Global existence and exponential stability for the micropolar fluid system

Global existence and exponential stability for the micropolar fluid system E. J. Villamizar-Roa∗& M. A. Rodr´ıguez-Bellido †‡ April 20, 2007 Abstract We consider the micropolar fluid system in a bounded domain of R3 and prove the existence and the uniqueness of a global strong solution with initial data being a perturbation of the stationary solution, whose existence is also obtained. We prove that these solutions converge uniformly to the stationary solutions with exponential decay rate. The technique of our analysis is the semigroups approach in Lp −spaces. Keywords: Micropolar fluids, strong solution, exponential decay rate, semigroup approach. AMS subject classification: 35A, 35B35, 35B40, 35B65, 35Q35, 76D03 1 Introduction In this paper we study the following equations for the motion of micropolar fluid in a bounded domain Ω⊂R3with smooth boundary ∂Ω:    ∂tu+(u·∇)u−(µ+µr)∆u+∇p=2µrrot w+f,in Ω×(0,∞), ∂tw+(u·∇)w−χ∆w−γ∇div w+4µrw=2µrrot u+g,in Ω×(0,∞), div u=0,in Ω×(0,∞), (1) where the unknowns are u=u(x,t)∈R3,w=w(x,t)∈R3and p=p(x,t)∈ Rwhich denote respectively, the linear velocity, the velocity of rotation of the fluid particles and the pressure, at a point (x,t)∈Ω×(0,∞). The terms f(x,t) and g(x,t) are given, and stand external sources of the linear and angular momentum of particles, respectively. µ,µr,c0,ca,cdare positive constants given to characterize isotropic properties of the fluid: µis the usual Newtonian viscosity; µr,c 0,c a,c dare new viscosities related to the asymmetry of the stress ∗E. J. Villamizar-Roa is supported by Colciencias, Colombia, Proyecto BID-III etapa and Universidad Industrial de Santander-UIS. †M. A. Rodr´ıguez-Bellido is partially supported by D.G.E.S. and M.C. y T. (Spain), Projet BFM2003-06446-C02-01 and by M.E.C. (Spain), Projet MTM2006-07932. ‡Corresponding author 1 tensor and consequently related to the appearance of the internal rotation field w. In order to shorten the equations, we denote χ=ca+cdand γ=c0+cd−ca. These constants satisfy c0+cd>c a. We use the standard notations ∇,∆, div and rot for the gradient, Laplacian, divergence and rotational operators respectively, and ∂tuto denote the time derivative. The ith component of (u·∇)win cartesian coordinates is given by [(u·∇)w]i= 3 � j=1 uj ∂wi ∂xj . Together with these equations we prescribe the following initial and boundary conditions u(x,0) = u0(x),w(x,0) = w0(x),in Ω,(2) u(x,t)=0,w(x,t)=0,on ∂Ω×(0,∞),(3) where, for the sake of simplicity in this exposition, we have taken homogeneous boundary conditions for both velocity and microrotational fields. The formulation and motivation of the equations (1)-(3) are found in Petrosyan [25], Condiff& Dalher [3], Eringen [4, 5] and Lukaszewicz [21], that treat the magneto-micropolar model which can be particularized to the micropolar model case. We observe that this model includes as a particular case the classical Navier-Stokes equations, which has been greatly studied (see the classical books by Ladyzhenskaya [17]; Temam [30] and the references therein). With respect to system (1)-(3), Lukaszewicz [19, 20] established the global existence of weak solutions and local strong solutions for (1)-(3) using linearization and an almost fixed point theorem. Using the spectral Galerkin method: Boldrini & Rojas-Medar [26] proved the global existence of weak solutions; the local strong solution was obtained by Rojas-Medar [27], and the global one by Ortega-Torres & Rojas-Medar [22]. The convergence rates by the spectral Galerkin method were established in Rojas-Medar [28]. Another work that showed the existence and uniqueness of strong solutions (local in time for any data and global in time for small data) is Ortega-Torres & Rojas-Medar [23]. However, those works were made only in the Hilbertian context. Recently, Yamaguchi [31] studied the problem (1)-(3) in a bounded domain using the semigroup approach in Lp,1<p<∞, showing the existence of global strong solutions for small data. Here, we study the global existence, asymptotic behavior and stability of the solutions for the micropolar fluid system. If we consider smallness conditions on the physical constants of the system, we prove the global existence of strong solutions for system (1)-(3) in different spaces for the linear and angular velocity (the equal case was considered by Yamaguchi [31]). In a second step, we show that such solutions converge uniformly to stationary solutions when t→∞ with exponential decay rate. Thus, we say that these stationary solutions are asymptotically stable. We consider the initial data (u0,w0)∈L3(Ω)×Lm(Ω), for m≥3/2.In particular, we show the existence of strong solutions in the space L3(Ω)×L3(Ω). To show the existence of stationary solutions for system (1)-(3) in the space L3(Ω)×Lm(Ω),we use a fixed point argument. On the other hand, we obtain estimates of type (Lp×Lq)−(Lr×Ls) for the analytic semigroup generated by 2 the linearization of the operator in a neighborhood of the stationary solution, which include the cases p�=qand r�=s, and consequently we prove the global existence and asymptotic behavior of the solutions of the system (1)-(3). In this sense, our results on the global existence generalize the results that Yamaguchi [31] studied for the initial value problem, only for the cases p=qand r=s. The existence of the semigroup is obtained through the estimates (Lp×Lq)− (Lr×Ls) for the resolvent of the linearized operator, whose behavior depends on exponents p,q,r,s. From the stability point of view, several papers are devoted to study of the stability of the stationary solutions for the particular case of the Navier-Stokes equations, see for instance, Kozono & Ozawa [15], Borchers & Miyakawa [1], Chen [2], Kozono & Ogawa [14], Kozono & Yamazaki [16]. In general, these studies have been made using a characterization of the fractional power of the associated linearized operator and estimates of kind Lp−Lrfor the semigroup. On the other hand, in the convection problem, which is a model of the hydrodynamic formed by a coupling between the Navier-Stokes equations with a equation for the temperature, Hishida [11, 12] analyzed the stability problem in bounded and exterior domains, in the framework of Lpand weak-Lpspaces, by means of Lp−Lrand weak-Lp−Lrestimates for the semigroup associated with the respective linearized operator. The works cited before motivated our analysis about the stability for the micropolar fluid system. By sections, this paper is organized as follows: In §2 we give some preliminary results, some definitions and state our principal results. In §3, we study the existence of a steady solution. The linearized operator is analyzed at §4. Finally, in §5 we prove the global existence of nonstationary solutions near a stationary solution and show that the nonstationary solution tends to the stationary solution as t→∞with a exponential rate. 2 Preliminaries and main results Let Ωbe a bounded domain of R3with smooth boundary ∂Ω.We consider the usual Sobolev spaces Wm,q(Ω) = {f∈Lq(Ω) : �∂kf�Lq(Ω)<∞,for |k|≤m}, for m≥1 and 1 ≤q≤∞with the norm denoted by �·� m,q.As it is usual, we denote by W1,p 0(Ω) the closure of C∞ 0(Ω) (functions of class C∞(Ω) with compact support) in W1,p(Ω).We denote by �·�{p,q}the norm in Lp(Ω)×Lq(Ω) and by �·�{p,q}→{r,s}(respectively, �·�p→r) the norm of a bounded operator of Lp(Ω)×Lq(Ω)→Lr(Ω)×Ls(Ω) (respectively, of Lp(Ω)→Lr(Ω)). We denote by C∞ 0,σthe space of the functions C∞ 0with free divergence and by Lp σ(Ω),1<p<∞, the closure of C∞ 0,σin the Lp(Ω)−norm. When the functions to be vectors, the space will appear in bold characters. It is well known the Helmholtz decomposition of vectorial field in Lp(see for instance Fujiwara & Morimoto [6]) Lp(Ω) = Lp σ(Ω)⊕{∇q:q∈W1,p(Ω)}. Let P=Ppbe the projector of Lp(Ω) in Lp σ(Ω).Then, the Stokes operator in 3 Lp σ(Ω),1<p<∞, is given by D(Ap)=Lp σ(Ω)∩W1,p 0(Ω)∩W2,p(Ω),A pu=−Pp∆u. Also, we denote the Laplace operator in Lq(Ω),1<q<∞,by D(Bq)=W1,q 0(Ω)∩W2,q(Ω),B qv=−∆v. It is well known that the Stokes operator generates a bounded analytic semigroup {e−tAp}t≥0,of class C0in Lp σ(Ω),1<p<∞, see Giga [8]). It is also a classical result the Laplace operator generates a bounded analytic semigroup {e−tBq}t≥0of class C0in Lq(Ω),1<q<∞.From now on, we sometimes drop the subscripts pand qattached to operators Ap,B qif there is no confusion. We will consider the following class of stationary solutions for the problem (1)-(3). Definition 2.1 Let 3/2≤m<∞and K>0.We say that a pair of functions (u,w)is a stationary solution of class (m, K)for (1)-(3), if (u,w)∈D(A3)×D(Bm),�Au�3+�Bw�m≤K, and (u,w)satisfies �(µ+µr)Au−2µrrot w+P{(u·∇)u}=0,in L3 σ(Ω), χBw+(u·∇)w−γ∇div w+4µrw−2µrrot u=0,in Lm(Ω).(4) Here, we have used that W1,m(Ω)⊂L3(Ω) together with the fact that P(rot w)= rot w.We observe that div rotw= 0 in Ω; and therefore, since w= 0 on ∂Ω, we have rot w·n= 0 on ∂Ω.Thus, rot w∈L3 σ(Ω).Moreover, we observe that W1,3(Ω)⊂Lm(Ω). The following theorem states the existence of the stationary solution of class (m, K). Theorem 2.2 For each m∈[3/2,∞)there exists ¯ δ=¯ δ(m)and K0>0such that if µ, χ≥¯ δ,then the problem (1)-(3) has a unique stationary solution (u,w) of class (m, K0). Remark 2.1 The existence of stationary solutions in bounded domains was studied by Lukaszewicz [18] (see also Lukaszewicz [21]). The proof was done using the Leray-Schauder’s Theorem. The existence of weak solutions is showed considering two problems, a problem for u,with wfixed and later, a problem for wwith ugiven. Under smallness conditions on the viscosity, the solution is unique. Regularity results can also be found. More precisely, the solution obtained by Lukaszewicz [21] belongs to V×H1 0(Ω)(when weak solutions are considered) and the space V×(H1 0(Ω)∩H2(Ω)) (when strong solutions are considered). We remark that V={v∈H1 0(Ω),div v=0}. 4 In order to establish the stability of the stationary solutions (u,w) given by Theorem 2.2, and give a global strong solution of (1)-(3), we make u=u+ � u,w=w+� w,p=¯p+�p, in (1)-(3), where ¯pis the associated pressure with (u,w) and (� u,� w,�p) denotes the perturbation of (u,w,¯p). Definition 2.3 A pair of functions (u,w)is a strong solution of (1)-(3) on [0,∞)if the perturbation (� u,� w)belongs to the class � u=u−u∈C([0,∞); L3 σ(Ω)) ∩C((0,∞); D(A3)) ∩C1((0,∞); L3 σ(Ω)), � w=w−w∈C([0,∞); Lm(Ω)) ∩C((0,∞); D(Bm)) ∩C1((0,∞); Lm(Ω)), for m∈[3/2,∞), and satisfies                ∂t� u+(µ+µr)A� u+P{(� u·∇)� u+(u·∇)� u+( � u·∇)u}− −2µrrot � w=0,t>0, ∂t� w+χB� w+( � u·∇)� w+(u·∇)� w+( � u·∇)w −γ∇div � w+4µr� w−2µrrot u=0,t>0, u(0) = u0−u,� w(0) = w0−w. (5) Our result on stability of stationary solution is the following. Theorem 2.4 Let 3/2≤m<∞and assume (u0,w0)∈L3 σ(Ω)×Lm(Ω).Then, there exists δ∈(0,¯ δ](where ¯ δis from Theorem 2.2) such that if γ+µr<δ, max �1 µ+µr,1 χ�<δ,then the stationary solution (u,w)given by Theorem 2.2 is stable, that is, there exists a positive constant �=�(µ, µr,χ,γ, m)such that if �u0−u�3+�w0−w�m<�, then the system (1)-(3) has a unique strong solution (u,w)on [0,∞). Moreover, this solution decays exponentially, that is, �u(t)−u�∞+�w(t)−w�∞=o(e−ζt), when t→∞,with ζ=ζ(µ, µr,χ,γ, m)some positive constant. 3 Proof of Theorem 2.2 In this section we build a stationary solution (u,w) of class (m, K). To this aim, we will use Banach’s Fixed Point Theorem. Thus, we focus our efforts in the building of a contractive operator F. This operator is defined as F: D(A3)×D(Bm)→D(A3)×D(Bm) is given by: F�u w�=  −1 µ+µr A−1{P{(u·∇)u}−2µrrot w} −1 χB−1{(u·∇)w+4µrw−γ∇div w−2µrrot u}  .(6) We observe that the system (4) is equivalent to [u,w]T=F[u,w]T,in D(A3)×D(Bm). Hence, the proof of Theorem 2.2 is a consequence of the following proposition. 5 Proposition 3.1 Let m∈[3/2,∞). There are positive constants ¯ δ=¯ δ(m) and K0=K0(µ, µr,χ,γ, m)>0such that if µ, χ≥¯ δ,then the operator Fis contractive on the complete metric space EK0={(u,w)∈D(A3)×D(Bm): �Au�3+�Bw�m≤K0}. To prove Proposition 3.1 we will use the following lemma, which follows from embedding properties obtained by Giga [10], the H¨older’s inequality and the Sobolev’s embedding. Lemma 3.2 Let 1< p, q, r < ∞be satisfying the relations 1 r>1 p−2 3,1 r>1 q−1 3,1 r≥1 p+1 q−1.(7) Then, the following estimates hold: for all u∈D(Ap),v∈D(Aq),w∈D(Bq) �P{(u·∇)v}�r≤C(p, q, r)�Au�p�Av�q,(8) �(u·∇)w�r≤C(p, q, r)�Au�p�Bw�q.(9) Proof of Proposition 3.1: We observe that from estimate (8) and the embedding W1,m(Ω)⊂L3(Ω) we obtain 1 µ+µr�P{u·∇u}−2µrrot w�3≤c1 µ+µr�Au�2 3+2µr µ+µr�rot w�3 ≤c1 µ+µr�Au�2 3+2c2µr µ+µr�Bw�m. (10) On the other hand, from the inequality (9) and the embedding W1,3(Ω)⊂ Lm(Ω) we have 1 χ�u·∇w+4µrw−γ∇div w−2µrrot u�m ≤c3 χ�Au�3�Bw�m+4µr χ�w�m+γ χ�∇div w�m+2µr χ�rot u�m ≤c3 χ�Au�3�Bw�m+4c4µr χ�Bw�m+c5γ χ�Bw�m+2c6µr χ�Au�3. (11) Therefore, if �Au�3+�Bw�m≤K0,we have � � �F[u,w]T� � �D(A3)×D(Bm)≤� C(�A¯ u�3+�B¯ w�m)+ � C1(�A¯ u�3+�B¯ w�m)2 ≤� CK0+� C1K2 0, being � C, � C1given by � C= max �2c2µr µ+µr+4c4µr χ+c5γ χ,2c6µr χ�,� C1= max �c1 µ+µr,c3 χ�. Let K0= (1 −� C)/� C1.Then, if µand χare large enough such that � C<1,we have F(EK0)⊂EK0(that is, EK0is an invariant set in D(A3)×D(Bm)). Remark 3.1 Observe that in Theorem 2.2, the hypothesis µand χlarge enough, can be replaced by the hypothesis µrand γsmall enough such that � C<1. 6 Using the arguments appearing in inequalities (10)-(11), we can infer that � � �F[u1,w1]T−F[u2,w2]T� � �D(A3)×D(Bm)≤ ≤� C2{�A(u1−u2)�3+�B(w1−w2)�m}, (12) for (u1,w1),(u2,w2)∈EK0,and a positive constant � C2=� C2(µ, µr,χ,γ, m, K0). Indeed, 1 µ+µr�P(u1·∇u1)−P(u2·∇u2)−2µrrot (w1−w2)�3 ≤c1 µ+µr�A(u1−u2)�3(�Au1�3+�Au2�3)+ 2c2µr µ+µr�B(w1−w2)�m ≤2c1K0 µ+µr�A(u1−u2)�3+2c2µr µ+µr�B(w1−w2)�m; and also, 1 χ�u1·∇w1−u2·∇w2+4µr(w1−w2)−γ∇div (w1−w2)−2µrrot (u1−u2)�m ≤c3 χ��A(u1−u2)�3�Bw1�m�+c3 χ��Au2�3�B(w1−w2)�m� +4c4µr χ�B(w1−w2)�m+c5γ χ�B(w1−w2)�m+2c6µr χ�A(u1−u2)�3 ≤c3K0 χ��A(u1−u2)�3+�B(w1−w2)�m� +4c4µr χ�B(w1−w2)�m+c5γ χ�B(w1−w2)�m+2c6µr χ�A(u1−u2)�3. So, we find (12) with � C2given by � C2= max �2c1K0 µ+µr +c3K0 χ+2c6µr χ,2c2µr µ+µr +c3K0 χ+4c4µr χ+c5γ χ�. We choose ¯ δ(m) such that for µ, χ≥¯ δ(m),we shall obtain � C2<1 (and therefore � C<1). With this choice, we have that Fis contractive, and, hence, the existence of a stationary solution is proved. 4 Analysis of the Linearized Operator In this section we will make an analysis of the linearized operator L=Lp,q with respect to the stationary solution (u,w) given by Theorem 2.2, being L=L0+L1, where: L0�u w�=�(µ+µr)Au χBw�, 7 L1�u w�=�P{(u·∇)u+(u·∇)u}−2µrrot w (u·∇)w+(u·∇)w+4µrw−γ∇div w−2µrrot u�, and whose definition domain is D(Lp,q)=D(Ap)×D(Bq), for p, q ∈(1,∞) suitably chosen. The following lemma provides one estimate of the operator L1. Lemma 4.1 Suppose that p, q ∈(1,∞)satisfy −1 3≤1 p−1 q≤1 3,1 p−1 q≤1−1 m,1 q>1 m−1 3.(13) Then, there exists a constant Cp,q(m)>0such that: � � �L1[u,w]T� � �{p,q}≤Cp,q(m)�(�Au�3+�Bw�m)�Au�p +�Au�3�Bw�q+µr�Au�p+(µr+γ)�Bw�q�, (14) for all (u,w)∈D(Ap)×D(Bq). Proof: The proof is a consequence of Lemma 3.2 together with the following embeddings W1,q ⊂Lp(Ω),W1,p ⊂Lq(Ω),D(Ap)⊂Lq(Ω). For 0 <ω<π/2,we put Σω={λ∈C:|arg λ|<π−ω}∪{0}.Then we have: Lemma 4.2 Let p, q be as in Lemma 4.1. Then for each ω∈(0,π/2),there exists a constant Mp,q(m, ω)such that � � �L1(λ+L0)−1[u,w]T� � �{p,q}≤Kp,q� � �[u,w]T� � �{p,q},(15) for all λ∈Σωand (u,w)∈Lp σ(Ω)×Lq(Ω),where Kp,q ≡Mp,q�(µr+γ)+�max �1 µ+µr ,1 χ��K0�, with K0the constant of Theorem 2.2. Proof: It is well known (see Giga [8]) that for each ω∈(0,π/2), Σω⊂ρ(−Ap)∩ρ(−Bq), with Kp(ω) = sup λ∈Σω (1 + |λ|)�(λ+A)−1�p→p<∞,(16) ˆ Kq(ω) = sup λ∈Σω (1 + |λ|)�(λ+B)−1�q→q<∞.(17) 8 It remains to estimate �A(λ+(µ+µr)A)−1u�pand �B(λ+χB)−1w�q. Observe that A(λ+(µ+µr)A)−1=1 µ+µr I−λ µ+µr (λ+(µ+µr)A)−1, and therefore: � � �A(λ+(µ+µr)A)−1u� � �p≤1 µ+µr�u�p+|λ| µ+µr� � �(λ+(µ+µr)A)−1� � �p→p�u�p. Now, rewriting (λ+(µ+µr)A)−1=1 µ+µr�λ µ+µr +A�−1 , we have: |λ| µ+µr� � �(λ+(µ+µr)A)−1� � �p→p=�|λ| µ+µr�1 µ+µr� � � � ��λ µ+µr +A�−1� � � � �p→p . If we call � λ=λ µ+µr , we have: 1 µ+µr|� λ|� � � ��� λ+A�−1� � � �p→p≤1 µ+µr Kp(ω). Estimates for �B(λ+χB)−1�qare similar. Remark 4.1 Observe that if λ∈Σω, then � λ=λ µ+µr∈Σω, because of arg λ= arg � λ. We recall that ρ(·) is the standard notation for the set resolvent and that �·�p→qdenotes the norm of the bounded linear operator defined from Lp(Ω) to Lq(Ω).Taking into account that: (λ+L0)−1�u w�=�(λ+(µ+µr)A)−1u (λ+χB)−1w�, by Lemma 4.1 and estimates (16)-(17) for � λ, we obtain: � � �L1(λ+L0)−1[u,w]T� � �{p,q} ≤Cp,q�(�Au�3+�Bw�m+µr)(1 + Kp(ω)) 1 (µ+µr)�u�p +(�Au�3+µr+γ)(1 + ˆ Kq(ω)) 1 χ�w�q�. Since (u,w) belongs to the class (m, K0),we have �Au�3+�Bw�m≤K0and thus � � �L1(λ+L0)−1[u,w]T� � �{p,q}≤Cp,q�(µr+K0)(1 + Kp(ω)) 1 (µ+µr)�u�p +(µr+γ+K0)(1 + ˆ Kq(ω)) 1 χ�w�q�. 9 where s0=�1 m+1 r0−1 3�−1,β=3 2�1 3−1 r0�=3 2�1 m−1 s0�>0.We consider the operator Υ:B→Bgiven by Υ�u w�(t)=e−tL �u0−u w0−w�−�t 0 e−(t−s)L�P(u·∇u) u·∇w�(s)ds. For J>0,we define the set BJ=�(u,w)∈B:� � �[u,w]T� � �B≤J�. We focus the proof on the obtention of bounds for Υ[u,w]Tin B. We can take a positive constant �=�(r0, m, µ, µr,γ,χ) such that if �u0−u�3+�w0− w�m<�,there exists a positive constant J=J(�u0−u�3,�w0−w�m) such that Υ(BJ)⊂B J, and Υis contractive on BJ. Since γ+µr<δ �,max �1 µ+µr,1 χ�<δ �,we are going to apply the results from Lemmas 5.2 and 5.3 for j=0,{r, s}={r0,s 0}and j=1,{r, s}={3,m}. Indeed, from Lemma 5.2 for j=0,{r, s}={r0,s 0}, we obtain: � � �e−tL [u0−u,w0−w]T� � �{r0,s0}≤Cζt−3 2 1 3−1 r0e−ζt� � �[u0−u,w0−w]T� � �{3,m}, which is equivalent to tβeζt� � �e−tL [u0−u,w0−w]T� � �{r0,s0}≤Cζ� � �[u0−u,w0−w]T� � �{3,m}. And from Lemma 5.2 for j=1,{r, s}={3,m}, we have: � � �∇e−tL [u0−u,w0−w]T� � �{3,m}≤Cζt−1 2e−ζt� � �[u0−u,w0−w]T� � �{3,m}. On the other hand, from Lemma 5.3 for j=0,{r, s}={r0,s 0}, we obtain: � � ��t 0 e−(t−s)L[P(u·∇u),u·∇w]Tds� � �{r0,s0} ≤�t 0� � �e−(t−s)L[P(u·∇u),u·∇w]Tds� � �{r0,s0}ds ≤Cζ�t 0 (t−s)−1/2e−ζ(t−s)�u�r0(�∇u�3+�∇w�m) ≤Cζt−βe−ζt� � �[u,w]T� � � 2 B. Analogously, for j=1,{r, s}={3,m}we have: � � �∇�t 0 e−(t−s)L[P(u·∇u),u·∇w]Tds� � �{3,m} ≤�t 0� � �∇e−(t−s)L[P(u·∇u),u·∇w]Tds� � �{3,m}ds ≤Cζ�t 0 (t−s)(−3/2)(1/r0−1/3)−1e−ζ(t−s)�u�r0(�∇u�3+�∇w�m)ds ≤Cζt−1/2e−ζt� � �[u,w]T� � � 2 B. 16 Therefore, from the above inequalities we have � � �Υ[u,w]T� � �B≤C0� � �[u0−u,w0−w]T� � �{3,m}+C1� � �[u,w]T� � � 2 B. So, taking J0=C0� � �[u0−u,w0−w]T� � �{3,m}<1 4C1 , we get: Υ(BJ)⊂(BJ),with J=1−√1−4C1J0 2C1 . Working in a similar manner, we can show that Υis contractive. Consequently, the system (30) has a solution in BJ,that satisfies the estimates (32)-(33) for {r, s}={r0,s 0}and the estimates (34)-(35) for {r1,s 1}={3,m}. The uniqueness proof is done in a standard way, so it is omitted. To see that the solution (� u,� w) of (30) satisfies the estimates (32)-(33) for the other values of r,s,r1,s1, we use again Lemma 5.2 and Lemma 5.3. Using arguments analogous of those of Fujita & Kato [7], Giga & Miyakawa [9], Kozono & Ogawa [13], we can show the H¨older continuity with respect to time for the nonlinear term in L3 σ(Ω)×Lα(Ω) and this allows us to show that (� u,� w) is, in fact, a strong solution for the system (29) in the sense given in Definition 2.3 (see Tanabe [29]), Theorem 3.3.4). Proof of Theorem 2.4: We take α=m=3/2 and we choose r0(m) satisfying (31). If χ+µr<δ �,max �1 µ+µr,1 χ�<δ �,then by Proposition 5.1, there exists a positive constant �=�(µ, µr,γ,χ, m, r0) such that if �u0− u�3+�w0−w�m<�,there exists a unique strong solution (� u,� w) for system (29) in the interval [0,∞).Therefore, (� u,� w) satisfies the estimates (32)-(35), being ζ>0 a number positive fix that satisfies (26) for {p, q}={3r0/(3 + r0), mr0/(m+r0)}and {3,m}. ¿From Nirenberg’s inequality and using the estimate for �∇� u(t)�r�for some r�>3,we obtain �� u(t)�∞≤C�∇� u(t)�a r��� u(t)�(1−a) 3≤Cζ� Ct−(3a/2)(1/3−1/r�)−a/2e−ζt, where asatisfies 0 = a(1/r�−1/3) + (1 −a)1/3. This implies that �u(t)�∞= o(e−ζt),when t→∞. Observe that we cannot apply the above argument to find the decay rate for the microrotation velocity w.In fact, in this case for m=3/2,s 0=�1 m+1 r0− 1 3�−1<3.To overcome this point, we apply Proposition 5.1 with another value for m. In fact, we choose m0and r0such that 3 2<3r0 2r0−3<m 0<3.The above inequality implies that 1/m0+1/r0<2/3,that is, �1 m0 +1 r0−1 3�−1>3.(42) From the estimates (32)-(35) with {r, s}={3,m 0},we can find a time T>0 big enough such that �� u(T)�3+�� w(T)�m0<�(µ, µr,χ,γ, m0,r 0), 17 provided that γ+µr<δ �(m0,r 0), max �1 µ+µr,1 χ�<δ �(m0,r 0),with � δ�(m0,r 0)≡min{δ3r0/(3+r0),m0r0/(m0+r0),δ 3,m0}. Let δ= min{� δ�(m0,r 0),δ�(m, r0)},where δ�is given as before and we assume: γ+µr<δ,max �1 µ+µr ,1 χ�<δ. Since (u(T),w(T)) ∈L3 σ(Ω)×Lm0(Ω),we apply Proposition 5.1 with initial condition equal to (u(T),w(T)) and m=m0.Consequently, we can deduce (32)-(35) and m=m0,with ζ>0 a fixed positive number satisfying (26) for {p, q}={3r0/(3 + r0),m 0r0/(m0+r0)}and {3,m 0}. Now, we can obtain the decay rates of �∇� w(t)�sfor some s>3 because (42) is verified. We observe that in this point we have used the uniqueness of solution. Consequently, we have �� w(t)�∞=o(e−ζt),when t→∞. 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