Asymptotic behaviour of a system of micropolar equations

Electronic Journal of Qualitative Theory of Differential Equations 2016, No. 15, 1–18; doi: 10.14232/ejqtde.2016.1.15 http://www.math.u-szeged.hu/ejqtde/ Asymptotic behaviour of a system of micropolar equations Pedro Marín-Rubio1,Mariano Poblete-Cantellano2, Marko Rojas-Medar3and Francisco Torres-CerdaB2 1Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Apdo. Correos 1160, 41080, Sevilla, Spain 2Dpto. de Matemática, Facultad de Ingeniería, Universidad de Atacama Copiapó, Chile 3Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile Received 10 September 2015, appeared 22 March 2016 Communicated by Gabriele Bonanno Abstract. This work is concerned with three-dimensional micropolar fluids flows in a bounded domain with boundary of class C∞. Based on the theory of dissipative systems, we prove the existence of restricted global attractors for local semiflows on suitable fractional phase spaces Zα p, namely for p∈(3, +∞)and α∈[1/2, 1). Moreover, we prove that all these attractors are in fact the same set. Previously, it is shown that the Lamé operator is a sectorial operator in each Lp(Ω)with 1 <p<+∞,p6=3/2 and therefore, it generates an analytic semigroup in these spaces. Keywords: micropolar fluids, local semiflows and restricted global attractors. 2010 Mathematics Subject Classification: 35Q35, 76D03. 1 Introduction and notation Let Ω⊂R3be an open, bounded set with smooth boundary ∂Ω, namely of class C∞; we consider the system of equations for the motion of micropolar fluid            ∂u ∂t+ (u·∇)u−(ν+χ)∆u+∇p=χrot w+f, in Ω×(0, T), div u=0, in Ω×(0, T), ∂w ∂t+ (u·∇)w−µ∆w−(µ+σ)∇div w+2χw=χrot u+g, in Ω×(0, T), (1.1) together with the following boundary and initial conditions            u=0 on ∂Ω×(0, T), u(x, 0) = u0(x)in Ω, w=0 on ∂Ω×(0, T), w(x, 0) = w0(x)in Ω, (1.2) BCorresponding author. Email: francisco.torr[email protected] 2P. Marín-Rubio, M. Poblete-Cantellano, M. Rojas-Medar and F. Torres-Cerda where u= (u1,u2,u3)is the velocity field, pis the pressure, and w= (w1,w2,w3)is the micro-rotational interpreted as the angular velocity field of rotational of particles. The fields f= (f1,f2,f3)and g= (g1,g2,g3)are external forces and moments respectively. The positive constants ν,χ,µ,σrepresent viscosity coefficients, νis the usual Newtonian viscosity and χis called the micro-rotational viscosity. We will assume that these constants satisfy µ>0 and 3σ+2µ>0. In the last ten years, much effort has been devoted to study the long time behaviour for the micropolar equations (see for instance [15,16,20]). Most of these papers deal with problems assuming that Ωis a bounded domain in R2and they conclude the existence of global attractor in an L2(Ω)-framework by standard techniques; whereas the case of Ωa subset in R3and Lp(Ω)-theory with 1 <p<+∞has not received so much attention. In the 3D problem, two issues appear to be possible obstructions to build a global attractor. For general data (u0,w0,f,g), there always exists a weak solution U(t) = (u,w), that is defined for all time t≥0. However, it is not known whether this solution is uniquely determined by the data. As a result, one cannot conclude that the mapping S(t):(u0,w0)→S(t)(u0,w0) = U(t) satisfies the semigroup property required for a semiflow. On the other hand, for “good” data (u0,w0,f,g), the initial value problem does have a unique strong solution on some interval [0, T). However, it is not known whether this strong solution continues to exist for all time. Nevertheless, with an alternative point of view, Carvalho, Cholewa, and Dlokto in a series of works (see [5–7]), propose bypasses to these issues for the Navier–Stokes equations and other initial boundary value problems for semilinear parabolic equations. In fact, the authors look at the problems as a sectorial equation in relevant Banach spaces (Lp(Ω), 1 <p<+∞), and then discuss to generate a local semiflow S(t)on a fractional phase space Zα p, afterwards applying adequate estimates, they choose a suitable metric space V⊂Zα pon which S(t) becomes a dissipative compact semigroup of global solutions. As a consequence, the existence of a global attractor Afor S(t)restricted to Vwill follow from a suitable estimate of the solutions in a Sobolev space. The goal of this paper is to prove, following the ideas contained in the above references, that the system (1.1) has a restricted global attractor for a local semiflow on Zα p(defined below) for α∈[1/2, 1)and p∈(3, +∞). The structure of the paper is as follows. After some notations introduced in this section, we recall some preliminary notions on the abstract formulation of the problem (Section 2), on conditions ensuring the existence of (local) solutions, discussions on how to turn them global, and on the concept and existence of (restricted) global attractor for a suitable local semiflow. The subsequent sections are devoted to follow this scheme. Namely, in Section 3it is shown that the Lamé operator is a sectorial operator in Lp(Ω)with 1 <p<+∞and therefore, it generates analytic semigroups in these spaces. Then, in Section 4the study of local and global solutions is carried out. Finally, our main result on existence of attractors in different spaces and the relation among them is stated in Section 5. In this paper we use the following notations, for 1 ≤p≤+∞,Lp(Ω)denotes the usual Lebesgue space over Ω,Wm,p(Ω)the usual Lp-Sobolev space of order m, and C∞ 0(Ω)is the set of all infinitely differentiable functions in Ωwith compact support in Ω. For function spaces of vector fields, we use the following symbols Lp≡Lp(Ω) = [Lp(Ω)]3, Wm,p≡Wm,p(Ω) = [Wm,p(Ω)]3. We define C∞ 0,σ(Ω):={v∈C∞ 0(Ω): div v=0 in Ω}, Micropolar fluids 3 and denote by Lσ pthe closure of C∞ 0,σ(Ω)in Lp(Ω). For notational simplicity, we denote the norms k·kLp(Ω)and k· kWm,p(Ω)by k · kpand k·km,p, respectively. For the differentiation of the vector field u= (u1,u2,u3)and the scalar field pwe use the following symbols: ∂jp=∂p ∂xj,∂tp=∂p ∂t,∇p= (∂1p,∂2p,∂3p), and div u= 3 ∑ j=1 ∂juj, rot u= (∂2u3−∂3u2,∂3u1−∂1u3,∂1u2−∂2u1). The identity operator will be denoted by I. In order to give an operator interpretation of the problem (1.1)–(1.2), we shall introduce the well known Helmholtz and Weyl decomposition. Let 1 <p<+∞. Then, the Banach space Lp(Ω)admits the Helmholtz and Weyl decomposition (cf. [9]) Lp=Lσ p⊕Gp(Ω), where ⊕denotes direct sum and Gp(Ω) = {∇ψ:ψ∈W1,p(Ω)}. Let P=Ppbe a continuous projection from Lp(Ω)into Lσ palong with Gp(Ω). Then the projection Phas the Lp-boundedness property kPukp≤Cpkukp. Denote U= (u,w)>and let us define the linear operator Ap=−(ν+χ)P∆0 0−µ∆−(µ+σ)∇div(1.3) with domain D(Ap) = (U=u w:u∈W2,p(Ω)∩W1,p 0∩Lσ p, w∈W2,p(Ω)∩W1,p 0(Ω)). Here we have used the fact that Prot w=rot w, since div rotw=0 in Ω. These facts imply that rot w∈Lσ p, because the space Lσ pis characterized (cf. [9]) as Lσ p={u∈Lp(Ω): div u=0 in Ω,n·u=0 on Ω}. We set Zp≡Lσ p×Lp, using the notation above, and state the following Cauchy problem in the Banach space Zp,(Ut+ApU=NU +F=:G(U), U(0) = U0,(1.4) where U0= (u0,w0)>,F= (Pf,g)>, and the nonlinear term is given by NU=−P(u·∇)u+χrot w −(u·∇w) + χrot u−2χw. Finally, suppose that Ais a sectorial operator and Re σ(A)>0 in a Banach space Z, and define for each α≥0, Zα=D(Aα)with the graph norm kzkα=kAαzk,z∈Zα. In the next sections, the role of Aoperator will be played by Ap. 4P. Marín-Rubio, M. Poblete-Cantellano, M. Rojas-Medar and F. Torres-Cerda 2 Preliminaries and background With the above notation, consider the Cauchy problem (1.4), and concerning this, let us enumerate some assumptions. (i) Ap:D(Ap)→Zpis a sectorial and positive operator (Re σ(Ap)>a>0) in Zp. (ii) For certain α∈[0, 1),G:Zα p→Zpis Lipschitz continuous on bounded subsets of Zα p=D(Aα p). (iii) The resolvent of Apis compact. It is known from the results of Henry [13] and Hale [12] that, under the assumptions (i) and (ii), to each U0∈Zα p, a unique local Zα p-solution U=U(t,U0)corresponds to (1.4) defined on a maximal interval of existence [0, τmax(U0)). More precisely, Ubelongs to C([0, τmax(U0)),Zα p)∩ C1((0, τmax(U0)),Zp),U(0) = U0,U(t)belongs to D(Ap)for each t∈(0, τmax(U0)), and the first equation in (1.4) holds in Zpfor all t∈(0, τmax(U0)). The assumptions (i), (ii) and (iii) are satisfied by the Stokes operator −(ν+χ)P∆(cf. Giga– Miyakawa [11]). Therefore, in order to show that Apsatisfies these assumptions, one only needs to prove that Lamé operator Lpw=−µ∆w−(µ+σ)∇div(w)is a sectorial, positive operator with compact resolvent. We will prove these facts in Section 3. After local existence has been proved (in Section 4), one must discuss the global existence of solutions. For instance, a subset Vα p⊂Zα p, for p∈(3, +∞)and α∈[1/2, 1), will be distinguished such that fractional solutions S(t)U0=U(t,U0)of (1.4) with U0∈Vα pare defined globally in the time. In addition, the existence of a restricted global attractor for the semigroup {S(t)}restricted to Vα p(see the definition just below) will be established. Definition 2.1. Let p∈(3, +∞),α∈[1/2, 1), and {S(t)}be a local semiflow defined on Zα p. We say that A ⊂ Zα pis a restricted global attractor for {S(t)}in Zα pif for some closed, nonempty subset Vof Zα p,S(t):V→V,(t≥0)is a global semiflow on Vsuch that Ais a global attractor for {S(t)}restricted to Vas stated in [12], that is, (i) S(t)A=Afor t≥0, (ii) Ais compact, (iii) Aattracts all trajectories starting at bounded subsets of V. The following result, given in [6,7] provides a useful criterion for the existence of restricted global attractor. Lemma 2.2. Let p ∈(3, +∞),α∈[1/2, 1),{S(t)}be a local semiflow on Zα p,and the resolvent of Apbe compact. Then, in order to prove the existence of a restricted global attractor for {S(t)}in Zα p,it suffices to show that there exists a Banach space Y ⊃D(Ap)and a nondecreasing function g:[0, +∞)→[0, +∞)for which the conjunction of conditions (a) and (b) stated below holds with some closed and positively invariant nonempty subset V of Zα p, where (a) ∃C>0,∀U0∈V,∀t∈(0, τmax(U0)), kS(t)U0kY≤C. (b) ∃θ∈[0, 1),∀U0∈V,∀t∈(0, τmax(U0)), kG(S(t)U0)kp≤g(kS(t)U0kY)(1+kS(t)U0kθ Zα p). Micropolar fluids 5 Now, let us recall some definitions (e.g. cf. [12, Chapter 3]) concerning the asymptotic behaviour of dynamical systems. Let Vbe a complete metric space, and S(t):V→Vbe a C0-semigroup on V. Denoting by [B]Vthe closure of a set Bin the space V, for any set B⊂V the two sets γ+(B)and ω(B)defined by γ+(B) = ∪t≥0S(t)B, ω(B) = ∩s≥0[∪t≥sS(t)B]V, are called, respectively, the positive orbit and the ω-limit set of B. Thus, an ω-limit set consists of all points v∈Vfor which there exist positive numbers tn%+∞and points vn∈Bwith S(tn)vn→vas n→+∞. Remark 2.3. The above requirements (a) and (b) ensure that local solutions corresponding to U0∈Vexist for all t≥0. If, in addition, U0∈Vimplies that U(t,U0)∈Vas long as it exists, then the relation S(t)U0=U(t,U0), defines on VaC0-semigroup of global Zα psolutions. By (a), {S(t)}is point dissipative (that is, there exists a nonempty, bounded set B⊂Vwhich attracts every point of V); and since the resolvent of Apis compact, S(t):V→Vis a compact map for each t>0, whence the existence of the attractor follows (cf. [12]). 3 On the Lamé operator The main goal of this section is to show that the Lamé operator Lpv=−µ∆v−(µ+σ)∇div(v), defined on domain D(Lp) = W2,p(Ω)∩W1,p 0(Ω)(1 <p<+∞), is a sectorial, positive operator with compact resolvent in Lp. Namely, we consider the following problem: −µ∆v−(µ+σ)∇div v+λv=h, v|∂Ω=0. (3.1) For this purpose, let us denote by ∆ϕ(η), where η∈Rand ϕ∈(0, π/2), the sector of the complex plane given by ∆ϕ(η) = nλ∈C:|arg(λ−η)|<π 2+ϕ,λ6=ηo. Before establishing our main result of this section, we state a useful lemma with a priori estimates of solutions. Lemma 3.1. Let be given p >1with p 6=3/2, λ∈Cwith Re(λ)>0, and h∈Lp.Then, there exists a constant 0<C=C(µ,σ)such that for any solution v∈D(Lp)of (3.1), the following inequalities hold: (|λ|p−C)kvkp p+1 C+1kvkp 2,p≤Ckhkp p,for p >1, p 6=3/2 (3.2) and kvk2,p≤C(|λ|khk2+khkp),p≥2. (3.3) Proof. Define w(x,t) = eλtv(x)and q(x,t) = eλth(x). Multiplying the equation (3.1) by eλtwe have that wsatisfies ∂tw−µ∆w−(µ+σ)∇div w=q(x,t), w|∂Ω=0, w(x, 0) = v(x), in Ω. (3.4) 6P. Marín-Rubio, M. Poblete-Cantellano, M. Rojas-Medar and F. Torres-Cerda Let us denote by W2−2/p,p=W2−2/p,p(Ω)the Slobodetskii–Besov space. As proved in [19], we have kwk(2,1) Lp(QT)+sup τ≤Tkw(x,τ)kW2−2/p,p≤C(kqkLp(QT)3+kvkW2−2/p,p), (3.5) where QT=Ω×(0, T), and kwk(2,1) Lp(QT)=k∂twkLp(QT)3+k∇(∇w)kLp(QT)27 +k∇wkLp(QT)9+kwkLp(QT)3. Let us recall that W(2,1) p(QT)is the space of distributions w∈Lp(0, T;W2,p(Ω)) such that ∂tw∈Lp(QT). This space, endowed with the norm kwk(2,1) Lp(QT), is a Banach space. The constant C=C(µ,σ,T)in (3.5) still depends on Tand it has the property that C(µ,σ,T1)≤C(µ,σ,T2)if T1≤T2. By standard arguments, we can replace this constant by a quantity C(µ,σ), which does not depend on T, and we arrive at ZT 0(k∂twkp p+kwkp 2,p)ds +sup τ≤Tkw(x,τ)kp W2−2/p,p ≤C(µ,σ)ZT 0(kqkp p+kwkp p)ds +kvkp W2−2/p,p,∀T>0. (3.6) Taking into account the change of variables done at the beginning, we deduce that vand h appearing in (3.1) satisfy |λ|pZT 0epRe(λ)tkvkp pdt +ZT 0epRe(λ)tkvkp 2,pdt ≤Ckvkp W2,p+ZT 0epRe(λ)tkhkp pdt +ZT 0epRe(λ)tkvkp pdt. Then, we obtain epRe(λ)T−1 pRe(λ)|λ|pkvkp p+kvkp 2,p≤C"kvkp 2,p+epRe(λ)T−1 pRe(λ)khkp p+kvkp p#. Taking Tsuch that C+1≤epRe(λ)T−1 pRe(λ), we deduce (3.2). In the case p=3/2, to the norm kvkW2−2/p,pthat appears on the right side of (3.6) we must add the term ZΩ|v|p (δ(x))2p−2dx1/p (δ(x)is the distance from xto ∂Ω). Therefore it is not possible to obtain the estimate (3.2) (see [14] for details). Next, for p≥2, we multiply the equation in (3.4) by wand integrate over Qt(0≤t≤T) to obtain Zt 0((wt,w) + µk∇wk2)ds + (µ+σ)Zt 0kdiv(w)k2 2ds =Zt 0(q,w)ds, hence wt∈L2(0, T;L2(Ω)) ⊂L2(0, T;H−1,2(Ω)),w∈L2(0, T;W1,2 0(Ω)). From this we obtain, with δ>0, kw(t)k2 2+ (2µ−δ)Zt 0k∇wk2ds ≤ kvk2 2+C(δ)Zt 0kqk2 2ds. (3.7) Micropolar fluids 7 On the other hand, from Gagliardo–Nirenberg inequality we have kw(t)kp≤δkw(t)k2,p+C(δ)kw(t)k1. (3.8) Returning to (3.6), using (3.8) and inserting (3.7) we have ZT 0(k∂twkp p+kwkp 2,p)ds +sup τ≤Tkw(x,τ)kp W2−2/p,p ≤C(µ,σ) Zt 0kqkp pds +Zt 0kqk2dsp/2 +kvkp W2−2/p,p!. From this estimate and using the same arguments above, we conclude (3.3). Lemma 3.2. The problem (3.1)has a unique weak solution v∈W1,2 0(Ω). Proof. We observe that the problem (3.1) is equivalent to the variational problem (To find v∈W1,2 0(Ω)such that µ(v,z) + (µ+σ)(div v, div z) + λ(v,z) = (h,z),∀z∈W1,2 0(Ω).(3.9) To prove that this problem has a solution, since W1,2 0(Ω)is a separable Hilbert space, we can consider a sequence of elements {zm}m≥1of W1,2 0(Ω)which is free and total in W1,2 0(Ω). For each fixed integer m≥1, we would like to define an approximation solution vmof (3.9) by (vm=∑m i=1ξi,mzi,ξi,m∈R, µ(∇vm,∇zk) + (µ+σ)(div vm, div zk) + λ(vm,zk) = (h,zk),∀k=1, . . . , m.(3.10) The equations (3.10) are the system of linear equations for ξ1,m, . . . , ξm,m, and the existence of a solution follows easily. The passage to the limit is a consequence of the following argument. We multiply (3.10) by ξk,mand sum from k=1, . . . , m; this gives µk∇vmk2+ (µ+σ)kdiv vmk2+λkvmk2= (h,vm), or µk∇vmk2+ (µ+σ)kdiv vmk2+λkvmk2≤1 2µkhk2 H−1+µ 2k∇vmk2. Thus, we obtain the a priori estimate k∇vmk2≤Ckhk2 H−1. (3.11) Since the sequence vmremains bounded in H1 0(Ω), there exist some v∈H1 0(Ω)and a subsequence m0→∞such that vm0→vin the weak topology of H1 0(Ω). (3.12) The injection of H1 0(Ω)into L2(Ω)is compact, so we have also vm0→vin the norm of L2(Ω). (3.13) With the convergences (3.12)–(3.13) it is easy to pass to the limit in (3.10) and thus to obtain the existence of a weak solution of (3.9). The uniqueness is proved in the standard way. 8P. Marín-Rubio, M. Poblete-Cantellano, M. Rojas-Medar and F. Torres-Cerda Theorem 3.3. Consider p >1, p 6=3/2. There exists a value η>0such that for each λ∈Cwith Re(λ)>−η, and each h∈Lp,the problem (3.1)has a unique solution v∈D(Lp). Moreover, there also exist constants ϕe∈(0, π/2)and M >0such that the resolvent set ρ(−Lp) contains the sector ∆ϕe(−η),and the following estimate holds k(λI+Lp)−1kp≤M |λ|+1,λ∈∆ϕe(−η). In the terminology of Henry [13], Lpis a sectorial operator. Proof. Before we proceed with the proof of all statements, let us observe some good properties of the Lamé operator when p=2. From Agranovich et al. [3], it is well known that L2is a closed densely defined self-adjoint operator with compact resolvent. Using the Gårding inequality, we can see that L2is a positive operator since δkvk2 H1(Ω)≤(L2v,v),∀v∈D(L2), with δ>0. From this, we deduce that the L2is a sectorial operator. The spectrum of L2 consists of isolated positive eigenvalues of finite multiplicity. Numbering them in nondecreasing order taking into account their multiplicities, we obtain a sequence {λj}+∞ j=1with the asymptotic behaviour λj∼Λ0j2/3 (Λ0a positive constant). Next, for a small enough constant e>0 and suitable values ηand ϕe(to be specified later) we show that the subset ∆ϕe(−η)is contained in the resolvent set ρ(−Lp)in several steps. For this, we define the following sectors in the complex plane (where the constant Cis given in Lemma 3.1): ∆0={λ∈C: Re(λ)>0, |λ| ≥ p √C+1}, ∆00 ={λ∈C: Re(λ)>0, |λ| ≤ p √C+1}, ∆e=λ∈C:|Re λ| |Im λ|<1 (1+e)M. Step I: We prove that ∆0⊂ρ(−Lp). First, assume that p≥2. To solve (λI+Lp)v=hfor any h∈Lp(Ω), we use the continuous injection of Lpinto L2. By Lemma 3.2, we can see that (3.1) has a unique weak solution in W1,2(Ω). In fact v∈W1,2 0(Ω), and it satisfies λZΩv·u+ZΩ Ξ(v):ε(u) = ZΩh·u,∀u∈W1,2 0(Ω), where Ξ(v) = σtr(ε(v))I+2µε(v) and ε(v) = 1/2 ∂jvi+∂ivj. As long as Ωhas smooth boundary, then v∈W2,2(Ω). Thus, vis a strong solution for the system. Using the Sobolev Lemma and interpolation, we have that v∈C(Ω)∩Lp(Ω), kvkC(Ω)≤CkL2vk, and therefore L2v=−λv+h∈Lp(Ω). From this, and using [18, Theorem 5.1, p. 301], we conclude that v∈D(Lp). Micropolar fluids 9 Next we consider the case 1 <p<2 with p6=3/2 and h∈Lp(Ω). We approximate h by a sequence (hn)∈C∞ 0(Ω). Each problem (λI+Lp)vn=hnhas a solution, applying the inequality (3.2) to (λI+Lp)(vn−vm) = hn−hm, we have (|λ|p−C)kvn−vmkp+1 C+1kvn−vmk2,p≤Ckhn−hmkp. Thus, vn→v∈W2,p(Ω)as n%+∞. Therefore, from the inequality (3.2) we deduce k(λI+Lp)vkp≥p r|λ|p−C Ckvkp, from which, the resolvent (λI+Lp)−1exists for all p>1 with p6=3/2. Using the resolvent series we obtain that ∆0is contained in the resolvent set of −Lpand k(λI+Lp)−1kp≤2(p−1)/pe c |λ|−e c,∀λ∈∆0, where C=e cp(e cis a positive constant). Step II: Now we prove that ∆00 ⊂ρ(−Lp). For p≥2, we solve as before (λI+Lp)v=hand from (3.3), the following estimate holds k(λI+Lp)−1kp≤C, Re λ>0. When 1 <p<2, p6=3/2, let us suppose that v1,v2∈D(Lp)are two solutions for (λI+Lp)v=h. Using the Sobolev embedding we obtain that v2−v1∈W2,1(Ω)⊂L3(Ω), (λI+Lp)(v2−v1) = 0. From this we have Lp(v2−v1) = −λ(v2−v1)in L2(Ω). By a regularity result in [18], we conclude that v2−v1∈D(L2), where uniqueness holds, which implies that v2=v1. Now, we claim that there exists a constant γ=γ(λ)>0 such that k(λI+Lp)vkp≥γ(λ)kvkp,∀v∈D(Lp). (3.14) Suppose this is false. Then there exists vn∈D(Lp),hn= (λI+Lp)vn∈Lp(Ω)such that, for all n,kvnkp=1, khnkp≤1/n. Since Lpvn=−λvn+hn∈Lp, using [18, Lemma 4.4, p. 301], we have that (for certain C0 and C00) kvnk2,p≤C0(|λ|kvnkp+|λ|khnkp+kvnkp)≤C00. Thus, kvnkW2,pis bounded. Taking a subsequence {vn0}converging weakly to some v∈D(Lp), we obtain that (λI+Lp)v=0. By the uniqueness of solution to any problem of type (3.1), it must be v=0, which contradicts kvkp=1. So, (3.14) holds. Now, approximate again h∈Lpby hn∈C∞ 0(Ω), and denote by vnthe unique solution of (λI+L2)v=hnand write L2(vm−vn) = (hm−hn)−λ(vm−vn). Using again [18, Lemma 4.4, p. 301], we have that kvm−vnk2,p≤C0(khm−hnkp+|λ|kvm−vnkp). (3.15) 16 P. Marín-Rubio, M. Poblete-Cantellano, M. Rojas-Medar and F. Torres-Cerda 5 Restricted global attractors in Zα p The purpose of this section is to show the existence of a restricted global attractor for {S(t)}. The construction of this attractor is based on the lemmas from the above section. Theorem 5.1. Let p ∈(3, +∞),α∈[1/2, 1)and F∈Lpbe sufficiently small, according to assumptions in Lemma 4.3. Denote by {S(t)}the local semiflow (where S(t)U0=U(t,U0)for t∈[0, τmax(U0)))of maximal fractional solutions of (1.4)defined on Zα p,and consider the constant κ given in Lemma 4.3. Then the following statements hold. (i) For Vα p= [γ+(BZ1/2 p(0, κ)) ∩Zα p]Zα p, all fractional solutions S(t)U0, with U0∈Vα p, are globally defined, S(t)Vα p⊂Vα p(t≥0),and the semigroup {S(t)}restricted to Vα phas a global attractor Aα,p. (ii) All attractors Aα,pwith p ∈(3, +∞),α∈[1/2, 1)coincide, i.e., Aα,p=ωZ1/2 p(BZ1/2 p(0, κ)). Proof. Due to the continuity of the semigroup, the estimate (4.5) remains valid for all U0in [γ+(BZ1/2 p(0, κ))]Z1/2 p. Using Lemma 2.2 with Y=Z1/2 p, we can choose Vα p= [γ+(BZ1/2 p(0, κ)) ∩Zα p]Zα p, and as (4.4) and (4.5) are the estimations required by Lemma 2.2, that is, conditions (b) and (a) respectively, we obtain the existence of the global restricted attractor Aα,pand therefore (i) is proved. Observe that Vα pfor α∈(1/2, 1)and p∈(3, +∞)is an unbounded, complete metric subspace of Zα p. When α=1/2 and p∈(3, +∞), then V1/2 p= [γ+(BZ1/2 p(0, κ))]Z1/2 pis a bounded, closed subset of Z1/2 p, and since S(t)is compact for t≥0 then, ωZ1/2 p(V1/2 p)is nonempty, compact, invariant and attracts V1/2 p(cf. [12, Theorem 4.2.2, Lemma 3.2.1]). Thus, the restricted attractor A1/2,pclaimed in part (i) coincides with ωZ1/2 p(V1/2 p). In order to prove (ii), observe that ωZ1/2 p(V1/2 p) = ∩s≥0[∪t≥sS(t)γ+(BZ1/2 p(0, κ))]Z1/2 p =∩s≥0[∪t≥s∪r≥tS(t)BZ1/2 p(0, κ)]Z1/2 p =∩s≥0[∪r≥sS(r)BZ1/2 p(0, κ)]Z1/2 p =ωZ1/2 p(BZ1/2 p(0, κ)). For p∈(3, +∞),ωZ1/2 p(BZ1/2 p(0, κ)) is a bounded set in V1/2 p. Using [13, Theorem 3.3.6] we have that S(t)(ωZ1/2 p(BZ1/2 p(0, κ))) is a bounded set in Vα pwith α∈[1/2, 1), and by invariance ωZ1/2 p(BZ1/2 p(0, κ)) ⊂ Aα,pfor 1/2 ≤α<1. To obtain the converse inclusion, it suffices to use the compactness of the embedding Zα p⊂Z1/2 pfor α∈(1/2, 1). Micropolar fluids 17 Acknowledgements P. Marín-Rubio was partially supported by MCI (Spain) MTM2015-63723-P and Junta de Andalucía Grant P12-FQM-1492. M. Poblete-Cantellano was partially supported by Universidad de Atacama, project 22256 DIUDA. M. Rojas-Medar was partially supported by Project MTM2012-32325, Spain and Fondecyt-Chile, Grant 1120260. F. Torres-Cerda was partially supported by Universidad de Atacama, project 22294 DIUDA. References [1] R. A. 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