Asymptotic behaviour of a system of micropolar equations

Asymptotic behaviour of a system of micropolar equations

Pedro Marín-Rubio

, Mariano Poblete-Cantellano

, Marko Rojas-Medar

and Francisco Torres-Cerda

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 sys- tems, we prove the existence of restricted global attractors for local semiflows on suit- able fractional phase spacesZ^α_p, namely forp∈(3,+_∞)andα∈[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 < +∞, p 6= 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 ⊂ _R³ be 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= χrotw+f, inΩ×(0,T), divu=_{0, in Ω}×(_0,T)_,

∂w

∂t + (u· ∇)w−_µ∆w−(µ+σ)∇divw+2χw=χrotu+g, inΩ×(0,T),

(1.1)

together with the following boundary and initial conditions















u=_{0 on}∂Ω×(_0,T)_, u(x, 0) =u₀(x) inΩ,

w=0 on∂Ω×(0,T), w(x, 0) =_w0(x) inΩ,

(1.2)

BCorresponding author. Email: francisco.torres@uda.cl

where u = (u₁,u₂, u₃) is the velocity field, p is 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)andg= (g₁, g₂, g₃)are external forces and moments respectively. The positive constants ν, χ, µ, σ represent viscosity coefficients, ν is the usual Newtonian viscosity and χ is 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 thatΩis a bounded domain in R² and they conclude the existence of global attractor in an L₂(_Ω)-framework by standard techniques; whereas the case of Ω a subset in R³ and L_p(_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 solutionU(t) = (u,w), that is defined for all timet ≥ 0. However, it is not known whether this solution is uniquely determined by the data. As a result, one cannot conclude that the mappingS(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(L_p(_Ω), 1< p<+_∞), and then discuss to generate a local semiflowS(t)on a fractional phase spaceZ^α_p, afterwards applying adequate estimates, they choose a suitable metric space V ⊂ Z^α_p on which S(t) becomes a dissipative compact semigroup of global solutions. As a consequence, the existence of a global attractor A for S(t) restricted to V will 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 onZ^α_p(defined below) forα∈[1/2, 1)andp∈ (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 (Section2), 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 Section3it 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 Section4the 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 Section5.

In this paper we use the following notations, for 1 ≤ p ≤ +_∞, L_p(_Ω) denotes the usual Lebesgue space overΩ,W^m,p(_Ω)_{the usual}L_p-Sobolev space of orderm, andC₀^∞(_Ω)_{is the set} of all infinitely differentiable functions inΩwith compact support in Ω. For function spaces of vector fields, we use the following symbols

Lp≡_Lp(_Ω) = [L_p(_Ω)]³, W^m,p≡W^m,p(_Ω) = [W^m,p(_Ω)]³. We define

C^∞_0,σ(_Ω)_:= {_v∈_C^∞₀ (_Ω)_{: divv}=_{0 in Ω}}_,

and denote byL^σ_p the closure ofC^∞_0,σ(_Ω)inL_p(_Ω).

For notational simplicity, we denote the norms k · k_Lp(Ω) and k · k_Wm,p(_Ω) by k · k_p and k · k_m,p, respectively. For the differentiation of the vector fieldu = (u₁,u₂,u₃)and the scalar field pwe use the following symbols: ∂_jp= _∂x^∂p

j,∂_tp= ^∂p_∂t,∇p= (∂₁p,∂₂p,∂₃p), and divu=

∑

∂_ju_j,

rotu= (∂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 spaceLp(_Ω)admits the Helmholtz and Weyl decomposition (cf. [9])

L_p =L^σ_p⊕G^p(_Ω), where⊕ denotes direct sum and

G^p(_Ω) ={∇ψ : ψ∈W^1,p(_Ω)}.

Let P = P_p be a continuous projection from Lp(_Ω) intoL^σ_p along with G^p(_Ω). Then the projectionPhas the Lp-boundedness property

kPuk_p ≤C_pkuk_p. DenoteU= (u,w)^>and let us define the linear operator

A_p =

−(ν+χ)_P∆ 0

0 −_µ∆−(µ+σ)∇div

(1.3) with domain

D(A_p) = (

U = u

w

: u∈_W^2,p(_Ω)∩_W^1,p₀ ∩_L^σ_p, w∈W^2,p(_Ω)∩W^1,p₀ (_Ω)

) . Here we have used the fact that Protw=rotw, since div rotw=0 inΩ.

These facts imply that rotw∈L^σ_p, because the spaceL^σ_pis characterized (cf. [9]) as L^σ_p= {u∈Lp(_Ω): divu=0 inΩ, n·u=0 onΩ}.

We setZ_p ≡L^σ_p×L_p, 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)

whereU0= (_u0,w0)^>_,F= (Pf,g)^>, and the nonlinear term is given by NU=

−P(u· ∇)u+χrotw

−(u· ∇w) +χrotu−2χw

.

Finally, suppose thatAis a sectorial operator and Reσ(A)>0 in a Banach spaceZ, and define for eachα≥0,Z^α =D(A^α)with the graph norm kzk_α = kA^αzk, z∈ Z^α. In the next sections, the role of Aoperator will be played by A_p.

2 Preliminaries and background

With the above notation, consider the Cauchy problem (1.4), and concerning this, let us enu- merate some assumptions.

(i) Ap :D(Ap)→Zp is a sectorial and positive operator (Reσ(Ap)>a >0) inZp.

(ii) For certain α ∈ [_{0, 1})_, G : Z^α_p → _Zp is Lipschitz continuous on bounded subsets of Z^α_p =D(A^α_p).

(iii) The resolvent of Ap is compact.

It is known from the results of Henry [13] and Hale [12] that, under the assumptions (i) and (ii), to eachU0∈Z^α_p, a unique localZ^α_p-solutionU=_U(t,U0)corresponds to (1.4) defined on a maximal interval of existence[0,τmax(U₀)). More precisely,Ubelongs toC([0,τmax(U₀)),Z^α_p)∩ C¹((0,τ_max(U₀)),Z_p), U(0) = U₀, U(t) belongs to D(A_p) for eacht ∈ (0,τ_max(U₀)), and the first equation in (1.4) holds in Zp for allt ∈(0,τ_max(_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 Ap satisfies these assumptions, one only needs to prove that Lamé operator L_pw = −_µ∆w−(µ+σ)∇div(w) is a sectorial, positive operator with compact resolvent. We will prove these facts in Section3.

After local existence has been proved (in Section4), one must discuss the global existence of solutions. For instance, a subset V_p^α ⊂ Z^α_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 ∈ V_p^α are defined globally in the time. In addition, the existence of a restricted global attractor for the semigroup{S(t)}restricted toV_p^α (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 Z^α_p. We say that A ⊂ Z^α_p is a restricted global attractor for {S(t)} in Z^α_p if for some closed, nonempty subsetV ofZ^α_p, S(t) : V → V, (t ≥ 0) is a global semiflow on V such thatAis a global attractor for {S(t)} restricted to V as stated in [12], that is, (i) S(t)A = A _for t ≥ _0, (ii)Ais compact, (iii)Aattracts all trajectories starting at bounded subsets ofV.

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 onZ^α_p, and the resolvent of Ap be compact. Then, in order to prove the existence of a restricted global attractor for {S(t)}in Z^α_p, it suffices to show that there exists a Banach space Y ⊃ D(A_p) 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 ofZ^α_p, where

(a) ∃C>0,∀U0 ∈V, ∀t∈ (0,τ_max(U0)),

kS(t)U₀k_Y ≤C.

(b) ∃θ ∈[0, 1), ∀U0 ∈V, ∀t∈(0,τmax(U0)),

kG(S(t)U₀)k_p ≤g(kS(t)U₀k_Y)(1+kS(t)U₀k^θ_Zα p).

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 → V be a C⁰-semigroup onV. Denoting by[B]_V the closure of a set Bin the spaceV, for any setB⊂V the two setsγ⁺(B)andω(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 theω-limit set ofB. Thus, anω-limit set consists of all points v ∈ Vfor which there exist positive numbers tn % +_{∞and points vn} ∈ Bwith S(tn)vn→v asn→+_∞.

Remark 2.3. The above requirements (a) and (b) ensure that local solutions corresponding to U₀ ∈Vexist for allt≥0. If, in addition,U₀ ∈Vimplies thatU(t,U₀)∈Vas long as it exists, then the relation S(t)U₀ = U(t,U₀), defines on V a C⁰-semigroup of globalZ^α_p solutions. By (a), {S(_t)} is point dissipative (that is, there exists a nonempty, bounded set B ⊂ V which attracts every point ofV); and since the resolvent of Apis compact,S(t):V →V is a compact map for eacht>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 operatorL_pv=−_µ∆v−(µ+σ)∇div(v), defined on domain D(L_p) =W^2,p(_Ω)∩W^1,p₀ (_Ω)(1< p <+∞), is a sectorial, positive opera- tor with compact resolvent inLp.

Namely, we consider the following problem:

−µ∆v−(µ+σ)∇_divv+_λv=_h,

v|_∂Ω =0. (3.1)

For this purpose, let us denote by∆ϕ(η), where η∈_R andϕ∈ (0,π/2), the sector of the complex plane given by

∆ϕ(η) =ⁿλ∈_C : |arg(λ−η)|< ^π

2 +ϕ, λ6=η o

.

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 p6=3/2,λ∈_Cwith Re(λ)>0,andh∈L_p.Then, there exists a constant0<_C= _C(_µ,σ)such that for any solutionv∈ D(L_p)_of (3.1), the following inequalities hold:

(|λ|^p−C)kvk^p_p+ ¹

C+1kvk_2,p^p ≤Ckhk^p_p, for p>1, p6=3/2 (3.2) and

kvk_2,p≤ C(|λ|khk₂+khk_p), p ≥2. (3.3) Proof. Definew(x,t) =e^λtv(x)andq(x,t) =e^λth(x). Multiplying the equation (3.1) bye^λt we have thatwsatisfies

∂tw−_µ∆w−(µ+σ)∇divw=q(x,t), w|_∂Ω =_0,

w(x, 0) =_v(x), inΩ.

(3.4)

Let us denote byW^2−2/p,p =W^2−2/p,p(_Ω)the Slobodetskii–Besov space. As proved in [19], we have

kwk⁽_L^2,1)

p(Q_T)+sup

τ≤T

kw(x,τ)k_W2−2/p,p ≤C(kqk_L

p(Q_T)³ +kvk_W2−2/p,p), (3.5) whereQ_T = _Ω×(0,T), and

kwk^(2,1)

Lp(QT)= k∂_twk

Lp(QT)³ +k∇(∇w)k

Lp(QT)²⁷+k∇wk

Lp(QT)⁹+kwk

Lp(QT)³.

Let us recall that W_p^(2,1)(Q_T) is the space of distributions w ∈ L_p(0,T;W^2,p(_Ω)) such that

∂_tw∈ L_p(Q_T). This space, endowed with the norm kwk⁽_L^2,1)

p(Q_T), is a Banach space.

The constant C = C(µ,σ,T) in (3.5) still depends on T and 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 quantityC(µ,σ), which does not depend onT, and we arrive at

Z _T

0

(k∂twk^p_p+kwk_2,p^p )ds+sup

τ≤T

kw(x,τ)k^p

W^2−2/p,p

≤C(µ,σ) _{Z T}

0

(kqk^p_p+kwk^p_p)ds+kvk^p

W^2−2/p,p

, ∀T >0. (3.6) Taking into account the change of variables done at the beginning, we deduce thatv and h appearing in (3.1) satisfy

|λ|^p

Z _T

0

e^pRe(λ)tk_vk^p_pdt+

Z _T

0

e^pRe(λ)tk_vk_2,p^p dt

≤C

kvk^p

W^2,p+

Z _T

0 e^pRe(λ)tkhk^p_pdt+

Z _T

0 e^pRe(λ)tkvk^p_pdt

. Then, we obtain

e^pRe(λ)T−1 pRe(λ)

|λ|^pk_vk^p_p+k_vk_2,p^p ≤C

"

k_vk_2,p^p + ^e

pRe(λ)_T−1

pRe(λ) k_hk^p_p+k_vk^p_p

# . TakingTsuch thatC+1≤ ^{epRe(λ)T−1}

pRe(λ) , we deduce (3.2).

In the case p=3/2, to the normkvk_W2−2/p,p that appears on the right side of (3.6) we must add the term

_Z

Ω

|v|^p (δ(x))^2p−2dx

1/p

(δ(x)is the distance fromxto∂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) bywand integrate over Qt (0≤ t ≤T) to obtain

Z _t

0

((wt,w) +µk∇wk₂)ds+ (µ+σ)

Z _t

0

kdiv(w)k²₂ds=

Z _t

0

(q,w)ds, hence

wt∈ L²(_0,T;L2(_Ω))⊂L²(_0,T;H^−1,2(_Ω))_{, w}∈L²(_0,T;W^1,2₀ (_Ω))_. From this we obtain, withδ>0,

kw(t)k²₂+ (2µ−δ)

Z _t

0

k∇wk₂ds≤ kvk²₂+C(δ)

Z _t

0

kqk²₂ds. (3.7)

On the other hand, from Gagliardo–Nirenberg inequality we have

kw(t)k_p ≤δkw(t)k_2,p+C(δ)kw(t)k₁. (3.8) Returning to (3.6), using (3.8) and inserting (3.7) we have

Z _T

0

(k∂_twk^p_p+kwk_2,p^p )ds+sup

τ≤T

kw(x,τ)k^p

W^2−2/p,p

≤C(µ,σ)

Z _t

0

kqk^p_pds+ Z _t

0

kqk₂ds p/2

+kvk^p

W^2−2/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 solutionv∈W^1,2₀ (_Ω).

Proof. We observe that the problem (3.1) is equivalent to the variational problem (To find v∈ W^1,2₀ (_Ω) such that

µ(v,z) + (µ+σ)(divv, divz) +λ(v,z) = (h,z), ∀z∈W^1,2₀ (_Ω). (3.9) To prove that this problem has a solution, since W^1,2₀ (_Ω)is a separable Hilbert space, we can consider a sequence of elements {z_m}_m≥1 of W^1,2₀ (_Ω)which is free and total inW^1,2₀ (_Ω). For each fixed integer m≥1, we would like to define an approximation solutionvm of (3.9) by

(vm =_∑^m_i=1ξ_i,mzi, ξ_i,m ∈_R,

µ(∇v_m,∇z_k) + (µ+σ)(divv_m, divz_k) +λ(v_m,z_k) = (h,z_k), ∀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) byξ_k,m and sum fromk=1, . . . ,m; this gives µk∇_vmk²+ (µ+σ)kdivvmk²+λk_vmk² = (h,vm), or

µk∇vmk²+ (µ+σ)kdivvmk²+λkvmk²≤ ¹

2µkhk²_H−1+ ^µ

2k∇vmk². Thus, we obtain the a priori estimate

k∇v_mk² ≤Ckhk²_H−1. (3.11) Since the sequence v_m remains bounded in H₀¹(_Ω), there exist somev∈ H₀¹(_Ω)and a subse- quence m⁰ →_{∞such that}

v_m⁰ →_v in the weak topology of H¹₀(_Ω)_{. (3.12)} The injection of H₀¹(_Ω)_intoL²(_Ω)is compact, so we have also

v_m⁰ →v in 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.

Theorem 3.3. Consider p > 1, p6= 3/2. There exists a value η> 0 such that for eachλ ∈ _Cwith Re(λ)> −η, and eachh∈Lp,the problem(3.1)has a unique solutionv∈D(L_p).

Moreover, there also exist constants ϕe∈(0,π/2)and M>0such that the resolvent setρ(−L_p) contains the sector∆ϕ_e(−η),and the following estimate holds

k(λI+L_p)⁻¹k_p ≤ ^M

|λ|+₁^{, λ}∈_∆ϕe(−η). In the terminology of Henry [13],L_p is 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 L₂ is a closed densely defined self-adjoint operator with compact resolvent. Using the Gårding inequality, we can see thatL₂is a positive operator since

δk_vk²_H1(Ω)≤(L₂v,v), ∀_v∈ D(L₂),

with δ > 0. From this, we deduce that the L₂ is a sectorial operator. The spectrum of L₂ consists of isolated positive eigenvalues of finite multiplicity. Numbering them in nonde- creasing order taking into account their multiplicities, we obtain a sequence{λ_j}⁺_j=^∞₁ with the asymptotic behaviourλ_j ∼ _Λ0j^2/3 (Λ0 a positive constant).

Next, for a small enough constante>0 and suitable valuesηandϕe(to be specified later) we show that the subset ∆ϕe(−η) is contained in the resolvent set ρ(−L_p) in several steps.

For this, we define the following sectors in the complex plane (where the constantCis given in Lemma3.1):

∆⁰ = {λ∈_C : Re(λ)>0,|λ| ≥ √^p

C+1},

∆⁰⁰= {λ∈_C : Re(λ)>0,|λ| ≤ √^p

C+1},

∆e=

λ∈_C: |Reλ|

|_Imλ| < ¹ (₁+e)M

. Step I:We prove that∆⁰ ⊂ ρ(−L_p).

First, assume thatp≥2. To solve(λI+L_p)v=hfor anyh∈Lp(_Ω), we use the continuous injection of L_p into L₂. By Lemma 3.2, we can see that (3.1) has a unique weak solution in W^1,2(_Ω). In factv∈W^1,2₀ (_Ω), and it satisfies

λ Z

Ωv·u+

Z

ΩΞ(v):ε(u) =

Z

Ωh·u, ∀u∈W^1,2₀ (_Ω), where

Ξ(v) =σtr(ε(v))I+2µε(v) and

ε(v) =1/2 ∂_jv_i+∂_iv_j .

As long as Ω has smooth boundary, then v ∈ _W^2,2(_Ω)_{. Thus, v} is a strong solution for the system. Using the Sobolev Lemma and interpolation, we have that v ∈ C(_Ω)∩Lp(_Ω), kvk_C(Ω) ≤ CkL₂vk, and therefore L₂v = −λv+h∈ L_p(_Ω). From this, and using [18, Theo- rem 5.1, p. 301], we conclude thatv∈D(L_p)_.

Next we consider the case 1 < p < 2 with p 6= 3/2 and h ∈ L_p(_Ω). We approximate h by a sequence (hn) ∈ C^∞₀(_Ω). Each problem(λI+L_p)vn = hn has a solution, applying the inequality (3.2) to (λI+L_p)(v_n−v_m) =h_n−h_m, we have

(|λ|^p−C)kvn−vmk_p+ ¹

C+1kvn−vmk_2,p ≤Ckhn−hmk_p.

Thus, vn→v∈W^2,p(_Ω)asn%+∞. Therefore, from the inequality (3.2) we deduce k(λI+L_p)vk_p≥ ^p

r|λ|^p−C C kvk_p,

from which, the resolvent(λI+L_p)⁻¹ exists for all p> 1 with p 6=3/2. Using the resolvent series we obtain that∆⁰ is contained in the resolvent set of−L_p and

k(λI+L_p)⁻¹k_p ≤ ²

(p−1)/p

ec

|λ| −_ec , ∀λ∈_∆⁰, whereC= _ec^p (ecis a positive constant).

Step II:Now we prove that∆⁰⁰ ⊂ρ(−L_p)_.

For p≥2, we solve as before(λI+L_p)v=hand from (3.3), the following estimate holds k(λI+L_p)⁻¹k_p ≤C, Reλ>0.

When 1 < p < 2, p 6= 3/2, let us suppose that v₁,v₂ ∈ D(L_p) are two solutions for (λI+L_p)v = h. Using the Sobolev embedding we obtain that v2−v₁ ∈ W^2,1(_Ω) ⊂ L3(_Ω), (λI+L_p)(v₂−v₁) =0. From this we have

L_p(v₂−v₁) =−λ(v₂−v₁) inL₂(_Ω).

By a regularity result in [18], we conclude that v2−_v1 ∈ D(L₂), where uniqueness holds, which implies thatv₂ =v₁.

Now, we claim that there exists a constantγ=γ(λ)>0 such that

k(λI+L_p)_vk_p ≥γ(λ)k_vk_p, ∀_v∈ D(L_p)_{. (3.14)} Suppose this is false. Then there exists vn∈ D(L_p),hn= (λI+L_p)vn ∈Lp(_Ω)such that, for alln,kv_nk_p=1,kh_nk_p≤1/n.

SinceL_pvn =−_λvn+_hn ∈ _Lp, using [18, Lemma 4.4, p. 301], we have that (for certainC⁰ andC⁰⁰)

kv_nk_2,p ≤C⁰(|λ| kv_nk_p+|λ| kh_nk_p+kv_nk_p)≤C⁰⁰.

Thus,k_vnk_W2,p is bounded. Taking a subsequence{_vn0}converging weakly to somev∈D(L_p), we obtain that (λI+L_p)v= 0. By the uniqueness of solution to any problem of type (3.1), it must bev=0, which contradicts kvk_p =1. So, (3.14) holds.

Now, approximate againh∈ _Lp byhn ∈C₀^∞(_Ω), and denote byvnthe unique solution of (λI+L₂)v=hnand write

L₂(v_m−v_n) = (h_m−h_n)−λ(v_m−v_n). Using again [18, Lemma 4.4, p. 301], we have that

k_vm−_vnk_2,p ≤C⁰(k_hm−_hnk_p+|λ| k_vm−_vnk_p)_{. (3.15)}

Since by (3.14), it holds that

γ(λ)k_vm−_vnk_p ≤ k(λI+L_p)(_vm−_vn)k_p,

we conclude thatvn →vinLp(_Ω). From this, together with the inequality (3.15), the sequence v_nconverges inW^2,p(_Ω)to the unique solutionv∈ D(L_p)of(λI+L_p)v=h, which concludes the proof of existence of solution to (3.1).

Finally, using the estimate [18, Lemma 4.4, p. 301] together with (3.14), we have kvk_2,p≤ Ckhk_p+

√p

C+1 δ khk_p

Thus(λI+L_p)⁻¹ exists in∆⁰⁰ and using the compactness of this set, we conclude k(λI+L_p)⁻¹k_p≤C.

Step III:We prove now that{λ∈_C : Re(λ) =0} ⊂ρ(−L_p).

For one such λ we proceed as follows. Indeed, we only need to take care to apply Lemma 3.1, where the condition Re(λ) > 0 appears. Consider a sequence of positive real numbers {ε_n}_n with lim_n→+_∞ε_n ↓ 0. Consider the problem ((λ+ε_n)I+L_p)v_n = h. By the above steps I and II, we have that {_vn} is bounded in W^2,p, whence a subsequence with a weak limitvexists, beingvsolution of(λI+L_p)v=h.

Finally, collecting all estimates in these steps we arrive at k(λI+L_p)⁻¹k_p ≤ ^M

|λ|+1, Reλ≥0.

Step IV:Now we prove that∆e⊂ρ(−L_p). The following argument shows thatρ(−L_p)⊃_∆e for some ϕe ∈ (0,^π₂). We consider the resolvent series and choose µsuch that Imµ = Imλ,

|Reµ| ≤ |Imλ|₍ ¹

1+e)M where λis given in ρ(−L_p)with Reλ=0,e>0. Moreover, k(µI+L_p)⁻¹k_p ≤

+_∞ n

∑

|µ−λ|ⁿk(λI+L_p)⁻¹kⁿ_p⁺¹

=

+_∞ l

∑

|µ−λ|^l−1k(λI+L_p)⁻¹k^l_p

≤

+_∞ l

∑

k(λI+L_p)⁻¹k_p(|µ−λ|k(λI+L_p)⁻¹k_p)^l−1

≤ k(λI+L_p)⁻¹k_p ¹

1− |µ−λ|k(λI+L_p)⁻¹k_p

≤ ^M

|λ|+₁ · ¹ 1− |µ−λ|_| ^M

λ|+1

. Observing that|λ−µ|= |Reµ|, we have

M

|λ|+1|λ−µ| ≤ ^M|Reµ|

|Imλ|+1 ≤ ¹ 1+e,

so 1

1−^|λ_|^−µ|M

λ|+1

≤ ¹+e e .

Using this estimate, we deduce that

k(µI+L_p)⁻¹k_p≤ ^M(1+e) (|λ|+1)e, but

|µ| ≤ |λ| s

1+

1 (1+e)M

2

. From this, we conclude that

k(µI+L_p)⁻¹k_p ≤ q

1+ (₍₁₊¹

e)M)^2M(1_e^+e)

|µ|+^q1+ (₍ ¹

1+e)M)² .

Thus, the resolvent setρ(−L_p)contains the sector∆e for someϕ_e∈ (_0,^π₂)_{, i.e.,} sinϕe≤

1 (1+e)M

q

1+ (₍₁₊¹

e)M) .

So far, we have shown that the resolvent ρ(−L_p) contains the sector ∆ϕe(0) = {λ ∈ _C :

|arg(λ)| ≤ ^π₂ +ϕ_e}.

Since the imaginary axis lies in the resolvent set of−L_p (Step III), we claim that for some η>0, the strip|Re(ξ)|<η,ξ ∈C, also lies the resolvent set of−L_p.

Indeed, if there would be no suchη >0, then there would exist a sequenceξ_k in σ(−L_p) (spectrum of−L_p) such that Reξ_k →0 and since

|Imξ_k| ≤ |Reξ_k|tan(ϕ_e),

a subsequence of {ξ_k}should converge toξ with Reξ =0. This is not possible, sinceσ(−L_p) is closed. Thus, there exists a value η > 0 such that ∆ϕe(−η) ⊂ ρ(−L_p). By [17, Chapter 3, p. 78], the following equivalence holds

∆ϕe(−η)⊂ ρ(−L_p)⇐⇒ _Σγ(η)⊂ρ(L_p)_, where

Σγ(η) ={λ∈_C:|arg(λ−η)|>γ, λ6= η}, γ= ^π 2 −ϕe. From [17, Lemma 31.6] one finds

k(λI− L_p)⁻¹k_p ≤ ^M

|λ−η|^{, λ}∈ _Σγ(η)_. Hence,L_p is sectorial onL_p.

Remark 3.4. Theorem 3.3 shows that −L_p generates an analytic semigroup e^−tLp in Lp(_Ω) where 1 < p < _∞, p 6= 3/2. Taking into account that Reσ(L_p) > δ > 0 for some δ, from Henry [13, Theorem 1.3.4], we also have the following estimates

ke^−tLpk_p≤Ce^−δt, t≥0, kL_pe^−tLpk_p≤C₁te^−δ, t>0, whereCandC₁are positive constants.

Remark 3.5.

(i) In the proof of the above theorem, we suppose that p 6= 3/2 since we do not know if the estimate (3.2) remains valid for p=3/2.

(ii) Let us observe that as a consequence of the facts in the Step I, it follows that if λ ∈ ρ(−L₂), then λ ∈ ρ(−L_p) for each p > 2. Moreover, from [4, Corollary IX.14] we deduce that the eigenfunctions of−L₂ belong toL_p(_Ω). Thus, the spectrumσ(−L₂)is contained in σ(−L_p)and thereforeσ(−L₂) =σ(−L_p). Following the argument in Agmon [2, pp. 131–132], we conclude that it is true for anyp >1.

Lemma 3.6. For0<α<1,1< p<_{∞, p}6=3/2, we have

Y^α = D(L^α_p)⊂ [L_p(_Ω),D(L_p)]_α ⊂W^2α,p(_Ω), where[·,·]_α denotes a complex interpolation space.

Proof. Since ∂Ω is C^∞, by [8] we conclude thatL^it_p, −_∞ < t < ∞, are bounded operators in Lp(_Ω). It is also known (see [21, Section 4.3.1]) that, for bounded set Ω satisfying the cone condition (in particular for the case under study here)

[L_p(_Ω),W^2,p(_Ω)]_α =W^2α,p(_Ω), α∈(0, 1).

Using the definition of the complex interpolation space, we have the embedding [_Lp(_Ω),D(L_p)]_α ⊂[_Lp(_Ω),W^2,p(_Ω)]_α, α∈(0, 1),

concluding the proof.

Remark 3.7. Compactness of the resolvent (λI+L_p)⁻¹ : L_p(_Ω) → L_p(_Ω) follows from the estimate in [18, Theorem 5.1] and the compactness of the embeddingW^2,p(_Ω),→ L_p(_Ω).

4 Existence of solutions and additional estimates

In this section we prove a series of lemmas, which will be required below to ensure the existence of local solutions and moreover, later for the existence of the attractors.

First at all, observe that from the results of the last section, we have that the operator A_p defined in (1.3) is a sectorial operator onZ_p= L^σ_p×L_p. For convenience, we use the notation S_p for the Stokes operators inLp(_Ω). From [7], we have that Re(σ(S₂))≥ (ν+χ)λ₁, where λ₁is the first eigenvalue of −_∆in L₂(_Ω)under homogeneous Dirichlet boundary conditions.

Similarly, from [3] we also have that Re(σ(L₂))≥µλ₁. Using the elliptic regularity theory we conclude that(σ(S_p))≥(ν+χ)λ₁ and Re(σ(L_p))≥µλ₁ for p>1. Thus, we can define

A^α_pU= [S^α_pu,L^α_pw],

the fractional powers of A_pwith α∈ [0, 1], on the domainsZ^α_p= X^α_p×Y^α_p, whereX^α_p =D(S_p^α) andY^α_p= D(L^α_p), and for each p∈(1,+_∞),α∈(0, 1]we have

kA^α_pe^−tApk_L(Zα

p,Zp) ≤C_α,pt^−αe^−λ1δt, (4.1) whereδ=min{(ν+χ),µ}.

Since S_p and L_p have compact resolvents, then the embeddings X^β_p ⊂ X^α_p, Y^β_p ⊂ Y^α_p (₀<α< β<_{1, 1}< p<+_∞)are compact.

Lemma 4.1. Consider the problem(1.4)and assume that p ∈ (3,+_∞), α∈ [1/2, 1),F∈ L_p. Then, the nonlinear term N : Z^α_p → Zp is Lipschitz continuous on bounded sets. In particular, to each U₀ ∈Z^α_p corresponds a unique local solutionU=U(t,U₀)to(1.4)on a maximal interval of existence [0,τ_max(U₀)).

Proof. Firstly, we prove that the nonlinear term N : Z^α_p → Zp is Lipschitz continuous on bounded sets. For this, we follow the argument in Cholewa et al. [5–7]. Let beU= (u,w),V= (_v,w)∈ O_{, where}Ois a bounded set inZ^α_p, then

kN(U)−N(V)k_p ≤ kP[(v· ∇)v−(u· ∇)u]k_Lp +χkrot(w−w)k_Lp +k(v· ∇)w−(_u· ∇)wk_Lp +χkrot(_v−u)k_Lp +2χk_w−_wk_Lp.

Using the Sobolev embedding (e.g. cf. [1]), if mp > N = 3, then W^m,p(_Ω) ,→ L_q(_Ω), for p ≤q≤+∞; choosingm=_{1 we have}

kP(_u· ∇)_vk_p ≤Ck_uk_∞k∇_vk_p≤ck_uk_1,pk_vk_1,p, and consequently

kP[(v· ∇)v−(u· ∇)u]k_Lp ≤Ckvk_1,pkv−uk_1,p+Ckv−uk_1,pkuk_1,p. (4.2) By a similar argument, we obtain

k(v· ∇w−(u· ∇)wk_p ≤Ckwk_1,pkv−uk_1,p+Ckw−wk_1,pkuk_1,p. (4.3) From (4.2) and (4.3) we deduce

k_N(_U)−_N(_V)k_p ≤C(k_uk_1,p,k_vk_1,p,k_wk_1,p,χ) k_v−_uk_1,p+k_w−_wk_1,p.

Now, we estimate the termskv−uk_1,pandkw−wk_1,p. It is known from general results for the Stokes operator (e.g. see [10,11]) that D(S^α_p),→ D(S^1/2_p ) with α ∈ [1/2, 1). Since D(S^1/2_p )is continuously injected inL^σ_p∩W^1,p(_Ω)(cf. Giga–Miyakawa [11, Proposition 1.4]), we conclude that kv−uk_1,p ≤ Ckv−uk_D(Sα

p), ∀α ∈ [1/2, 1). Using Lemma 3.6 with α = 1/2, we have k_w−_wk_1,p ≤ Ck_w−_wk

D(L^1/2_p ) and therefore we conclude thatNis Lipschitz continuous on bounded sets ofZ^α_p.

Now, the existence of local solutions follows from the general results in [13, Chapter 3], mentioned in Section 2. So, we have a local semiflow {S(t)} (where S(t)U₀ = U(t,U₀)) for t∈[_0,τ_max(_U0))of maximal fractional solutions of (1.4) defined onZ^α_p.

Lemma 4.2. Under the above assumptions and notation, the fractional local solution satisfies the estimate(b)in Lemma2.2.

Proof. Observe that

k_N(_U) +_Fk_p ≤ k_uk_1,pk_Uk_1,p+2χk_Uk_1,p+Ck_Fk_p

≤ kUk²_1,p+2χkUk_1,p+CkFk_p. On the other hand,

kUk_1,p =k(u,w)k^1/4_1,pk(u,w)k^3/4_1,p

≤C_pkUk^1/4_Zα

p kUk^3/4

Z^1/2p .