22 (1988), 93–118. 10.1051/m2an/1988220100931Search in Google Scholar

[27] D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993), 875–887. 10.1512/iumj.1993.42.42039Search in Google Scholar

[28] Q. Ma, S. Wang, and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J. 51 (2002), 1541–1559. 10.1512/iumj.2002.51.2255Search in Google Scholar

[29] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. A: Math. Phys. Eng. Sci. 463 (2007), 163–181. 10.1098/rspa.2006.1753Search in Google Scholar

[30] J. García-Luengo, P. Marín-Rubio, J. Real, and J. C. Robinson, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing, Discrete Contin. Dyn. Syst. 34 (2014), 203–227. 10.3934/dcds.2014.34.203Search in Google Scholar

[31] G. Liu, Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid, J. Math. Fluid Mech. 19 (2017), 623–643. 10.1007/s00021-016-0299-9Search in Google Scholar

[32] W. Sun and Y. Li, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows with minimally regular force and moment, Commun. Math. Sci. 16 (2018), 1043–1065. 10.4310/CMS.2018.v16.n4.a6Search in Google Scholar

[33] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer, New York, 1975. Search in Google Scholar

[34] C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. 10.1017/CBO9780511546754Search in Google Scholar

[35] W. Sun, The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay, Electron. Res. Arch. 28 (2020), no. 3, 1343–1356. 10.3934/era.2020071Search in Google Scholar

[36] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, 2002. 10.1090/coll/049Search in Google Scholar

[37] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 199710.1007/978-1-4612-0645-3Search in Google Scholar

[38] P. E. Kloeden, J. A. Langa, and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal. 6 (2007), 937–955. 10.3934/cpaa.2007.6.937Search in Google Scholar

Received: 2024-10-20

Revised: 2025-05-07

Accepted: 2025-06-10

Published Online: 2025-07-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2025-0174

Keywords for this article

magneto-micropolar fluid equations; pullback attractor; flattening property

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