Pullback Attractors for a 2D-Navier-Stokes Model In an Infinite Delay Case

Pullback attractors for a 2D−Navier-Stokes model in an infinite delay case Pedro Mar´ın-Rubio1, Jos´e Real1& Jos´e Valero2 1Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain e-mails: [email protected], [email protected] 2Universidad Miguel Hern´andez, Centro de Investigaci´on Operativa Avda. Universidad s/n, Elche (Alicante), 03202, Spain e-mail: jv[email protected] October 27, 2010 Abstract We prove the existence of solutions for a Navier-Stokes model in dimension two with an external force containing infinite delay effects in the weighted space Cγ(H). After that, under additional suitable assumptions we prove the existence and uniqueness of a stationary solution and the exponential decay of the solution of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions. Keywords: Navier-Stokes equations; infinite delays; pullback attractors. Mathematics Subject Classifications (2000): 35K55, 35Q30, 34D45, 34K20. 1 Introduction and statement of the problem The Navier-Stokes equations govern the motion of usual fluids like water, air, oil, etc. These equations have been the object of numerous works (e.g. see [17, 23] and references cited therein), since the first paper was published by Leray [16]. On other hand, delay terms appear naturally for instance as effects in wind tunnels experiments (cf. [18]). Very recently, Caraballo & Real [4, 5, 6] developed a full theory of existence, stability of solutions and global attractors for Navier-Stokes models including some hereditary characteristics in several ways (fixed, variable and distributed delays). This study has been continued by some other authors, e.g. [22, 9, 19, 21]. While in other fields, such as reaction-diffusion equations (cf. [24]), infinite-delay effects have been considered, to our knowledge it has not been studied so thoroughly for the Navier-Stokes equations yet. 1 Our purpose is to study the following problem. Let Ω ⊂R2be an open and bounded set with boundary Γ (not necessarily smooth) and consider (arbitrary) values τ < T in R and the following functional Navier-Stokes problem:          ∂u ∂t −ν∆u+P2 i=1 ui ∂u ∂xi =f(t)− ∇p+g(t, ut) in (τ, T )×Ω, div u= 0 in (τ, T )×Ω, u= 0 on (τ, T )×Γ, u(τ+r, x) = φ(r, x), r ∈(−∞,0], x ∈Ω. (1) where we assume that ν > 0 is the kinematic viscosity, uis the velocity field of the fluid, pthe pressure, u0the initial velocity field, fa non-delayed external force field, ganother external force containing some hereditary characteristic and φ(s−τ) the initial datum in the interval of time (−∞, τ]. The structure of the paper is the following: in this section we introduce some functional spaces useful for the establishment of the abstract variational formulation of the problem, and some assumptions on the delay operator (and an example of the existence of such kind of operators, as well). In Section 2 we prove the existence and uniqueness of solution for (1) by an energy method among other compactness arguments; we also analyze continuity properties of the solutions with respect to initial data. In Section 3 a simpler model –the stationary problem– is studied, proving the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem toward the stationary solution under suitable assumptions as well. Finally, Section 4 is devoted to generalize the above results on asymptotic behaviour, proving under more general assumptions the existence of pullback attractors in two different frameworks, those of fixed bounded sets and in a universe of time-dependent families defined by a tempered condition. To start with, we consider some usual abstract spaces. Let V=u∈(C∞ 0(Ω))2: divu= 0, and let Hbe the closure of Vin (L2(Ω))2with the norm | · |,and inner product (·,·), where for u, v ∈(L2(Ω))2, (u, v) = 2 X j=1 ZΩ uj(x)vj(x)dx. Also, Vwill be the closure of Vin (H1 0(Ω))2with the norm k·k associated to the inner product ((·,·)),where for u, v ∈(H1 0(Ω))2, ((u, v)) = 2 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. It follows that V⊂H≡H0⊂V0,where the injections are dense and continuous, and, in fact, compact. We will use k·k∗for the norm in V0and h·,·i for the duality between 2 V0and V. Define the trilinear form bon V×V×Vby b(u, v, w) = 2 X i,j=1 ZΩ ui ∂vj ∂xi wjdx∀u, v, w ∈V. Let Xbe a Banach space. The notation BX(a, r) will be used to denote the open ball of center aand radius rin X. Given a function u: (−∞, T ]→X, for each t≤Twe denote by utthe function defined on (−∞,0] by the relation ut(s) = u(t+s), s ∈(−∞,0]. There are several phase spaces which allow us to deal with infinite delays (cf. [11]). One of them is to consider, for any γ > 0, the space Cγ(H) = ϕ∈C((−∞,0]; H) : ∃lim s→−∞ eγsϕ(s)∈H, which is a Banach space with the norm kϕkγ:= sup s∈(−∞,0] eγs|ϕ(s)|. Hereafter we will use this phase space (with a suitable γ > 0) for our problem. In order to state the problem in the correct framework, let us first establish some initial assumptions on some terms in the equation. We will assume that f∈L2(τ, T;V0). For the term g, in which the delay is present, we assume that g: [τ, T]×Cγ(H)→ (L2(Ω))2satisfies (g1) For any ξ∈Cγ(H) the mapping [τ, T]3t7→ g(t, ξ)∈(L2(Ω))2is measurable. (g2) g(·,0) = 0. (g3) There exists a constant Lg>0 such that for any t∈[τ, T ] and all ξ, η ∈Cγ(H), |g(t, ξ)−g(t, η)| ≤ Lgkξ−ηkγ. Remark 1. (i) Condition (g2) is not really a restriction, since otherwise, if |g(·,0)| ∈ L2(τ, T ),we could redefine ˆ f(t) = f(t) + g(t, 0) and ˆg(t, ·) = g(t, ·)−g(t, 0).In this way the problem is exactly the same and ˆ fand ˆgsatisfy the required assumptions. (ii) Conditions (g2) and (g3) imply that |g(t, ξ)| ≤ Lgkξkγ, so that |g(·, ξ)| ∈ L∞(τ, T ). An example of an operator satisfying assumption (g3) is given here. 3 Example 2. We will consider on the Banach space Cγ(H)the operator g: [τ, T]× Cγ(H)→(L2(Ω))2defined as follows: for all t∈[τ, T], ξ ∈Cγ(H)we take g(t, ξ)as the element of (L2(Ω))2given by g(t, ξ) (x) := Z0 −∞ G(t, s, ξ(s) (x))ds, for a.e. x∈Ω, where the function G: [τ, T]×R−×R2→R2satisfies the following assumptions: (a)G(t, s, 0) = 0 ∀(t, s)∈[τ, T ]×R−. (b)There exists a function κ:R−→R+such that kG(t, s, u)−G(t, s, v)kR2≤κ(s)ku−vkR2∀u, v ∈R2,∀(t, s)∈[τ, T]×R−. (c)The function κsatisfies that κ(·)e−(γ+ϑ)·∈L2(R−)for certain ϑ > 0. We check now that gsatisfies the assumption given in (g3), and also, using (a)above, we obtain that it is well defined as a map with values in (L2(Ω))2: |g(t, ξ)−g(t, η)|2 ≤ZΩZ0 −∞ κ(s)kξ(s)(x)−η(s)(x)kR2ds2 dx =ZΩZ0 −∞ κ(s)e−(γ+ϑ)se(γ+ϑ)skξ(s)(x)−η(s)(x)kR2ds2 dx ≤ZΩZ0 −∞ κ2(s)e−2(γ+ϑ)sdsZ0 −∞ e2(γ+ϑ)skξ(s)(x)−η(s)(x)k2 R2dsdx =Z0 −∞ κ2(s)e−2(γ+ϑ)sdsZΩZ0 −∞ e2(γ+ϑ)skξ(s)(x)−η(s)(x)k2 R2dsdx =kκ(·)e−(γ+ϑ)·k2 L2(R−)Z0 −∞ ZΩ e2(γ+ϑ)skξ(s)(x)−η(s)(x)k2 R2dxds ≤ kκ(·)e−(γ+ϑ)·k2 L2(R−)"sup s∈(−∞,0] e2γs ZΩ kξ(s)(x)−η(s)(x)k2 R2dx#Z0 −∞ e2ϑsds =kκ(·)e−(γ+ϑ)·k2 L2(R−)kξ−ηk2 γ 1 2ϑ =L2 gkξ−ηk2 γ. Now we will give the definition of weak solutions for problem (1). Definition 3. A weak solution of (1) in the interval (−∞, T],with initial datum φ∈ Cγ(H),is a function u∈C((−∞, T]; H)∩L2(τ, T;V)with uτ=φsuch that for all v∈V, and t∈(τ, T), d dt(u(t), v) + ν((u(t), v)) + b(u(t), u(t), v) = hf(t), vi+ (g(t, ut), v),(2) where the equation must be understood in the sense of D0(τ, T ). 4 Remark 4. If uis a weak solution of (1) in the sense given above, then usatisfies an energy equality. Namely: |u(t)|2+ 2νZt s ku(r)k2dr =|u(s)|2+ 2 Zt s [hf(r), u(r)i+ (g(r, ur), u(r))]dr ∀s, t ∈[τ, T]. 2 Existence of solutions In this section we establish the existence of weak solutions for (1) by a compactness method using a Faedo-Galerkin scheme. Let us denote λ1= inf v∈V\{0} kvk2 |v|2>0. Our main result is the following Theorem 5. Take γsuch that νλ1<2γ. If f∈L2(τ, T ;V0), g : [τ, T]×Cγ(H)→ (L2(Ω))2satisfying the assumptions (g1)–(g3), and φ∈Cγ(H)are given, then there exists a unique weak solution of (1). Proof. The uniqueness of solution can be obtained in the following way: consider two solutions, uand vof (1) with the same initial data, and denote w=u−v. We note that |b(u(t), u(t), u(t)−v(t)) −b(v(t), v(t), u(t)−v(t))| =|b(u(t), u(t), u(t)−v(t)) ∓b(v(t), u(t), u(t)−v(t)) −b(v(t), v(t), u(t)−v(t))| =|b(w(t), u(t), w(t))|, where we have used the standard property b(v, w, w) = 0. We shall take into account the estimate kuk2 (L4(Ω))2≤2−1/2|u|kuk,(3) used already in [9], which is a slight improvement of Lemma 3.3 in [23, p.291] (the proof follows the same lines of that lemma using a remark given in [15]). Then we have |b(w, u, w)|≤kwk2 (L4(Ω))2kuk ≤2−1/2|w|kwkkuk ∀u, v ∈V. Then, from the equation satisfied by wand the integration by parts formula, we obtain for all t∈[τ, T ] that |w(t)|2+ 2νZt τ kw(s)k2ds =−2Zt τ b(w(s), u(s), w(s))ds + 2 Zt τ (g(s, us)−g(s, vs), w(s))ds ≤21/2Zt τ |w(s)|kw(s)kku(s)kds + 2LgZt τ kwskγ|w(s)|ds. 5 Observe that w(θ) = 0 if θ≤τ. Therefore, kwskγ= sup θ≤0 eγθ|w(s+θ)| ≤sup θ∈[τ−s,0] |w(s+θ)|,for τ≤s≤T. So, it yields that |w(t)|2+ 2νZt τ kw(s)k2ds ≤21/2Zt τ |w(s)|kw(s)kku(s)kds + 2LgZt τ sup r∈[τ,s] |w(r)||w(s)|ds ≤νZt τ kw(s)k2ds +1 2νZt τ ku(s)k2|w(s)|2ds + 2LgZt τ sup r∈[τ,s] |w(r)|2ds, where we have used the Young inequality. Now we deduce that sup r∈[τ,t] |w(r)|2≤1 2ν+ 2LgZt τ (1 + ku(s)k2) sup r∈[τ,s] |w(r)|2ds, whence the Gronwall lemma finishes the proof of uniqueness. For the existence, we split the proof in several steps. Step 1: A Galerkin scheme. Let us consider {vj} ⊂ V, the orthonormal basis of Hof all the eigenfunctions of the Stokes problem in Ω with homogeneous Dirichlet boundary conditions. Denote Vm=span[v1, . . . , vm] and consider the projector Pmu= Pm j=1(u, vj)vj. Define also um(t) = m X j=1 αm,j(t)vj, where the upper script mwill be used instead of (m) for short since no confusion is possible with powers of u, and where the coefficients αm,j are required to satisfy the following system: d dt(um(t), vj) + ν((um(t), vj)) + b(um(t), um(t), vj) =hf(t), vji+ (g(t, um t), vj),1≤j≤m, (4) and where the equations are understood in the sense of D0(τ, T),and the initial conditions are um(τ+s) = Pmφ(s) for s∈(−∞,0]. The above system of ordinary functional differential equations with infinite delay fulfills the conditions for existence and uniqueness of local solution of [12, Th.1.1, p.36]. Next, we will deduce a priori estimates that assure that the solutions do exist for all time [τ, T ]. 6 Step 2: A priori estimates. Multiplying (4) by αm,j,summing up and using Poincar´e’s inequality, we obtain 1 2 d dt|um(t)|2+νλ1 2|um(t)|2+ν 2kum(t)k2 ≤ hf(t), um(t)i+ (g(t, um t), um(t)) ≤ kf(t)k∗kum(t)k+Lgkum tkγ|um(t)| ≤ν 4kum(t)k2+kf(t)k2 ∗ ν+Lgkum tk2 γ. Hence |um(t)|2+Zt τ e−νλ1(t−s)ν 2kum(s)k2ds ≤e−νλ1(t−τ)|u(τ)|2+Zt τ 2e−νλ1(t−s)(kf(s)k2 ∗/ν +Lgkum sk2 γ)ds. (5) Further kum tk2 γ≤max sup θ∈(−∞,τ−t] e2γθ|φ(θ+t−τ)|2, sup θ∈[τ−t,0] e2γθ−νλ1(t−τ+θ)|u(τ)|2 +e2γθ Zt+θ τ 2e−νλ1(t+θ−s)(kf(s)k2 ∗/ν +Lgkum sk2 γ)ds. On the one hand sup θ∈(−∞,τ−t] eγθ|φ(θ+t−τ)|= sup θ≤0 eγ(θ−(t−τ))|φ(θ)| =e−γ(t−τ)kφkγ. On the other, as we are assuming that νλ1<2γ, sup θ∈[τ−t,0] e2γθ−νλ1(t−τ+θ)|u(τ)|2≤e−νλ1(t−τ)|u(τ)|2 and sup θ∈[τ−t,0] e2γθ Zt+θ τ 2e−νλ1(t+θ−s)(kf(s)k2 ∗/ν +Lgkum sk2 γ)ds ≤Zt τ 2e−νλ1(t−s)(kf(s)k2 ∗/ν +Lgkum sk2 γ)ds. Collecting these inequalities we deduce kum tk2 γ≤e−νλ1(t−τ)kφk2 γ+Zt τ 2e−νλ1(t−s)(kf(s)k2 ∗/ν +Lgkum sk2 γ)ds. 7 By the Gronwall lemma we have kum tk2 γ≤e−(νλ1−2Lg)(t−τ)kφk2 γ+ 2ν−1Zt τ e−(λ1ν−2Lg)(t−s)kf(s)k2 ∗ds. Then we obtain the following estimates: there exists a constant C, depending on some constants of the problem (namely, λ1, ν, Lgand f), and on τ, T and R > 0, such that kum tk2 γ≤C(τ, T, R)∀t∈[τ, T],kφkγ≤R, ∀m≥1.(6) In particular, this implies that {um}is bounded in L∞(τ, T;H).(7) Now, it follows from (5) and (6) that e−νλ1(T−τ)ν 2ZT τ kum(s)k2ds ≤ZT τ e−νλ1(T−s)ν 2kum(s)k2ds ≤ |u(τ)|2+ZT τ 2e−νλ1(T−s)(kf(s)k2 ∗/ν +Lgkum sk2 γ)ds ≤R2+ZT τ 2e−νλ1(T−s)(kf(s)k2 ∗/ν +LgC(τ, T, R))ds, (8) so that we conclude the existence of another constant (relabelled the same) C(τ, T, R) such that kumk2 L2(τ,T ;V)≤C(τ, T, R)∀m. (9) Then, as (3) implies |b(u, u, v)|=| − b(u, v, u)| ≤ kuk2 (L4(Ω))2kvk ≤2−1/2|u|kukkvk,∀u, v ∈V, from (4) we obtain k(um)0k∗≤νkumk+ 2−1/2|um|kumk+kfk∗+λ−1/2 1|g(t, um t)|, which combined with Remark 1, (6), (7) and (9) implies that {(um)0}is bounded in L2(τ, T;V0).(10) Step 3: Approximation in Cγ(H)of the initial datum. For the initial datum φ∈Cγ(H),we have used the projections in the Galerkin scheme in Step 1. Let us check that Pmφ→φin Cγ(H).(11) 8 Indeed, if not, there would exist ε > 0 and a subsequence, that we relabel the same, such that eγθm|Pmφ(θm)−φ(θm)|> ε. (12) One can assume that θm→ −∞. Otherwise, if θm→θ, then Pmφ(θm)→φ(θ),since |Pmφ(θm)−φ(θ)| ≤ |Pmφ(θm)−Pmφ(θ)|+|Pmφ(θ)−φ(θ)| → 0 as m→+∞.But with θm→ −∞ as m→+∞,if we denote x= lim θ→−∞ eγθφ(θ),we obtain that eγθm|Pmφ(θm)−φ(θm)| =|Pm(eγθmφ(θm)) −eγθmφ(θm)| ≤ |Pme(γθmφ(θm)) −Pmx|+|Pmx−x|+|x−eγθmφ(θm)| → 0. This is a contradiction with (12), so (11) holds. Step 4: Energy method and compactness results. Now we combine some wellknown compactness results with an energy method to pass to the limit in a subsequence of {um}to obtain a solution of (1). From the assumptions on the operator gand Step 2 we deduce using the Compactness Theorem (cf. [17]) that there exist a subsequence (which we relabel the same) {um},an element u∈L∞(τ, T ;H)∩L2(τ, T;V) with u0∈L2(τ, T;V0),and ξ∈L2(τ, T; (L2(Ω))2) such that            um∗ * u weakly star in L∞(τ, T ;H), um* u weakly in L2(τ, T ;V), (um)0* u0weakly in L2(τ, T;V0), um→ustrongly in L2(τ, T ;H) g(·, um ·)* ξ weakly in L2(τ, T ; (L2(Ω))2). (13) Observe that u∈C([τ, T ]; H) (cf. [23, p.261]). Using (13) we also deduce that we can assume that um(t)→u(t) in H a.e. t ∈(τ, T ),(14) which is not enough for our purposes. However, we can obtain convergence for all t∈[τ, T ] with a little more effort and in a more general sense. Observe that um(t)−um(s) = Zt s (um)0(r)dr in V0,∀s, t ∈[τ, T ], and by (10) we have that {um}is equi-continuous on [τ, T] with values in V0. Since the injection of Vinto His compact, by [1, Th.VI.4, p.90] the injection of H into V0is compact too. So, from (6) and the equi-continuity in V0,by the Ascoli-Arzel`a theorem we have that um→uin C([τ, T ]; V0).(15) This, jointly with (6), allows us to claim that for any sequence {tm} ⊂ [τ, T ],with tm→t, one has um(tm)* u(t) weakly in H, (16) where we have used (15) in order to identify which is the weak limit. 9 Proof. It is analogous to that in [5, Th.3.1]. To be precise, in that reference an abstract constant CΩcorresponding to the continuous injection of H1 0(Ω) into L4(Ω) appears. This constant, which can be obtained from (3), is made explicit here, namely CΩ= (2λ1)−1/4 (see also [23, 15, 9]). Theorem 9. Assume the assumptions in Theorem 5 with gand findependent of time, and suppose moreover that (37) holds. Then there exists a value 0< λ < 2γsuch that, denoting for brevity u(t) = u(t; 0, φ)the solution of (1) with τ= 0 and φ∈Cγ(H),and w(t) = u(t)−u∗,with u∗the unique stationary solution given by Theorem 8, the following estimates hold for all t≥0: |w(t)|2≤e−λt |w(0)|2+Lg 2γ−λkφ−u∗k2 γ,(38) kwtk2 γ≤max e−2γtkφ−u∗k2 γ, e−λt |w(0)|2+Lg 2γ−λkφ−u∗k2 γ.(39) Proof. Considering equations (34) for u(t) and (35) for u∗,one has d dt(w(t), v) + ν((w(t), v)) + b(u(t), u(t), v)−b(u∗, u∗, v) = (g(ut)−g(u∗), v). From the energy equality and the fact that b(u(t), u(t), w)−b(u∗, u∗, w) = b(w(t), u∗, w(t)), combined with the Lipschitz condition on g, and introducing an exponential term eλt with a positive value λto be fixed later on, we obtain d dt(eλt|w(t)|2)≤eλt(λ|w(t)|2−2νkw(t)k2+ 2|b(w(t), w(t), u∗)|+ 2Lgkwtkγ|w(t)|). From (36) and estimating bin the same way as in Proposition 6 we obtain |b(w(t), u∗, w(t))| ≤ (2λ1)−1/2 ν−λ−1 1Lg kfk∗kw(t)k2. Hence, using a Young inequality with δ > 0 to be fixed later on, we conclude that d dt(eλt|w(t)|2) ≤eλt(−2ν+λλ−1 1+2(2λ1)−1/2 ν−λ−1 1Lg kfk∗+δλ−1 1Lg)kw(t)k2+Lg δeλtkwtk2 γ. Therefore, integrating from 0 to t, we have eλt|w(t)|2≤ |w(0)|2+Lg δZt 0 eλskwsk2 γds +(−2ν+λλ−1 1+2(2λ1)−1/2 ν−λ−1 1Lg kfk∗+δλ−1 1Lg)Zt 0 eλskw(s)k2ds. (40) 16 In order to control the term Rt 0eλskwsk2 γds, we proceed as follows. Zt 0 eλs sup θ≤0 e2γθ|w(s+θ)|2ds =Zt 0 eλs max{sup θ≤−s e2γθ|w(s+θ)|2,sup θ∈[−s,0] e2γθ|w(s+θ)|2}ds =Zt 0 max{e−(2γ−λ)skφ−u∗k2 γ,sup θ∈[−s,0] e(2γ−λ)θeλ(s+θ)|w(s+θ)|2}ds. So, if λ≤2γ, using the above equality in (40), we obtain eλt|w(t)|2≤ |w(0)|2+Lg δkφ−u∗k2 γZt 0 e(λ−2γ)sds +−2ν+λλ−1 1+2(2λ1)−1/2kfk∗ ν−λ−1 1Lg +δλ−1 1Lg+Lg(λ1δ)−1 ×Zt 0 max r∈[0,s](eλrkw(r)k2)ds. Observe that the (optimal) choice of δ= 1 makes δ+δ−1minimum and therefore the coefficient of the last integral is negative with a suitable choice of λ∈(0,2γ) by (37). So, we can omit this term and deduce that eλt|w(t)|2≤ |w(0)|2+Lg 2γ−λ(1 −e(λ−2γ)t)kφ−u∗k2 γ, whence (38) follows. Finally, (39) can be deduced in the following way: kwtk2 γ= sup θ≤0 e2γθ|w(t+θ)|2 = max{sup θ∈(−∞,−t] e2γθ|φ(t+θ)−u∗|2,max θ∈[−t,0] e2γθ|w(t+θ)|2} = max{e−2γtkφ−u∗k2 γ,max θ∈[−t,0] e2γθ|w(t+θ)|2}. The second term can be estimated using (38) and that e(2γ−λ)θ≤1 when θ≤0. 4 Existence of pullback attractors Our goal in this section is to prove more general results on the asymptotic behaviour of problem (1) than those shown in the above section. Namely, we will establish for a suitable semi process related to problem (1) that we can assure the existence of a pullback attractor under less restrictive assumptions than those in Theorem 8 and Theorem 9. Observe that if the terms fand gin (1) do not depend on time, the problem becomes autonomous and the notions of pullback attractor and global attractor lead in fact to the same object. 17 In fact, we split the analysis in two subsections. The first one is devoted to pullback attractors of fixed bounded sets, which is the most usual framework. In a second subsection, we extend the previous result to the more recent framework of pullback attractors in a universe of families of time dependent sets with a tempered growth condition, following the ideas of [2, 20]. 4.1 Pullback attractor for fixed bounded sets In order to proceed, we start with some standard notions related to Dynamical Systems. Definition 10. Given a metric space (X, d),a semi process Uon Xis a bi-parametric family of mappings U(t, τ) : X→Xfor −∞ < τ ≤t < +∞,with the following properties: (i) U(t, τ)∈C(X;X)for all t≥τ. (ii) U(τ, τ) =Id (the identity map) for all τ∈R. (iii) U(t, τ) = U(t, r)U(r, τ)for all −∞ < τ ≤r≤t < +∞. Next we recall some useful concepts in order to study the asymptotic behaviour of a semi process. (Although slightly different from the classical ones, this allows us to simplify significantly some requirements in order to ensure the existence of attractors, as can be seen e.g. in [3, Th.17].) Definition 11. For a semi process Udefined on a metric space (X, d),a family b B0= {B0(t) : t∈R}of subsets of Xis said to be pullback absorbing for bounded sets if for any bounded set Bof X, and any tthere exists a time τ(B, t)such that U(t, τ)B⊂B0(t)∀τ≤τ(B, t). The semi process Uis said to be b B0−pullback asymptotically compact if for any t, and any sequences {τn},{xn} ⊂ Xwith τn≤t, limn→+∞τn=−∞,and xn∈B0(τn), the sequence {U(t, τn)xn}is relatively compact in X. Definition 12. Finally, a family A={A(t) : t∈R}is said to be a pullback attractor for the semi process Uif (i) Each A(t)is a compact subset of Xfor all t∈R. (ii) It is invariant, i.e. U(t, τ)A(τ) = A(t)∀τ≤t, (iii) It attracts bounded sets in a pullback sense, i.e. given a bounded Bof X, lim τ→−∞ dist(U(t, τ)B, A(t)) = 0,∀t∈R, where dist(C1, C2)denotes the Hausdorff semi-distance between two sets C1and C2,that is, dist(C1, C2) = sup x∈C1 inf y∈C2 d(x, y). The proof of the following result, which is a slight variant of Theorem 1.1 in [7], can be found in [20]. 18 Theorem 13. Consider a family b B0={B0(t) : t∈R}of nonempty subsets of Xand a process Uon Xthat is b B0-asymptotically compact, and assume also that b B0is pullbackabsorbing for U. Then, the family of sets A={A(t) : t∈R}given by A(t) = [ Bbounded Λ(B, t) X ,(41) where Λ(B, t) := \ s≤t[ τ≤s U(t, τ)B X ∀t∈R, is a pullback attractor for U. Remark 14. Observe that the pullback attractor defined by (41) is the minimal family of closed sets that attracts all bounded sets, i.e. if ˜ A={˜ A(t)}also attracts bounded sets in a pullback sense and ˜ A(t)is closed for all t, then A(t)⊂˜ A(t). We add here a result concerning the connectedness of the global attractor. Proposition 15. Let Xbe a connected metric space. Assume that the semi process U satisfies additionally that for every tand x∈Xthe map (−∞, t]3τ→U(t, τ)xis continuous. If Upossesses a pullback attractor A, then A(t)is connected for every t∈R. Proof. Assume that for some tthe set A(t) is not connected. Then A(t) = A1∪A2, where Aiare compact non-empty disjoint sets, and there exist two open disjoint sets Ui for which Ai⊂Ui. We define the following disjoint sets: Xi={x∈Xi:U(t, τ)x⊂Ui,∀τ≤T(x)}. We shall obtain a contradiction if we prove that Xiare non-empty open sets and X1∪X2= X, as Xis connected. First we state that X1∪X2=X. For an arbitrary x∈Xthere exists T=T(x) such that B:= ∪τ≤TU(t, τ)x⊂U1∪U2. The continuity of (−∞, t]3τ→U(t, τ)ximplies that Bis connected. Hence, either B⊂U1or B⊂U2. Thus, xbelongs to one of the sets Xi. On the other hand, each Xiis non-empty. Indeed, as Aiis non-empty, there exists at least one bounded set Bsuch that Ui∩Λ(B, t)6=∅. There exists T(B) for which U(t, τ)B⊂U1∪U2for all τ≤T. On the other hand, Ui∩Λ(B, t)6=∅implies the existence of a sequence yn=U(t, τn)xn∈U(t, τn)B,τn→ −∞, such that yn∈Ui. If we take nfor which τn≤T, then the continuity of (−∞, t]3τ→U(t, τ)ximplies that xn∈Xi. Thus, Xiis non-empty. Finally, we check that Xiare open sets. Let x∈Xiand Bbe a bounded neighborhood of x. There exists T≥0 such that U(t, τ)B⊂U1∪U2, and also U(t, τ)x∈U1,∀τ≤T. Since the map x→U(t, T)xis continuous, there exists a neighborhood O⊂Bof xsuch that U(t, T)Obelongs to U1.Arguing as before for each y∈O, we get U(t, τ)y∈U1, for all τ≤T. Hence O⊂X1. Since xis arbitrary, Xiis open. Now, by the results of Section 2 we are able to define correctly a semi process Uand obtain some properties according to the above results. 19 Proposition 16. Assume that f∈L2 loc(R;V0)and that g:R×Cγ(H)→(L2(Ω))2 satisfies the assumptions (g1)–(g3) for all τ < T. Suppose also that νλ1<2γ. Then, the bi-parametric family of maps U(t, τ) : Cγ(H)→Cγ(H),with τ≤t, given by U(t, τ)φ=ut where uis the unique solution of (1), defines a semi process on Cγ(H). Proof. It is a consequence of Theorem 5 and Proposition 6. Lemma 17. Under the assumptions of Proposition 16, the following estimates hold for a solution to (1) for all t≥τ: kutk2 γ≤e−(νλ1−2Lg)(t−τ)kφk2 γ+2 νZt τ e−(νλ1−2Lg)(t−s)kf(s)k2 ∗ds, (42) ν 2Zt τ ku(s)k2ds ≤eνλ1(t−τ)|u(τ)|2+e2Lg(t−τ)kφk2 γ+ 2ν−1e−νλ1τZt τ eνλ1skf(s)k2 ∗ds +2ν−1e2Lgt−νλ1τZt τ e(νλ1−2Lg)skf(s)k2 ∗ds. (43) Proof. The uniform estimates that we require for the solutions which define the process Uare analogous to those provided in the proof of Theorem 5, but there with the Galerkin approximations. For the sake of brevity, we only sketch the main ideas: Multiplying (2) by u, by the Young inequality, splitting the term related to the delay in the initial datum and the evolution of the solution for t≥τ, and using that νλ1<2γ, we arrive to kutk2 γ≤e−νλ1(t−τ)kφk2 γ+Zt τ 2e−νλ1(t−s)(kf(s)k2 ∗/ν +Lgkusk2 γ)ds. By the Gronwall lemma we conclude that (42) holds. In the middle of the above manipulations we have omitted a positive term in the left hand side, namely kuk2 L2(τ,T ;V).Indeed, analogously as we did in (8), we have e−νλ1(t−τ)ν 2Zt τ ku(s)k2ds ≤ |u(τ)|2+Zt τ 2e−νλ1(t−s)(kf(s)k2 ∗/ν +Lgkusk2 γ)ds. Combining this with estimate (42), applying the Fubini theorem and rearranging coefficients, we conclude (43). From now on we will assume that 2Lg< νλ1,(44) and Z0 −∞ e(νλ1−2Lg)skf(s)k2 ∗ds < +∞.(45) 20 Remark 18. If we assume that f∈L2 loc(R;V0),assumption (45) is equivalent to Zt −∞ e−(νλ1−2Lg)(t−s)kf(s)k2 ∗ds < +∞,∀t∈R. Corollary 19. Under the assumptions of Proposition 16, if moreover conditions (44) and (45) are satisfied, then the family b B0={B0(t) : t∈R},with B0(t) = BCγ(H)(0, ρ(t)), where ρ2(t) = 1 + 2ν−1Zt −∞ e−(νλ1−2Lg)(t−s)kf(s)k2 ∗ds, (46) is pullback absorbing for bounded sets for the semi process U. Proof. It follows immediately from Lemma 17. Proposition 20. Under the assumptions of Corollary 19, the semi process Uis b B0- asymptotically compact. Proof. Our proof relies on an energy method, analogous to that employed in Step 4 in the proof of Theorem 5. Again, to avoid unnecessary repetitions, we only sketch the main ideas. Let t0∈R, un(·) be a sequence of solutions in their respective intervals [τn, t0],with initial data φn∈B0(τn) = BCγ(H)(0, ρ(τn)), where τn→ −∞ as n→+∞.Consider the sequence ξn=un t0.Then we will prove that this sequence is relatively compact in Cγ(H). Step 1: For brevity let us denote σ=νλ1−2Lg. Consider two arbitrary values 0 < T < T. We will prove that ξn|[−T ,0] is relatively compact in C([−T, 0]; H).It follows from (42) and (45) that there exists n0(t0, T) such that τn≤t0−Tfor n≥n0(t0, T),and kun tk2 γ≤R(t0, T)∀t∈[t0−T, t0],∀n≥n0(t0, T),(47) where R(t0, T) = 1 + 2 νe−σ(t0−T)Zt0 −∞ eσskf(s)k2 ∗ds, so that |un(t)|2≤R(t0, T),∀t∈[t0−T, t0],∀n≥n0(t0, T), kun t0−Tk2 γ≤R(t0, T),∀n≥n0(t0, T).(48) Let yn(·) = un(·+t0−T). Then, for each n≥1 such that τn< t0−T, the function yn(·) is a solution on [0, T] of a similar problem to (1), namely with freplaced by ˜ f(s) = f(t0−T+s) and greplaced by ˜g(s, ·) = g(t0−T+s, ·),and with yn 0=un t0−T, yn T=un t0=ξn. Then kyn 0kγsatisfies the estimate in (48), for all n≥n0(t0, T).From (43) we have kynk2 L2(0,T ;V)≤K(t0, T). 21 Hence, {yn}nis bounded in L∞(0, T;H) and L2(0, T;V), and {(yn)0}nis bounded in L2(0, T;V0). Thus, up to a subsequence (relabelled the same), for some function y(·) we have that yn∗ * y weakly star in L∞(0, T;H), yn* y weakly in L2(0, T;V), (yn)0* y0weakly in L2(0, T;V0), yn→ystrongly in L2(0, T;H), yn(t)→y(t) a.e. t∈(0, T). Moreover, reasoning as in the proof of Theorem 5 we obtain yn(tn)* y(t0) weakly in H, if tn→t0∈[0, T]. Also, by (g3) and (47) we obtain Zt 0 |˜g(s, yn s)|2ds ≤Ct, where C > 0 does not depend either on nor t, and also ˜g(·, yn ·)* ξ weakly in L2(0, T; (L2(Ω))2), Zt s |˜g(r, yn r)|2dr ≤C(t−s), Zt s |ξ(s)|2ds ≤lim inf n→+∞Zt s |˜g(r, yn r)|2dr ≤C(t−s), ∀0≤s≤t≤T. Then, in a standard way, one can prove that y(·) is the unique weak solution to the problem        ut−ν∆u+ (u· ∇)u=−∇p+˜ f(t) + ξ(t), div u = 0, u|∂Ω= 0, u(0, x) = y(0, x), x ∈Ω. (49) Multiplying the equation in (1) by un, the equation in (49) by y, and integrating we obtain the energy inequality 1 2|z(t)|2+νZt s kz(r)k2dr ≤1 2|z(t)|2+Zt s h˜ f(r), z(r)idr +C(t−s),0≤s≤t≤T, where z=ynor z=y. Then the maps Jn, J : [0, T]→Rdefined by J(t) = 1 2|y(t)|2−Zt 0 h˜ f(r), y(r)idr −Ct, Jn(t) = 1 2|yn(t)|2−Zt 0 h˜ f(r), yn(r)idr −Ct, are non-increasing and continuous. Analogously as we did in Step 4 of the proof in Theorem 5, for a fixed t0>0,using a sequence {˜ tk}with ˜ tk%t0,we are able to establish the convergence of the norms, 22 and therefore, jointly with the weak convergence already proved, deduce that yn→yin C([δ, T]; H), for any δ > 0. Now, as we had T > T, we obtain that ξn→ψin C([−T, 0]; H), where ψ(s) = y(s+T), for s∈[−T, 0].Repeating the same procedure for 2T, 3T, etc. for a diagonal subsequence (relabelled the same) we can obtain a continuous function ψ: (−∞,0] →Hand a subsequence such that ξn→ψin C([−T, 0]; H) on every interval [−T, 0]. Moreover, since for a fixed T > 0, un(s+t0),with s∈[−T, 0],satisfies the estimate (48) for any n≥n0(t0, T),it is clear that we also have |ψ(s)|2≤1 + MeσT ,∀s∈[−T, 0],∀T > 0,(50) where M=2 νe−σt0Zt0 −∞ eσskf(s)k2 ∗ds. Step 2: We claim that ξnconverges to ψin Cγ(H).Indeed, we have to see that for every  > 0 there exists nsuch that sup s∈(−∞,0] |ξn(s)−ψ(s)|2e2γs ≤∀n≥n.(51) Fix T>0 such that max{e−2γTε, Meσe(σ−2γ)T} ≤ /8,and take n≥n0(t0, T) such that |ξn(s)−ψ(s)|2e2γs ≤∀s∈[−T,0],and τn≤t0−T,∀n≥n. (This is possible since the convergence of ξnto ψholds in compact intervals of time.) So, in order to prove (51) we only have to check that sup s∈(−∞,−T] |ξn(s)−ψ(s)|2e2γs ≤∀n≥n. By (50) and the choice of T,it is not difficult to check that for all k∈N∪ {0}that for all s∈[−(T+k+ 1),−(T+k)] it holds that e2γs|ψ(s)|2≤e−2γ(T+k)(1 + Meσ(T+k+1)) =e−2γTe−2γk +Meσe(σ−2γ)Tek(σ−2γ) ≤/8 + /8 =/4. So, it suffices to prove the following: sup s∈(−∞,−T] e2γs|ξn(s)|2≤/4∀n≥n. We remember that ξnhas two parts: ξn(s) = φn(s+t0−τn),if s∈(−∞, τn−t0), un(s+t0),if s∈[τn−t0,0]. Thus, the proof is finished if we prove that max (sup s∈(−∞,τn−t0) e2γs|φn(s+t0−τn)|2,sup s∈[τn−t0,−T] e2γs|un(s+t0)|2)≤/4. 23 The first term above can be estimated as follows: sup s≤τn−t0 e2γs|φn(s+t0−τn)|2= sup s≤τn−t0 e2γ(s+t0−τn)e2γ(τn−t0)|φn(s+t0−τn)|2 =e2γ(τn−t0)kφnk2 γ ≤e2γ(τn−t0)ρ2(τn) ≤e2γ(τn−t0)+Me(2γ−σ)(τn−t0)≤/4, thanks to the choice of n.And finally, for the second term, we have sup s∈[τn−t0,−T] e2γs|un(s+t0)|2= sup θ∈[τn−t0+T,0] e2γ(θ−T)|un(t0−T+θ)|2 ≤e−2γTkun t0−Tk2 γ ≤e−2γTR(t0, T) =e−2γT+Me(σ−2γ)T ≤/4, where we have used (48) with T=T. Joining all the above statements we may conclude our main result of this section. Theorem 21. Assume that f∈L2 loc(R;V0)satisfying (45) and g:R×Cγ(H)→(L2(Ω))2 satisfying the assumptions (g1)–(g3) are given. Also, suppose that 2Lg< νλ1<2γ. Then, the semi process Udefined in Cγ(H)associated to (1) has a pullback attractor A={A(t)}. Moreover, every A(t)is connected in Cγ(H). Proof. The existence of the pullback attractor is a direct consequence of Theorem 13, Proposition 16, Corollary 19, and Proposition 20. The connectedness follows from Propositions 15 and 7 and the fact that the space Cγ(H) is connected. 4.2 Pullback D−attractor In this section we briefly recall some concepts of Dynamical Systems when the universe of objects that can be attracted is not fixed but composed of time-depending families (e.g. cf. [2, 20]). This approach is useful in some situations when the proof of existence of the pullback attractor in the sense of Definition 11 is unclear. Although we have obtained successfully Theorem 21, we will check that with the same effort we may obtain also a pullback attractor in this new sense. Let be given D, a nonempty class of sets parameterized in time b D={D(t) : t∈R}, with D(t)⊂Xfor all t∈R.The main concepts, adapted to those in Definition 11, are the following. Definition 22. The semi process Uon Xis said to be pullback D−asymptotically compact if for any t∈R,any b D={D(t) : t∈R}∈D,any sequence τn→ −∞,and any sequence xn∈D(τn),the sequence {U(t, τn)xn}is relatively compact in X. It is said that b D0={D0(t) : t∈R} ∈ D is pullback D−absorbing for the process Uif for any t∈Rand any b D∈ D,there exists a τ0(t, b D)≤tsuch that U(t, τ)D(τ)⊂D0(t)for all τ≤τ0(t, b D). 24 Remark 23. If there exists a family b D0={D0(t) : t∈R}∈Dsuch that the semi process Uon Xis pullback b D0−asymptotically compact (in the sense given in Definition 11), and besides b D0is pullback D−absorbing for the process U, then Uis pullback D−asymptotically compact. Indeed, given any b D∈ D,one only has to introduce a sequence ˜τm→ −∞ and choose suitably a diagonal sequence of times {τn(m)}mrelated to b Dto ensure that the process maps initial data of some D(τn(m))in the appropriate time sections D0(˜τm). Now, we have the following result (cf. [20, Th.18] and also [2, Th.7]): Theorem 24. Assume that b D0={D0(t) : t∈R} ∈ D is pullback D−absorbing for U and also that Uis pullback b D0−asymptotically compact. Then, the family AD={AD(t) : t∈R}⊂P(X)defined by AD(t) = Λ( b D0, t) := \ s≤t[ τ≤s U(t, τ)D0(τ) X , t ∈R, has the following properties: (a)the set AD(t)is compact for any t∈R; (b)ADis pullback D−attracting, i.e. lim τ→−∞ dist(U(t, τ)D(τ),AD(t)) = 0 for all b D∈ D; (c)ADis invariant, i.e. U(t, τ)AD(τ) = AD(t)for all τ≤t; (d)the following equality holds: AD(t) = [ b D∈D Λ( b D, t) X for t∈R. The family AD,called the global pullback D−attractor for the semi process U, is minimal in the sense that if b C={C(t) : t∈R}⊂P(X)is a family of closed sets such that lim τ→−∞ dist(U(t, τ)B(τ), C(t)) = 0, then AD(t)⊂C(t). Remark 25. (i) If we assume that D0(t)is closed for all t∈R,and the family Dis inclusion-closed (i.e. if b D∈ D,and b D0={D0(t) : t∈R}⊂P(X)with D0(t)⊂D(t) for all t, then b D0∈ D), then the pullback D−attractor ADbelongs to D,and is the only family in Dsatisfying properties (a)–(c)above. (ii) When Theorem 24 can be applied, and the universe Dcontains all bounded subsets of X, then the family Agiven by (41) is a pullback attractor for the semi process Uin the sense of Definition 11, and the following relation exists between Aand the attractor ADgiven in Theorem 24: A(t)⊂ AD(t),∀t∈R. If, moreover, for some T∈Rthe set ∪t≤TD0(t)is bounded, then A(t) = AD(t),∀t≤T. 25