Pullback Attractors for Three-Dimensional Non-Autonomous Navier-Stokes-Voigt Equations

Pullback attractors for three dimensional non-autonomous Navier-Stokes-Voigt equations Julia Garc´ıa-Luengo, Pedro Mar´ın-Rubio∗ , Jos´e Real Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla (Spain) e-mails: [email protected] [email protected] [email protected] December 13, 2011 Abstract In this paper we consider a non-autonomous Navier-Stokes-Voigt model, to which a continuous process can be associated. We study the existence and relationship between minimal pullback attractors for this process in two different frameworks, namely, for the universe of fixed bounded sets, and also for another universe given by a tempered condition. Since the model does not have a regularizing effect, to obtaining asymptotic compactness for the process is a more involved task. We prove this in a relatively simple way just by using an energy method. Our results simplify –and in some aspects generalize– some of those obtained previously for the autonomous and non-autonomous cases, since for example in Section 4, regularity is not required for the boundary of the domain and the force may take values in V0.Under additional suitable assumptions, regularity results for these families of attractors are also obtained, via bootstrapping arguments. Finally, we also conclude some results concerning the attraction in the D(A) norm. Keywords: Navier-Stokes-Voigt system; pullback attractors. Mathematics Subject Classifications (2010): 35Q30, 35Q35, 35B40, 35B41, 76F20. 1 Introduction and setting of the problem Let Ω ⊂R3be a bounded domain with, unless otherwise indicated, smooth enough (e.g., C2) boundary ∂Ω. We consider the following problem for a system of non-autonomous Navier-Stokes-Voigt equations,            ∂ ∂t u−α2∆u−ν∆u+ (u· ∇)u+∇p=f(t) in Ω ×(τ, ∞), div u= 0 in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ) = uτ(x), x ∈Ω, (1.1) where u= (u1, u2, u3) is the unknown velocity field of the fluid and pis the unknown pressure, and we are given the kinematic viscosity ν > 0, a length scale parameter α > 0, characterizing the elasticity of the fluid (in the sense that the ratio α2/ν describes the reaction time that is required for the fluid to respond to the applied force), an initial velocity field uτat the initial time τ∈R,and an external ∗Corresponding author. Tel.: +34 954559909; fax: +34 954552898. 1 force term f, depending on time. The Navier-Stokes-Voigt (NSV) model of viscoelastic incompresible fluid was introduced by Oskolkov in [20], gives an approximate description of the Kelvin-Voigt fluid (see [21, 14]), and recently was proposed as a regularization of the 3D-Navier-Stokes equation for the purpose of direct numerical simulations in [1]. The extra regularizing term −α2∆utchanges the parabolic character of the equation, which makes it so that in 3D the problem is well-posed (forward and backward), but one does not observe any immediate smoothing of the solution, as expected in parabolic PDEs. Moreover, the generated semigroup is only asymptotically compact, similarly to damped hyperbolic systems. One of the studied topics about the problem is the inviscid question in some different senses. It is also worth to observe that when ν= 0,the inviscid equation that one recovers is the simplified Bardina subgrid scale model of turbulence. The relationship between the original and inviscid models was also addressed in [1]. On other hand, some questions on the inviscid regularization have been recently used for the study of a 2D surface quasi-geostrophic model (cf. [12]). The long-time dynamics of the autonomous model was studied by Kalantarov [9] and Kalantarov and Titi [11]. Namely, the existence of global compact attractor was proved, and estimates on its fractal and Hausdorff dimensions, and upper bounds on the number of determining modes were given. Other related results are the G´evrey regularity of the global attractor (again for the autonomous model) when the force term is analytic of G´evrey type, and the establishment of similar statistical properties (and invariant measures) as the 3D-Navier-Stokes equations (cf. [10, 15, 22]). On the other hand, the analysis of data in real applications indicates in general that the force term in applied fluid mechanic problems might not be autonomous, but non-autonomous. In this sense, there exist some different approaches in order to study the asymptotic behaviour, as that of uniform attractors (e.g., cf. [8] and [4, 26] and the references therein for earlier papers); trajectory attractors (again cf. [4] and the references therein); and more recently cocycle and pullback attractors (e.g., cf. [24, 25, 6, 13]) for both deterministic and random cases. Our main goal in this paper is to obtain sufficient conditions such that the minimal pullback attractors for the process associated to (1.1) do exist. This can be done in different universes (see [18] for a comparison of these concepts, and [17, 19, 7] for some applications), namely, the classical one of fixed bounded sets, and more recently, in the framework of a general class of families, composed by time-dependent sets, given by a tempered growth condition at −∞. As commented before, the difference of this model in comparison with the standard 2D-Navier- Stokes (NS) model is that there exists a regularizing effect in the Navier-Stokes model (in 2D), while not here. For NS a continuous energy method can be applied thanks to the extra estimates that holds in higher norms (e.g., cf. [19]), which does not seem to hold for the Navier-Stokes-Voigt model. Some of the proofs in the previously cited references about NSV (e.g., cf. [11]) rely on splitting the problem in two, one with exponential decay, and the other with good asymptotic properties in the domain of a suitable fractional power of the Stokes operator. However, we will provide a simpler proof, which does not require the above-mentioned technicalities, but a sharp use of the energy equality, and the energy method used by Rosa in [23]. Moreover, it is worth to point out that our results in Section 4 do not use the regularity assumption on ∂Ω at all, and the force term may take values in V0instead of in L2 as appears in [11]. We may also cite in this non-autonomous framework the paper [29], where the existence of uniform attractor for a Navier-Stokes-Voigt model is studied. However, there appears the same treatment with 2 the fractional powers of the Stokes operator, and they require more regularity in the non-autonomous case that we need here. As a second goal, we analyze some additional properties of the obtained attractor. Namely, extra regularity is deduced by using a bootstrapping argument, which now does rely on fractional powers of the Stokes operator, similarly as done in [11] for the autonomous case. Attraction in D(A) norm is also proved by using the energy method as before and previous results on strong solutions. The structure of the paper is the following. In Section 2 we recall some definitions of classical abstract functional spaces, useful for the establishment of our problem in an abstract setting. For completeness, we include the proof on existence and uniqueness of weak solution for problem (1.1), and a regularity property. Two continuous dependence results with respect to the initial datum, in the strong and weak senses, are also provided. In Section 3 we present a brief summary on abstract results in order to ensure the existence of minimal pullback attractor in a general universe. Moreover, we point out some relations between two possible families of attractors, each of them associated to the two cited universes, that of fixed bounded sets, and another one given by a tempered condition. This will be applied in Section 4. Namely, we prove there that the conditions in order to ensure the existence of pullback attractors in each suitable universe are fulfilled. To be more precise, both –pullback absorbing and pullback asymptotic compactness properties– are obtained from a rather general condition on the V0norm of f, square integrable in (−∞,0) with an exponential weight. As a consequence the two announced families of pullback attractors, and relations among them, are obtained. In Section 5 regularity results for the obtained attractors will be deduced thanks to splitting the solution in sum of two for two different problems, using carefully a bootstrapping argument that involves fractional powers of the Stokes operator. Finally, in Section 6 the problem of attraction in D(A) norm is studied, and indeed under suitable assumptions, all attractors are proved to coincide. 2 Existence and uniqueness of solution In this section we analyze existence, uniqueness, and regularity properties of the solutions to problem (1.1). At least, part of these results may be found in [20], but for convenience of the reader, they are developed here. In order to proceed, we need initially to pose the problem in an abstract setting, recalling some definitions of functional spaces, operators, and some of their properties (for the details see [28]). To start, we consider the usual spaces in the variational theory of Navier-Stokes equations: H, the closure of V={u∈(C∞ 0(Ω))3: div u= 0}in (L2(Ω))3with norm |·| ,and inner product (·,·),and V, the closure of Vin (H1 0(Ω))3with norm k·k ,and inner product ((·,·)). We will use k·k∗for the norm in V0and h·,·i for the duality hV0, V i.We consider every element h∈Has an element of V0, given by the equality hh, vi= (h, v) for all v∈V. It follows that V⊂H⊂V0,where the injections are dense and compact. Define the linear continuous operator A:V→V0as hAu, vi= ((u, v)) ∀u, v ∈V. Let us denote D(A) = {u∈V:Au ∈H}. Observe that by the regularity of ∂Ω, one has that D(A) = (H2(Ω))3∩V, and Au =−P∆ufor all u∈D(A),is the Stokes operator (Pis the orthoprojector from (L2(Ω))3onto H). The set of eigenvalues of A, repeated according to their multiplicities, is an infinite sequence {λj}j≥1, with 0 < λ1≤λ2≤. . . , and limj→∞ λj=∞.We will denote by {wj}j≥1⊂D(A) the Hilbert basis of Hof all the normalized eigenfunctions of the Stokes operator A (Awj=λjwj, and |wj|= 1). For the fractional powers of A, we have the following inclusions with continuous injection (cf. [27, 3 Ch.III, Lem.2.4.2, Lem.2.4.3]) D(Aβ)⊂(L6/(3−4β)(Ω))3,∀0≤β < 3/4,(2.1) D(A3/4)⊂(Lp(Ω))3,∀1≤p < ∞,(2.2) and D(Aβ)⊂(L∞(Ω))3,∀3/4< β ≤1.(2.3) Let us define b(u, v, w) = 3 X i,j=1 ZΩ ui ∂vj ∂xi wjdx, for every functions u, v, w : Ω →R3for which the right-hand side is well defined. In particular, bhas sense for all u, v, w ∈V, and is a continuous trilinear form on V×V×V, i.e., there exists a constant C1>0 such that |b(u, v, w)| ≤ C1kukkvkkwk ∀u, v, w ∈V. (2.4) Important properties concerning bare that b(u, v, w) = −b(u, w, v),∀u, v, w ∈V, (2.5) b(u, v, v)=0,∀u, v ∈V, (2.6) and, using Agmon inequality (e.g., cf. [5]), we can assure that there exists a constant C2>0 such that |b(u, v, w)| ≤ C2|Au|1/2kuk1/2kvk|w|,∀u∈D(A), v ∈V, w ∈H. (2.7) For any u∈V, we will use B(u) to denote the element of V0given by hB(u), wi=b(u, u, w),∀w∈V. Thus, by (2.4), kB(u)k∗≤C1kuk2,∀u∈V, (2.8) and in particular, by (2.7) and the identification of H0with H, if u∈D(A), then B(u)∈H, with |B(u)| ≤ C2|Au|1/2kuk3/2,∀u∈D(A).(2.9) In fact, from (2.3), one also deduces that if u∈D(Aβ) with 3/4< β ≤1, then B(u)∈H, and more exactly |B(u)| ≤ C(β)|Aβu|kuk,∀u∈D(Aβ),∀3/4< β ≤1.(2.10) Analogously, if 0 ≤β < 3/4,from (2.1) one obtains that if u∈D(Aβ)∩V,B(u)∈D(Aβ−3/4), and more exactly |Aβ−3/4B(u)| ≤ C(β)|Aβu|kuk,∀u∈D(Aβ)∩V, ∀0< β < 3/4.(2.11) Finally, in the case β= 3/4,from (2.2) one can see that if u∈D(A3/4), then B(u)∈D(A−δ) for all δ > 0, and more exactly |A−δB(u)| ≤ C(3/4,δ)|A3/4u|kuk,∀u∈D(A3/4),∀δ > 0. Before studying (1.1), we treat the autonomous equation u+α2Au =g. From Lax-Milgram lemma, we know that for each g∈V0there exists a unique ug∈Vsuch that ug+α2Aug=g. (2.12) 4 The mapping C:u∈V7→ u+α2Au ∈V0is linear and bijective, with C−1g=ug.From (2.12), one has |ug|2+α2kugk2≤ kgk∗kugk,and in particular, kugk ≤ α−2kgk∗,i.e., kC−1gk ≤ α−2kgk∗,∀g∈V0.(2.13) Observe that by the definition of D(A), we also have that C−1(H) = D(A),and reasoning as for the obtention of (2.13), we deduce that |Aug|=α−2|g−ug| ≤2α−2|g|,∀g∈H. (2.14) Assume that uτ∈Vand f∈L2 loc(R;V0). Definition 1 It is said that uis a weak solution to (1.1) if ubelongs to L2(τ, T;V)for all T > τ, and satisfies d dt(u(t) + α2Au(t)) + νAu(t) + B(u(t)) = f(t),in D0(τ, ∞;V0),(2.15) and u(τ) = uτ.(2.16) Remark 2 If u∈L2(τ, T ;V)for all T > τ and satisfies (2.15), then the function vdefined by v(t) = u(t) + α2Au(t)t > τ, (2.17) belongs to L2(τ, T;V0)for all T > τ, and by (2.8),v0=dv dt ∈L1(τ, T ;V0)for all T > τ. Consequently, v∈C([τ, ∞); V0),and therefore, by (2.13),u∈C([τ, ∞); V).In particular, (2.16) has a sense. Moreover, again by (2.8) and (2.15),v0∈L2(τ, T ;V0)for all T > τ, and therefore, as u0=C−1v0, we deduce that u0∈L2(τ, T;V)for all T > τ. From these considerations, it is clear that uis a weak solution to (1.1) if and only if u∈C([τ, ∞); V), u0∈L2(τ, T ;V)for all T > τ, and u(t) + α2Au(t) + Zt τ (νAu(s) + B(u(s))) ds =uτ+α2Auτ+Zt τ f(s)ds (equality in V0), for all t≥τ. We have the following energy equality for the solutions to (1.1). Lemma 3 If uis a weak solution to (1.1), then 1 2 d dt(|u(t)|2+α2ku(t)k2) + νku(t)k2=hf(t), u(t)i,a.e. t > τ. (2.18) Proof. We know from Remark 2 that u∈W1,2(τ, T ;V) and v∈W1,2(τ, T ;V0) for all T > τ, where vis given by (2.17). Thus, d dthv(t), u(t)i=hv0(t), u(t)i+hv(t), u0(t)i,a.e. t > τ. (2.19) But observing that Cis self-adjoint, and using the fact that v(t) = Cu(t) and v0(t) = Cu0(t),we have hv(t), u0(t)i=hv0(t), u(t)i.Therefore, by (2.19), we have d dthv(t), u(t)i= 2hv0(t), u(t)i,a.e. t > τ. From this identity, taking into account (2.6) and (2.15), we obtain (2.18). With respect to the existence and uniqueness of solution to (1.1), we have the following result. 5 Theorem 4 Let f∈L2 loc(R;V0)be given. Then, for each τ∈Rand uτ∈V, there exists a unique weak solution u=u(·;τ, uτ)of (1.1). Moreover, if f∈L2 loc(R;H)and uτ∈D(A), then the weak solution u=u(·;τ, uτ)of (1.1) satisfies u∈C([τ, ∞); D(A)),u0∈L2(τ, T;D(A)) for all T > τ, (2.20) and 1 2 d dt(ku(t)k2+α2|Au(t)|2) + ν|Au(t)|2+ (B(u(t)), Au(t)) = (f(t), Au(t)),a.e. t > τ. (2.21) Proof. We divide the proof in four steps, according to the claims of existence, uniqueness, regularity, and the energy equality (2.21). Uniqueness. Let u(1) and u(2) be two weak solutions to (1.1), corresponding to the same data f,τand uτ. Let us denote ˆu=u(1) −u(2),and ˆv= ˆu+α2Aˆu. We have that ˆv∈C([τ, ∞); V0), with ˆv(t) = −νZt τ Aˆu(s)ds −Zt τ (B(u(1)(s)) −B(u(2)(s))) ds for all t≥τ. (2.22) Observe that by (2.4), kB(u(1)(s)) −B(u(2)(s))k∗ = sup w∈V,kwk=1 |b(u(1)(s)−u(2)(s), u(1)(s), w)−b(u(2)(s), u(2)(s)−u(1)(s), w)| ≤C1(ku(1)(s)k+ku(2)(s)k)ku(1)(s)−u(2)(s)k. Thus, if we fix an arbitrary T > τ, and denote RT=C1maxs∈[τ,T ](ku(1)(s)k+ku(2)(s)k),we have kB(u(1)(s)) −B(u(2)(s))k∗≤RTku(1)(s)−u(2)(s)kfor all s∈[τ, T ].(2.23) Then, as kAˆu(s)k∗=kˆu(s)k,from (2.22) and (2.23) we deduce that kˆv(t)k∗≤(ν+RT)Zt τ kˆu(s)kds for all t∈[τ, T], and therefore, by (2.13), kˆu(t)k ≤ α−2(ν+RT)Zt τ kˆu(s)kds for all t∈[τ, T]. From this inequality, by Gronwall’s lemma, we deduce that kˆu(t)k= 0 for all t∈[τ, T],and therefore, the uniqueness of weak solution to (1.1) holds. Existence. We can prove the existence of weak solution to (1.1) reasoning as in [1, pp. 844–846], but then with this method of proof, we do not know how to obtain the regularity result (2.20). We will proceed by using a Galerkin scheme. Let {wj}j≥1⊂D(A) be the Hilbert basis of Hformed by all the normalized eigenfunctions of the Stokes operator Aintroduced before. Let f∈L2 loc(R;V0), τ∈R,and uτ∈V, be given. 6 For each integer m≥1,let define um(t) = m X j=1 γm,j(t)wj, where the coefficients γm,j are required to satisfy the system d dt(um(t) + α2Aum(t), wj) = −hνAum(t) + B(um(t)) −f(t), wji,a.e. t>τ, 1 ≤j≤m, (2.24) and the initial condition um(τ) = Pmuτ, where Pmuτ=Pm j=1(uτ, wj)wj,is the orthogonal (in Hand in V) projection of uτonto the space Vm=span[w1, . . . , wm]. The above system of ordinary differential equations fulfills the conditions of the Picard theorem for existence and uniqueness of local solution. Next, we will deduce a priori estimates that assure that the solutions umdo exist for all time t∈[τ, ∞). Multiplying in (2.24) by γm,j(t),summing from j= 1 to j=m, and taking into account (2.6), we obtain that a.e. t > τ, d dt(|um(t)|2+α2kum(t)k2)+2νkum(t)k2= 2hf(t), um(t)i ≤νkum(t)k2+ν−1kf(t)k2 ∗, and in particular, |um(t)|2+α2kum(t)k2≤ |Pmuτ|2+α2kPmuτk2+ν−1Zt τ kf(s)k2 ∗ds, for all t≥τ, and any m≥1. Observe that as uτ∈V, one has that |Pmuτ|≤|uτ|,kPmuτk≤kuτk,and lim m→∞ kuτ−Pmuτk= 0. Thus, the sequence {um}m≥1is bounded in C([τ, T ]; V) for all T > τ. Now observe that by (2.24), vm=Cumsatisfies d dt(vm(t)) = ˜ Pm(−νAum(t)−B(um(t)) + f(t)),a.e. t > τ, (2.25) where h˜ Pmg, wi=hg, Pmwi ∀g∈V0, w ∈V. Consequently, as k˜ PmkL(V0)≤1 for all m≥1,we deduce that the sequence {dvm/dt}m≥1is bounded in L2(τ, T ;V0) for all T > τ, and therefore, taking into account that dum/dt =C−1(dvm/dt), we have that the sequence {dum/dt}m≥1is bounded in L2(τ, T;V) for all T > τ. Thus, by the compactness of the injection of Vinto Hand the Ascoli-Arzel`a theorem, we deduce that there exist a subsequence {um0}m0≥1⊂ {um}m≥1and a function u∈W1,2(τ, T ;V) for all T > τ, such that                                um0∗ * u weakly-star in L∞(τ, T;V), um0→ustrongly in C([τ, T ]; H), um0→ua.e. in Ω ×(τ, T), dum0 dt *du dt weakly in L2(τ, T ;V), dvm0 dt =C dum0 dt !*Cdu dt weakly in L2(τ, T ;V0), (2.26) 7 for all T > τ. As in particular H1 0(Ω) ⊂L4(Ω) with continuous injection, for each 1 ≤i, j ≤3, the product um0 ium0 jof the corresponding components of um0is bounded in L∞(τ, T;L2(Ω)),for all T > τ, and by (2.26), um0 ium0 j→uiuja.e. in Ω ×(τ, T). So, by [16, Ch.1, Lem.1.3], we deduce that um0 ium0 j* uiuj weakly in L2(Ω ×(τ, T )),for all T > τ. Therefore, taking into account (2.5), if w∈L2(τ, T;V), ZT τ hB(um0(t)), w(t)idt =−ZT τ b(um0(t), w(t), um0(t)) dt =− 3 X i,j=1 ZT τZΩ um0 i(x, t)um0 j(x, t)∂wj ∂xi (x, t)dx dt → − 3 X i,j=1 ZT τZΩ ui(x, t)uj(x, t)∂wj ∂xi (x, t)dx dt =ZT τ hB(u(t)), w(t)idt. Hence, B(um0)* B(u) weakly in L2(τ, T;V0),for all T > τ. From all the convergences above, and (2.25), we can take limits and we obtain that usatisfies (2.15). Observe that u(τ) = lim m0→∞ um0(τ) = lim m0→∞ Pm0uτ=uτ.Thus, uis the weak solution to (1.1). Regularity. Assume now that uτ∈D(A) and f∈L2 loc(R;H). Multiplying in (2.24) by λjγm,j(t),and summing from j= 1 to j=m, we obtain that a.e. t > τ, d dt(kum(t)k2+α2|Aum(t)|2)+2ν|Aum(t)|2=−2 (B(um(t)), Aum(t)) + 2 (f(t), Aum(t)) .(2.27) But by (2.9) and Young inequality, 2|(B(um(t)), Aum(t)) | ≤ Cνkum(t)k6+ν|Aum(t)|2, where Cν= 27C4 2(16ν3)−1. Also, 2|(f(t), Aum(t)) | ≤ ν|Aum(t)|2+ν−1|f(t)|2. Thus, observing that |APmuτ|≤|Auτ|and kPmuτk≤kuτk,from (2.27) we deduce in particular that α2|Aum(t)|2≤ kuτk2+α2|Auτ|2+ν−1Zt τ |f(s)|2ds +Cν(t−τ) sup s∈[τ,t] kum(s)k6, for all t≥τ, and any m≥1. Consequently, as {um}m≥1is bounded in C([τ, T ]; V),we have that {um}m≥1is bounded in C([τ, T ]; D(A)),for all T > τ, and therefore, extracting a subsequence weakly-star convergent in L∞(τ, T ;D(A)),we see that u∈L∞(τ, T;D(A)),for all T > τ. But then, v=u+α2Au ∈L∞(τ, T ;H),with v0=−νAu −B(u) + f∈L2(τ, T;H),for all T > τ, and therefore, v∈C([τ, ∞); H). Thus, Au =α−2(v−u)∈C([τ, ∞); H),i.e., u∈C([τ, ∞); D(A)). Moreover, as v0∈L2(τ, T;H),by (2.14), then u0=C−1v0∈L2(τ, T;D(A)),for all T > τ. 8 Identity (2.21). If uτ∈D(A) and f∈L2 loc(R;H), we have seen that u∈W1,2(τ, T ;D(A)) and v=Cu∈ W1,2(τ, T ;H),for all T > τ. Then, d dt|v(t)|2= 2(v0(t), v(t)),a.e. t > τ, and d dt(u(t), v(t)) = 2(u(t),Cu0(t)) = 2(u(t), v0(t)),a.e. t > τ. Thus, d dt(Au(t), v(t)) = α−2d dt(v(t)−u(t), v(t)) = 2(v0(t), Au(t)),a.e. t > τ. From this equality, we have (2.21). Remark 5 Observe that in the above proof, using the uniqueness of solution for the problem, for any T > τ the whole sequence of the Galerkin approximations satisfies that umconverges to uin C([τ, T ]; H), and actually, all convergences in (2.26), except the third one, hold for the whole sequence. Analogously, one also deduces that for any t∈[τ, T],um(t)* u(t)weakly in V. Moreover, if uτ∈D(A)and f∈L2 loc(R;H), then in fact for any T > τ the sequence umconverges to uin C([τ, T ]; V),and weakly-star in L∞(τ, T;D(A)),for any t∈[τ, T], um(t)* u(t)in D(A), and the sequence dum/dt converges to du/dt weakly in L2(τ, T ;D(A)). Now, we establish a result on the sequential weak continuity of the solutions to (1.1) with respect to the initial datum uτ. Theorem 6 Let f∈L2 loc(R;V0)and τ < t be given. Consider a sequence {uτ,n} ⊂ Vweakly converging to uτin V. Then, the following convergences hold for the sequence of solutions u(·;τ, uτ,n)toward the solution u(·;τ, uτ). u(·;τ, uτ,n)∗ * u(·;τ, uτ)weakly-star in L∞(τ, t;V), u(·;τ, uτ,n)→u(·;τ, uτ)strongly in C([τ, t]; H), u(t;τ, uτ,n)* u(t;τ, uτ)weakly in V. Moreover, if f∈L2 loc(R;H)and the sequence {uτ,n} ⊂ D(A)converges weakly to uτin D(A),then, in fact, u(·;τ, uτ,n)∗ * u(·;τ, uτ)weakly-star in L∞(τ, t;D(A)), u(·;τ, uτ,n)→u(·;τ, uτ)strongly in C([τ, t]; V), u(t;τ, uτ,n)* u(t;τ, uτ)weakly in D(A). Proof. The proof can be done analogously to that of Theorem 4, since the a priori estimates follow exactly the same. The fact that the whole sequence satisfies the above convergences is a consequence of the uniqueness of solution for the problem (cf. Remark 5). 9 and          z∈C([τ, ∞); V), d dt(z(t) + α2Az(t)) + νAz(t) = f(t)−B(u(t)),in D0(τ, ∞;V0), z(τ)=0, (5.2) respectively. The existence and uniqueness of weak solution to (5.1) and to (5.2) can be obtained reasoning as in the proof of Theorem 4. For the problem (5.1) we have the following result. Lemma 25 For any τ∈R, uτ∈Vand σfulfilling the assumption (4.2), the solution y=Y(·, τ)uτ of (5.1) satisfies kY(t, τ)uτk2≤(1 + α−2λ−1 1)eσ(τ−t)kuτk2for all t≥τ. (5.3) Proof. It is analogous to the proof of (4.3), and we omit it. For the study of the problem (5.2), we will make use of the following lemma. Lemma 26 Assume that g∈L2 loc(R;D(A−β)) with 0≤β≤1/2. Then, for each τ∈Rand σ satisfying the assumption (4.2), the unique solution zof the problem          z∈C([τ, ∞); V), d dt(z(t) + α2Az(t)) + νAz(t) = g(t),in D0(τ, ∞;V0), z(τ)=0, (5.4) satisfies z∈C([τ, ∞); D(A1−β)),(5.5) and |A1−βz(t)|2≤α−2ε−1Zt τ eσ(s−t)|A−βg(s)|2ds for all t≥τ, (5.6) where εis given by (4.4). Proof. We give a formal proof, the rigorous one should be made using the Galerkin approximations constructed with the basis {wj}j≥1of eigenfunctions of the Stokes operator A. Multiplying in (5.4) by A1−2βz(t), we obtain 1 2 d dt |A(1−2β)/2z(t)|2+α2|A1−βz(t)|2+ν|A1−βz(t)|2= (A−βg(t), A1−βz(t)), a.e. t > τ. Thus, d dt neσt |A(1−2β)/2z(t)|2+α2|A1−βz(t)|2o+ 2νeσt|A1−βz(t)|2 =σeσt |A(1−2β)/2z(t)|2+α2|A1−βz(t)|2+ 2eσt(A−βg(t), A1−βz(t)), a.e. t > τ. (5.7) Now, using that 2eσt|(A−βg(t), A1−βz(t))| ≤ εeσt|A1−βz(t)|2+ε−1eσt|A−βg(t)|2,and |A1−βz(t)|2=|A1/2(A(1−2β)/2z(t))|2 ≥λ1|A(1−2β)/2z(t)|2, from (5.7) and the fact that z(τ)=0,we obtain (5.6). Now, from (5.6) we have that v=z+α2Az and its derivative v0belong to L2(τ, T;D(A−β)) for any T > τ. So, it holds that v∈C([τ, ∞); D(A−β)),whence using the mapping C,(5.5) follows. Now we can prove the following regularity result for the pullback attractors in Vnorm. 16 Theorem 27 Assume that f∈L2 loc(R;D(A−β)) for some 0≤β≤1/2,and that sup r≤0Zr r−1 kf(s)k2 ∗ds < ∞.(5.8) Then: (1) If falso satisfies Z0 −∞ eσs|A−βf(s)|2ds < ∞,(5.9) for some 0< σ < 2ν(λ−1 1+α2)−1, and        sup r≤0Zr r−1 |A−1/4−βf(s)|2ds < ∞,if 0< β < 1/4, sup r≤0Zr r−1 |A−δf(s)|2ds < ∞for some 0< δ < 1/4,if β= 0, (5.10) then the pullback attractor ADV max =ADV Ffulfills that [ t1≤t≤t2 ADV max (t)= [ t1≤t≤t2 ADV F(t)is a bounded subset of D(A1−β), for any t1< t2.(5.11) (2) If falso satisfies sup r≤0Zr r−1 |A−βf(s)|2ds < ∞,(5.12) then for any 0< σ < 2ν(λ−1 1+α2)−1, [ t≤t2 ADV σ(t) = [ t≤t2 ADV F(t)is a bounded subset of D(A1−β), for any t2∈R.(5.13) Proof. Let us fix t∈Rand v∈ ADV σ(t) = ADV F(t). By (4.14), (5.8) and Remark 23, we see that [ r≤t ADV σ(r)⊂BV(0,e R1/2 σ(t)),(5.14) where e Rσ(t) = 1 + α−2ε−1supr≤t(e−σr Rr −∞ eσskf(s)k2 ∗ds),with εgiven by (4.4). Let {τn}n≥1⊂(−∞, t] be a sequence with τn→ −∞ as n→ ∞.By the invariance of ADV σ, for each n≥1 there exists uτn∈ ADV σ(τn) such that v=U(t, τn)uτn, and therefore, v=Y(t, τn)uτn+Z(t, τn)uτn. From (5.3) and (5.14) we deduce that kY(t, τn)uτnk → 0 as n→ ∞.Thus, lim n→∞ kZ(t, τn)uτn−vk= 0.(5.15) Let us denote un(r) = U(r, τn)uτnfor r≥τnand n≥1.By (5.14) and the invariance of ADV σ, un(r)∈ ADV σ(r)⊂BV(0,e R1/2 σ(t)),∀τn≤r≤t, ∀n≥1.(5.16) Now we distinguish three cases. Case 1. If 1/4≤β≤1/2. 17 In this case, from (2.11), the continuous injection of Vin D(A3/4−β) and (5.16), we deduce that |A−βB(un(r))| ≤ C(3/4−β)|A3/4−βun(r)|kun(r)k ≤e C(3/4−β)kun(r)k2 ≤e C(3/4−β)e Rσ(t),∀τn≤r≤t, ∀n≥1. Thus, if we assume (5.9), from Lemma 26 we obtain that |A1−βZ(t, τn)uτn|2≤Mσ,β(t),(5.17) where Mσ,β(t) = 2α−2ε−1Zt −∞ eσ(s−t)|A−βf(s)|2ds +σ−1e C2 (3/4−β)e R2 σ(t). From (5.15), (5.17) and the weak lower semi-continuity of the norm, we deduce that vbelongs to BD(A1−β)(0, M1/2 σ,β (t)),and therefore (5.11) holds. Moreover, if fsatisfies (5.12), then (5.13) holds, and more exactly, [ t≤t2 ADV σ(t)⊂BD(A1−β)(0,f M1/2 σ,β (t2)),for all t2∈R,(5.18) where f Mσ,β(t2) = 2α−2ε−1sup t≤t2Zt −∞ eσ(s−t)|A−βf(s)|2ds +σ−1e C2 (3/4−β)e R2 σ(t2). Case 2. If 0 < β < 1/4. In this case, if fsatisfies (5.10), as 1/4<1/4 + β < 1/2,from (5.18) we have that [ r≤t ADV σ(r)⊂BD(A3/4−β)(0,f M1/2 σ,1/4+β(t)). Thus, by (2.11) and (5.16), we obtain that |A−βB(un(r))| ≤ C(3/4−β)|A3/4−βun(r)|kun(r)k ≤C(3/4−β)f M1/2 σ,1/4+β(t)e R1/2 σ(t),∀τn≤r≤t, ∀n≥1. Thus, if we assume (5.9), from Lemma 26 we deduce that |A1−βZ(t, τn)uτn|2≤Rσ,β(t),(5.19) where Rσ,β(t) = 2α−2ε−1Zt −∞ eσ(s−t)|A−βf(s)|2ds +σ−1C2 (3/4−β)f Mσ,1/4+β(t)e Rσ(t). Again, from (5.15), (5.19) and the weak lower semi-continuity of the norm, we deduce that vbelongs to BD(A1−β)(0, R1/2 σ,β (t)),and therefore (5.11) holds. Moreover, if fsatisfies (5.12), then (5.13) holds, and more exactly, [ t≤t2 ADV σ(t)⊂BD(A1−β)(0,e R1/2 σ,β (t2)),for all t2∈R,(5.20) where e Rσ,β(t2) = 2α−2ε−1sup t≤t2Zt −∞ eσ(s−t)|A−βf(s)|2ds +σ−1C2 (3/4−β)f Mσ,1/4+β(t2)e Rσ(t2). 18 Case 3. If β= 0. In this case, if fsatisfies (5.10), as 0 < δ < 1/4,from (5.20) we see that [ r≤t ADV σ(r)⊂BD(A1−δ)(0,e R1/2 σ,δ (t)). So, by (2.10) and (5.16), we deduce that |B(un(r))| ≤ C(1−δ)|A1−δun(r)|kun(r)k ≤C(1−δ)e R1/2 σ,δ (t)e R1/2 σ(t),∀τn≤r≤t, ∀n≥1. Thus, if we assume (5.9), from Lemma 26 we deduce that |AZ(t, τn)uτn|2≤Rσ,δ,0(t),(5.21) where Rσ,δ,0(t)=2α−2ε−1Zt −∞ eσ(s−t)|f(s)|2ds +σ−1C2 (1−δ)e Rσ,δ(t)e Rσ(t). Again, from (5.15), (5.21) and the weak lower semi-continuity of the norm, we deduce that v∈BD(A)(0, R1/2 σ,δ,0(t)), and therefore (5.11) holds. Moreover, if fsatisfies (5.12), then (5.13) holds, and more exactly, [ t≤t2 ADV σ(t)⊂BD(A)(0,e R1/2 σ,δ,0(t2)),for all t2∈R, where e Rσ,δ,0(t2)=2α−2ε−1sup t≤t2Zt −∞ eσ(s−t)|f(s)|2ds +σ−1C2 (1−δ)e Rσ,δ(t2)e Rσ(t2). 6 Attraction in D(A)norm By the previous results, when f∈L2 loc(R;H), the restriction to D(A) of the process Udefined by (4.1) is a process on D(A). Now, we will prove that under suitable assumptions on f, we can obtain the existence of minimal pullback attractors for Uon D(A). Proposition 28 Assume that f∈L2 loc(R;H)is given. Then, the restriction to D(A)of the biparametric family of maps U(t, τ),with τ≤t, given by (4.1), is a closed process on D(A). Proof. It is a consequence of Theorem 4 and Theorem 6. Remark 29 Observe that, by Remark 7, Urestricted to D(A)is in fact a continuous process on D(A). For the obtention of a pullback absorbing family for the process Urestricted to D(A), we first have the following result. Lemma 30 Assume that f∈L2 loc(R;H)satisfies (5.8). Then, for any τ∈R, uτ∈D(A), 0< σ < 2ν(λ−1 1+α2)−1, and 0< σ < σ/3,(6.1) 19 the solution u=u(·;τ, uτ)of (1.1) satisfies ku(t)k2+α2|Au(t)|2≤eσ(τ−t)(kuτk2+α2|Auτ|2)+2ε−1Zt τ eσ(s−t)|f(s)|2ds +4CεC3 σ(σ−3σ)−1e−3σ(t−τ)kuτk6+M3 t,σ(6.2) for all t≥τ, where ε > 0is given by (4.4), Cε= 27C4 2(2ε3)−1,(6.3) Cσ=α−2max (α2+λ−1 1),ν−σ 2(λ−1 1+α2)−1,(6.4) and Mt,σ = sup r≤tZr −∞ eσ(s−r)kf(s)k2 ∗ds. (6.5) Proof. Let τ∈R, uτ∈D(A), σ and σsatisfying (6.1) be fixed. From Lemma 17 we deduce in particular that u=u(·;τ, uτ) satisfies ku(s)k2≤Cσeσ(τ−s)kuτk2+Mt,σ,∀τ≤s≤t. (6.6) On the other hand, by (2.21), d dt(eσtku(t)k2+α2eσt|Au(t)|2)+2νeσt|Au(t)|2+ 2eσt(B(u(t)), Au(t)) =σeσtku(t)k2+α2σeσt|Au(t)|2+ 2eσt(f(t), Au(t)),a.e. t > τ. Thus, taking into account that ku(t)k2≤λ−1 1|Au(t)|2, 2|(B(u(t)), Au(t)) | ≤ 2C2ku(t)k3/2|Au(t)|3/2 ≤Cεku(t)k6+ε 2|Au(t)|2, and 2|(f(t), Au(t)) | ≤ ε 2|Au(t)|2+2 ε|f(t)|2, we deduce that ku(t)k2+α2|Au(t)|2≤eσ(τ−t)(kuτk2+α2|Auτ|2)+2ε−1Zt τ eσ(s−t)|f(s)|2ds +CεZt τ eσ(s−t)ku(s)k6ds for all t≥τ. From this inequality and (6.6), we easily obtain (6.2). Definition 31 For any σ, σ > 0,we will consider the universe DD(A) σ∩ DV σformed by the class of all families of nonempty subsets b D={D(t) : t∈R}⊂P(D(A)) such that lim τ→−∞ eστ sup v∈D(τ) |Av|2= lim τ→−∞ eστ sup v∈D(τ) kvk2= 0. Accordingly to the notation introduced in Section 3, DD(A) Fwill denote the class of families b D= {D(t) = D:t∈R}with Da fixed nonempty bounded subset of D(A). 20 Remark 32 Observe that for any σ, σ > 0,DD(A) F⊂ DD(A) σ∩ DV σand that both universes are inclusion-closed. As a consequence of Lemma 30, we have the following result. Corollary 33 Assume that f∈L2 loc(R;H)satisfies (5.8) and Z0 −∞ eσs|f(s)|2ds < ∞,(6.7) for some 0< σ < 2ν(λ−1 1+α2)−1. Then, for any 0< σ < σ/3, the family b Dσ,σ ={Dσ,σ(t) : t∈R}defined by Dσ,σ(t) = BD(A)(0, R1/2 σ,σ (t)),(6.8) the closed ball in D(A)of center zero and radius R1/2 σ,σ (t),where Rσ,σ(t) = α−21+2ε−1Zt −∞ eσ(s−t)|f(s)|2ds + 4CεC3 σ(σ−3σ)−1M3 t,σ,(6.9) with ε, Cε,Cσand Mt,σ, given by (4.4),(6.3),(6.4) and (6.5), respectively, is pullback DD(A) σ∩ DV σ−absorbing for the restriction to D(A)of the process Ugiven by (4.1) (and therefore DD(A) F− absorbing too). Now, we prove that the process Uis pullback DD(A) σ∩DV σ−asymptotically compact. We will apply, with obvious necessary changes, the same energy method used in the proof of Lemma 21. Lemma 34 Assume that f∈L2 loc(R;H)satisfies (5.8) and (6.7) for some 0< σ < 2ν(λ−1 1+α2)−1. Then, for any 0< σ < σ/3,the restriction to D(A)of the process Udefined by (4.1) is pullback b D−asymptotically compact for any b D∈ DD(A) σ∩ DV σ. Proof. Let us fix 0 < σ < σ/3.Let b D∈ DD(A) σ∩ DV σ, t ∈R, τn→ −∞ with τn≤tand uτn∈D(τn) for all n, be given. We must prove that the sequence {U(t, τn)uτn}is relatively compact in D(A).By Corollary 33, for each integer k≥0,there exists τb D(k)≤t−ksuch that U(t−k, τ)D(τ)⊂Dσ,σ(t−k),∀τ≤τb D(k).(6.10) From this and a diagonal argument, we can extract a subsequence {uτn0}⊂{uτn}such that U(t−k, τn0)uτ0 n* wkweakly in D(A),∀k≥0,(6.11) where wk∈Dσ,σ(t−k). Now, applying Theorem 6 on each fixed interval [t−k, t] we obtain that w0=D(A)−weak lim n0→∞ U(t, τn0)uτn0 =D(A)−weak lim n0→∞ U(t, t −k)U(t−k, τn0)uτn0 =U(t, t −k)hD(A)−weak lim n0→∞ U(t−k, τn0)uτn0i =U(t, t −k)wk. In particular, observe that |Aw0| ≤ lim infn0→∞ |AU(t, τn0)uτn0|.We will prove now that it also holds that lim sup n0→∞ |AU(t, τn0)uτn0| ≤ |Aw0|,(6.12) 21 which combined with (6.11) for k= 0,will imply the convergence in the strong topology of D(A),and the asymptotic compactness. Observe that, as we already used in Lemma 30, for any pair (τ, uτ) with uτ∈D(A),the solution u(·;τ, uτ),for short denoted u(·),satisfies the differential equality (2.21). Since 0 < σ < 2ν(λ−1 1+α2)−1,we observe that [[·]],with [[v]]2= (2ν−α2σ)|Av|2−σkvk2,defines an equivalent norm to |·|D(A)in D(A). We integrate (2.21) in the interval [t−k, t] for the solutions U(·, τn0)uτn0,which yields kU(t, τn0)uτn0k2+α2|AU(t, τn0)uτn0|2 =kU(t, t −k)U(t−k, τn0)uτn0k2+α2|AU(t, t −k)U(t−k, τn0)uτn0|2 =e−σk kU(t−k, τn0)uτn0k2+α2|AU(t−k, τn0)uτn0|2 +2 Zt t−k eσ(s−t)(f(s), AU(s, t −k)U(t−k, τn0)uτn0)ds −2Zt t−k eσ(s−t)(B(U(s, t −k)U(t−k, τn0)uτn0), AU(s, t −k)U(t−k, τn0)uτn0)ds −Zt t−k eσ(s−t)[[U(s, t −k)U(t−k, τn0)uτn0]]2ds. (6.13) From (6.11) and Theorem 6, we have that U(·, t −k)U(t−k, τn0)uτn0→U(·, t −k)wkstrongly in C([t−k, t]; V), and also U(·, t −k)U(t−k, τn0)uτn0* U(·, t −k)wkweakly in L2(t−k, t;D(A)). Then, it is not difficult to see that lim n0→∞ Zt t−k eσ(s−t)(B(U(s, t −k)U(t−k, τn0)uτn0), AU(s, t −k)U(t−k, τn0)uτn0)ds =Zt t−k eσ(s−t)(B(U(s, t −k)wk), AU(s, t −k)wk)ds. (6.14) Also, as eσ(·−t)f(·)∈L2(t−k, t;H),it yields lim n0→∞ Zt t−k eσ(s−t)(f(s), AU(s, t −k)U(t−k, τn0)uτn0)ds =Zt t−k eσ(s−t)(f(s), AU(s, t −k)wk)ds. (6.15) Finally, as Rt t−keσ(s−t)[[v(s)]]2ds defines an equivalent norm in L2(t−k, t;D(A)),we also deduce from above that Zt t−k eσ(s−t)[[U(s, t −k)wk]]2ds ≤lim inf n0→∞ Zt t−k eσ(s−t)[[U(s, t −k)U(t−k, τn0)uτn0]]2ds. (6.16) From (6.10), (6.11) with k= 0,the compactness of the injection of D(A) into V, and (6.13), (6.14)– (6.16), we conclude that kw0k2+α2lim sup n0→∞ |AU(t, τn0)uτn0|2 ≤e−σk(λ−1 1+α2)Rσ,σ(t−k)+2Zt t−k eσ(s−t)(f(s), AU(s, t −k)wk)ds −2Zt t−k eσ(s−t)(B(U(s, t −k)wk), AU(s, t −k)wk)ds −Zt t−k eσ(s−t)[[U(s, t −k)wk]]2ds. 22 Now, taking into account that w0=U(t, t −k)wk,integrating again in (2.21), we obtain kw0k2+α2|Aw0|2=e−σk(kwkk2+α2|Awk|2)+2Zt t−k eσ(s−t)(f(s), AU(s, t −k)wk)ds −2Zt t−k eσ(s−t)(B(U(s, t −k)wk), AU(s, t −k)wk)ds −Zt t−k eσ(s−t)[[U(s, t −k)wk]]2ds. Comparing the above two expressions, we conclude that kw0k2+α2lim sup n0→∞ |AU(t, τn0)uτn0|2 ≤e−σk(λ−1 1+α2)Rσ,σ(t−k) + kw0k2+α2|Aw0|2−e−σk kwkk2+α2|Awk|2. But from (6.9), we have that lim k→∞ e−σkRσ,σ(t−k)=0,so (6.12) holds. In general, the pullback absorbing family b Dσ,σ defined by (6.8) does not belong to DD(A) σ∩ DV σ, and we do not know if Uis pullback b Dσ,σ−asymptotically compact. Thus, we cannot apply Theorem 11 to the family b Dσ,σ. Nevertheless we can prove the following result. Theorem 35 Assume that f∈L2 loc(R;H)satisfies (5.8) and (6.7) for some 0< σ < 2ν(λ−1 1+α2)−1. Then, for any 0< σ < σ/3, the family of sets Xσ,σ(t) = [ b D∈DD(A) σ∩DV σ ΛD(A)(b D, t) D(A) t∈R,(6.17) has the following properties: (a) lim τ→−∞ distD(A)(U(t, τ)D(τ), Xσ,σ(t)) = 0 for all t∈Rand any b D∈ DD(A) σ∩ DV σ(pullback attraction). (b) It is minimal in the sense that if b C={C(t) : t∈R}⊂P(D(A)) is a family of closed subsets of D(A)such that lim τ→−∞ distD(A)(U(t, τ)D(τ), C(t)) = 0 for all t∈Rand any b D∈ DD(A) σ∩ DV σ, then Xσ,σ(t)⊂C(t)for all t∈R. (c) U(t, τ)Xσ,σ(τ) = Xσ,σ(t)for all τ≤t(invariance). Proof. The assertion (a) is an easy consequence of Proposition 9 and Lemma 34. For the proof of (b), assume that b C={C(t) : t∈R}⊂P(D(A)) is a family of closed subsets of D(A) such that lim τ→−∞ distD(A)(U(t, τ)D(τ), C(t)) = 0 for all t∈Rand any b D∈ DD(A) σ∩ DV σ. Now, let us fix t∈R. In this case, it is easy to see that, for any u∈ΛD(A)(b D, t), with b D∈ DD(A) σ∩ DV σ, one has that distD(A)(x, C(t)) = 0. Thus, as C(t) is closed in D(A), we deduce that ΛD(A)(b D, t)⊂C(t), and therefore, Xσ,σ(t)⊂C(t). Finally, let τ≤tbe fixed. In order to prove (c) we observe that by Proposition 9, we also have that U(t, τ)ΛD(A)(b D, τ)=ΛD(A)(b D, t) for any b D∈ DD(A) σ∩ DV σ.(6.18) If y∈Xσ,σ(t), there exist two sequences {b Dn}⊂DD(A) σ∩ DV σand {yn} ⊂ D(A), such that yn∈ ΛD(A)(b Dn, t) and y=D(A)−lim n→∞ yn. But by (6.18), yn=U(t, τ)xn, with xn∈ΛD(A)(b Dn, τ)⊂ 23 Xσ,σ(τ). By Corollary 33, we can also deduce that Xσ,σ(τ)⊂BD(A)(0, R1/2 σ,σ (τ)), and therefore, by the compactness of the injection of D(A) into V,Xσ,σ(τ) is a compact subset of V. Thus, there exists a subsequence {xn0} ⊂ {xn}such that xn0→x∈Xσ,σ(τ) in V. But then, as Uis a closed process on V,y=U(t, τ)x, and this proves that Xσ,σ(t)⊂U(t, τ)Xσ,σ(τ). The reverse inclusion can be proved analogously. Under the additional assumption sup r≤0Zr r−1 |f(s)|2ds < ∞,(6.19) the pullback absorbing family b Dσ,σ defined by (6.8) does belong to DD(A) σ∩ DV σ, whence we can apply Theorem 11, and actually we have the following result. Theorem 36 Assume that f∈L2 loc(R;H)satisfies (6.19). Then, for any 0< σ < 2ν(λ−1 1+α2)−1 and 0< σ < σ/3, we have that: (a) The family of sets Xσ,σ(t)defined by (6.17) is the minimal pullback DD(A) σ∩ DV σ-attractor, and in fact is a family of compact subsets of D(A). (b) Xσ,σ(t) = ADV F(t)for all t∈R. (c) Indeed, ADV Fis the unique family of closed subsets for the norm of D(A)in any universe of the form DD(A) σ∩DV σthat is invariant for Uand attracts any b D∈ DD(A) σ∩DV σin the pullback sense. Proof. Let us fix 0 < σ < 2ν(λ−1 1+α2)−1and 0 < σ < σ/3. Observe that under the above assumptions on f, the family b Dσ,σ ={Dσ,σ(t) : t∈R}defined by (6.8)–(6.9) belongs to DD(A) σ∩ DV σ. Therefore, the assertion (a) is a direct consequence of Theorem 11, Proposition 28, Corollary 33, and Lemma 34. Now, let us fix t∈R. It is evident that by (6.19), Xσ,σ(t)⊂[ b D∈DV σ ΛD(A)(b D, t) D(A) ⊂[ b D∈DV σ ΛD(A)(b D, t) V =ADV σ(t) = ADV F(t). On the other hand, again by (6.19), from Theorem 27 we have that [ r≤t ADV F(r) is a bounded subset of D(A), and therefore, distD(A)(U(t, τ)[ r≤t ADV F(r), Xσ,σ(t)) ≤distD(A)(U(t, τ)[ r≤t ADV F(r),ΛD(A)([ r≤t ADV F(r), t)). From this inequality, Proposition 9, Lemma 34, and the invariance of ADV F, we deduce that distD(A)(ADV F(t), Xσ,σ(t)) = 0, and therefore ADV F(t)⊂Xσ,σ(t).Thus, (b) is proved. Finally, (c) is a direct consequence of Remark 12. Remark 37 Observe that in particular, if f∈L2 loc(R;H)satisfies (6.19), by Remark 14 the minimal attractor ADD(A) F does exist, and it also coincides with the family ADV F.Moreover, this last family attracts in the pullback sense in the norm of D(A)to all the families of the universe DD(A),V max =[ 0<σ <2ν(λ−1 1+α2)−1 0<σ <σ/3 DD(A) σ∩ DV σ. 24 Acknowledgments This work has been partially supported by Ministerio de Ciencia e Innovaci´on (Spain) under project MTM2008-00088, and the Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa), Proyecto de Excelencia P07-FQM-02468. J.G.-L. is a fellow of Programa de FPU del Ministerio de Educaci´on (SPAIN). The authors would like to thank the referees for their helpful comments and suggestions. References [1] Y. P. Cao, E. M. Lunasin, and E. S. Titi, Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci. 4(2006), 823–848. [2] T. Caraballo, G. Lukaszewicz, and J. 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