THE BOUNDEDNESS AND UPPER SEMICONTINUITY OF THE PULLBACK ATTRACTORS FOR A 2D MICROPOLAR FLUID FLOWS WITH DELAY

In this paper, two properties of the pullback attractor for a 2D non-autonomous micropolar fluid flows with delay on unbounded domains are investigated. First, we establish the H1-boundedness of the pullback attractor. Further, with an additional regularity limit on the force and moment with respect to time t, we remark the H2-boundedness of the pullback attractor. Then, we verify the upper semicontinuity of the pullback attractor with respect to the domains.


1.
Introduction. The micropolar fluid model is a qualitative generalization of the well-known Navier-Stokes model in the sense that it takes into account the microstructure of fluid [7]. The model was first derived in 1966 by Eringen [4] to describe the motion of a class of non-Newtonian fluid with micro-rotational effects and inertia involved. It can be expressed by the following equations: ∂u ∂t − (ν + ν r )∆u − 2ν r rotω + (u · ∇)u + ∇p = f, ∂ω ∂t − (c a + c d )∆ω + 4ν r ω + (u · ∇)ω −(c 0 + c d − c a )∇divω − 2ν r rotu =f , where u = (u 1 , u 2 , u 3 ) is the velocity, ω = (ω 1 , ω 2 , ω 3 ) is the angular velocity field of rotation of particles, p represents the pressure, f = (f 1 , f 2 , f 3 ) andf = (f 1 ,f 2 ,f 3 ) stand for the external force and moment, respectively. The positive parameters ν, ν r , c 0 , c a and c d are viscous coefficients. Actually, ν represents the usual Newtonian viscosity and ν r is called microrotation viscosity. Micropolar fluid models play an important role in the fields of applied and computational mathematics. There is a rich literature on the mathematical theory of micropolar fluid model. Particularly, the existence, uniqueness and regularity of solutions for the micropolar fluid flows have been investigated in [6]. Extensive studies on long time behavior of solutions for the micropolar fluid flows have also been done. For example, in the case of 2D bounded domains: Lukaszewicz [7] established the existence of L 2 -global attractors and its Hausdorff dimension and fractal dimension estimation. Chen, Chen and Dong proved the existence of H 2 -global attractor and uniform attractor in [1] and [2], respectively. Lukaszewicz and Tarasińska [9] investigated the existence of H 1 -pullback attractor. Zhao, Sun and Hsu [18] established the existence of L 2 -pullback attractor and H 1 -pullback attractor of solutions for a universe given by a tempered condition, respectively. For the case of 2D unbounded domains: Dong and Chen [3] investigated the existence and regularity of global attractors. Zhao, Zhou and Lian [19] established the existence of H 1 -uniform attractor and further gave the inclusion relation between L 2 -uniform attractor and the H 1 -uniform attractor. Sun and Li [15] verified the existence of pullback attractor and further investigated the tempered behavior and upper semicontinuity of the pullback attractor. More recently, Sun, Cheng and Han [14] investigated the existence of random attractors for 2D stochastic micropolar fluid flows.
As we know, in the real world, delay terms appear naturally, for instance as effects in wind tunnel experiments (see [10]). Also the delay situations may occur when we want to control the system via applying a force which considers not only the present state but also the history state of the system. The delay of partial differential equations (PDE) includes finite delays (constant, variable, distributed, etc) and infinite delays. Different types of delays need to be treated by different approaches.
In this paper, we consider the situation that the velocity component u 3 in the x 3 -direction is zero and the axes of rotation of particles are parallel to the x 3 axis, be an open set with boundary Γ that is not necessarily bounded but satisfies the following Poincaré inequality: There exists λ 1 > 0 such that λ 1 ϕ 2 L 2 (Ω) ≤ ∇ϕ 2 L 2 (Ω) , ∀ϕ ∈ H 1 0 (Ω). (2) Then we discuss the following 2D non-autonomous incompressible micropolar fluid flows with finite delay: whereᾱ := c 0 + 2c d > 0, x := (x 1 , x 2 ) ∈ Ω ⊆ R 2 , u := (u 1 , u 2 ), g andg stand for the external force containing some hereditary characteristics u t and ω t , which are defined on (−h, 0) as follows where h is a positive fixed number, and To complete the formulation of the initial boundary value problem to system (3), we give the following initial boundary conditions: For problem (3)-(5), Sun and Liu established the existence of pullback attractor in [16], recently.
The first purpose of this work is to investigate the boundedness of the pullback attractor obtained in [16]. We remark that García-Luengo, Marín-Rubio and Real [5] proved the H 2 -boundedness of the pullback attractors of the 2D Navier-Stokes equations in bounded domains. Motivated by [5] and following its main idea, we generalize their results to the 2D micropolar fluid flows with finite delay in unbounded domains. Compared with the Navier-Stokes equations (ω = 0, ν r = 0), the micropolar fluid flow consists of the angular velocity field ω, which leads to a different nonlinear term B(u, w) and an additional term N (u) in the abstract equations (13). In addition, the time-delay term considered in this work also increases the difficulty.
The second purpose of this work is to investigate the upper semicontinuity of the pullback attractor with respect to the domain Ω. To this end, using the arguments in [15,17], we first let {Ω m } ∞ m=1 be an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that ∞ m=1 Ω m = Ω. Then we consider the Cauchy problem (3)- (5) in Ω m . We will conclude that there exists a pullback attractor A H(Ωm) for the problem (3)-(5) in each Ω m . Finally, we establish the upper semicontinuity by showing lim Throughout this paper, we denote the usual Lebesgue space and Sobolev space by L p (Ω) and W m,p (Ω) endowed with norms · p and · m,p , respectively. Especially, we denote H m (Ω) := W m,2 (Ω).
(·, ·)− the inner product in L 2 (Ω), H or H, ·, · − the dual pairing between V and V * or between V and V * . Throughout this article, we simplify the notations · 2 , · H and · H by the same notation · if there is no confusion. Furthermore, Following the above notations, we additionally denote The rest of this paper is organized as follows. In section 2, we make some preliminaries. In section 3, we investigate the boundedness of the pullback attractor. In section 4, we prove the upper semicontinuity of the pullback attractor with respect to the domains.

2.
Preliminaries. In this section, for the sake of discussion, we first introduce some useful operators and put problem (3)-(5) into an abstract form. Then we recall some important known results about the non-autonomous micropolar fluid flows.
To begin with, we define the operators A, B(·, ·) and N (·) by What follows are some useful estimates and properties for the operators A, B(·, ·) and N (·), which have been established in works [11,13]. (2) The operator B(·, ·) is continuous from V × V to V * . Moreover, for any u ∈ V and w ∈ V , there holds Lemma 2.2.
(1) There are two positive constants c 1 and c 2 such that (2) There exists a positive constant α 0 which depends only on Ω, such that for (3) There exists a positive constant c(ν r ) such that In addition, where δ 1 := min{ν,ᾱ}.
According to the definitions of operators A, B(·, ·) and N (·), equations (3)-(5) can be formulated into the following abstract form: where Before recalling the known results for problem (13), we need to make the following assumptions with respect to F and G.
There exists a constant L G > 0 such that for any t ∈ R and any ξ, η ∈ L 2 (−h, 0; H), (iv) There exists C G ∈ (0, δ 1 ) such that, for any t ≥ τ and any w, v ∈ L 2 (τ − h, t; H), In order to facilitate the discussion, we denote 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 can construct the universe D γ in the following.
We denote by D γ the class of all families H centered at zero with radius ρ D (t).
Based on the above assumptions, we can recall the global well-posedness of solutions and the existence of pullback attractor of problem (13).
Remark 2.1. According to Proposition 2.1, the biparametric mapping defined by U (t, τ ) : w in , φ in (s) → w(t; τ, w in , φ in (s)), w t (s; τ, w in , φ in (s)) , ∀ t ≥ τ, (15) generates a continuous process in E 2 H and E 2 V , respectively, which satisfies the following properties: Proposition 2.2. (Existence of pullback attractor, see [16]) Under the Assumption 2.1 and Assumption 2.2, there exists a pullback attractor A H = A H (t) t ∈ R for the process {U (t, τ )} t≥τ that satisfies the following properties: • Compactness: for any t ∈ R, A H (t) is a nonempty compact subset of E 2 H ; Finally, we introduce a useful lemma, which plays an important role in the proof of higher regularity of the pullback attractor.
Lemma 2.4. (see [12]) Let X, Y be Banach spaces such that X is reflexive, and the inclusion X ⊂ Y is continuous. Assume that {w n } n≥1 is a bounded sequence in L ∞ (τ, t; X) such that w n w weakly in L q (τ, t; X) for some q ∈ [1, +∞) and w ∈ C([τ, t]; Y ). Then w(t) ∈ X and 3. Boundedness of the pullback attractor for the universe D γ . This section is devoted to investigating the boundedness of the pullback attractor for the universe D γ given by a tempered condition in space E 2 H . To this end, we consider the Galerkin approximation of the solution w(t) of system (13), which is denoted by w n (t) = w n (t; τ, w in , φ in (s)) = n j=1 ξ nj (t)e j , w nt (·) = w n (t + ·), where the sequence {e j } ∞ j=1 is an orthonormal basis of H and formed by eigenvectors of the operator A, that is, for all j ≥ 1, where the eigenvalues {λ j } j≥1 of A are real number that we can order in such a way 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ j ≤ · · · , λ j → +∞ as j → ∞.
Next we verify the following estimates of the Galerkin approximate solutions defined by (16).
With the above lemma, we are ready to conclude this section with the following H 1 -boundedness of the pullback attractor A H for the universe D γ .

4.
Upper semicontinuity of the pullback attractor. In this section, we concentrate on verifying the upper semicontinuity of the pullback attractor A H obtained in Propositon 2.2 with respect to the spatial domain. To this end, first we let {Ω m } ∞ m=1 be an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that ∞ m=1 Ω m = Ω. Then we consider the system (3) in each Ω m and define the operators A, B(·, ·) and N (·) as previous (in (6)) with the spatial domain Ω replaced by Ω m . Further we can formulate the weak version of problem (3)-(5) as follows: On each bounded domain Ω m , the well-posedness of solution can be established by Galerkin method and energy method, one can refer to [7]. Moreover, the solution w m (·) depends continuously on the initial value w in m with respect to H(Ω m ) norm.
In the following, we investigate the relationship between the solutions of system (30) and (13). Indeed, we devoted to proving the solutions w m of system (30) converges to the solution of system (13) as m → ∞. To this end, for w m ∈ H(Ω m ), we extend its domain from Ω m to Ω by setting Next, using the same proof as that of Lemma 8.