Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume XX, Number 0xx, XXXXXX 20xx pp. – SOME NEW REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR 2D NAVIER-STOKES EQUATIONS WITH DELAYS Julia Garc´ ıa-Luengo, Pedro Mar´ ın-Rubio & Jos´ e Real∗ Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080–Sevilla, Spain (Communicated by XXXXX) Abstract. In this paper we strengthen some results on the existence and properties of pullback attractors for a 2D Navier-Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271– 297]. Actually, we prove that under suitable assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Moreover, thanks to regularity results, the attraction from different phase spaces also happens in C([−h, 0]; V). Finally, from comparison results of attractors, and under an additional hypothesis, we establish that all these families of attractors are in fact the same object. 1. Introduction and statement of the problem. Let Ω ⊂R2be an open bounded set with smooth enough boundary ∂Ω, and consider an arbitrary initial time τ∈R, and the following functional Navier-Stokes problem:                    ∂u ∂t −ν∆u+ (u· ∇)u+∇p=f(t) + g(t, ut) in Ω ×(τ, ∞), div u= 0 in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ) = uτ(x), x ∈Ω, u(x, τ +s) = φ(x, s), x ∈Ω, s ∈(−h, 0), (1) where ν > 0 is the kinematic viscosity, u= (u1, u2) is the velocity field of the fluid, pis the pressure, fis a non-delayed external force field, gis another external force with some hereditary characteristics, and uτand φ(x, s −τ) are the initial data in τand (τ−h, τ) respectively, where h > 0 is the time of memory effect. For each t≥τ, we denote by utthe function defined a.e. on (−h, 0) by the relation ut(s) = u(t+s), a.e. s∈(−h, 0). 2010 Mathematics Subject Classification. Primary: 35B41, 35Q30, 37L30. Key words and phrases. 2D Navier-Stokes equations; delay terms; pullback attractors. ∗Deceased on January 27th, 2012. The first two coauthors dedicate this paper to the memory of their coauthor Jos´e Real. 1 2 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL The importance of physical models for fluid mechanic problems including delay terms is related, for instance, to real applications where devices to control properties of fluids (temperature, velocity, etc.) are inserted in domains and make a local influence on the behaviour of the system (e.g., cf. [13] for a wind-tunnel model). The study of Navier-Stokes models including delay terms –existence, uniqueness, stationary solutions, exponential decay, and other asymptotic properties such as the existence of attractors– was initiated in the references [3,4,5], and after that, many different questions, as dealing with unbounded domains, and models (for instance in three dimensions for modified terms) have been addressed (e.g., cf. [10,17,21,19,14,20,11,15,16] among others). In the recent paper [9], we have treated a relaxation on the assumptions for the delay operator involved, removing conditions related to the control of the L2norm of the delay terms (see assumptions (IV) and (V) below). Although this implies to restrict the phase space to continuous functions instead of square integrable in time, the delay functions driving the delayed time within this theory can be taken just measurable, without any additional assumption as continuity nor C1with bounded derivative, as usual in the literature. Moreover, in [9] we were also able to establish attraction in a higher norm (namely, H1instead of L2) making a sharp use of regularization of the equations in dimension two and by energy methods. Relationships among attractors in different metrics was successfully carried out there, too. Our goal in this paper is to keep all usual conditions for the delay operator (including (IV) and (V)) and to compare both kind of attractors, for both possibilities of phase spaces (continuous in time, or just square integrable in time). Observe that in the autonomous framework this issue would be almost immediate since one inclusion is clear by continuous embedding, and the other is obtained after an elapsed time as long as the memory effect. However, in the non-autonomous case (that we are dealing with) this is not the case at all. Using the theory of attraction for universes (cf. [1,2,18]) we deal with different families and under different metrics. Namely, we consider universes of fixed (in time) bounded sets and also time-dependent families given by a tempered condition when time goes to −∞. Moreover, we also improve some results previously obtained in the literature (cf. [5]) since we can deal with the phase space V×L2(−h, 0; V) and not only H×L2(−h, 0; H).Finally, from comparison results of attractors and under an additional assumption, we establish that all these families of attractors are in fact the same object. The structure of the paper is the following. We continue this section with the abstract setting of the problem, general definitions and some well-known results on existence of weak and strong solutions and regularity properties. In Section 2 we recall the basic theory of pullback attractors for non-autonomous dynamical systems within the framework of universes, and comparison results, when different metrics are involved, are also given. Section 3 is devoted to establish all possible attractors for different phase-spaces but taking into account the L2norm in space. Our main results, established in the higher norm H1(in space), are given in Section REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 3 4. In these two last sections, energy methods (introduced in this context by Rosa in [23]) are used to prove asymptotic compactness in the respective universes. As said before, relationships among all these objects are obtained. To set our problem in the abstract framework, we consider the following usual function spaces: V=u∈(C∞ 0(Ω))2: div u= 0, H= 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, V= 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. We will use k · k∗for the norm in V0and h·,·i for the duality between V0and V. 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 continuous, and, in fact, compact. Now, we define the operator A:V→V0as hAu, vi= ((u, v)) ∀u, v ∈V. Let us denote D(A) = {u∈V:Au ∈H}. By the regularity of ∂Ω, one has that D(A)=(H2(Ω))2∩V, and Au =−P∆ufor all u∈D(A) is the Stokes operator (Pis the ortho-projector from (L2(Ω))2onto H). On D(A) we consider the norm |·|D(A)defined by |u|D(A)=|Au|. Observe that on D(A) the norms k·k(H2(Ω))2and |·|D(A)are equivalent (see [6] or [25]), and D(A) is compactly and densely injected in V. Let us define b(u, v, w) = 2 X i,j=1 ZΩ ui ∂vj ∂xi wjdx, for every functions u, v, w : Ω →R2for 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. Some useful properties concerning bthat we will use in the next sections are the following (see [22] or [24]): b(u, v, w) = −b(u, w, v) for all u,v,w∈V, which also implies that b(u, v, v) = 0 for all u,v∈V. Moreover, there exists a constant C1>0, only dependent on Ω, such that (recall that we are in dimension two) |b(u, v, w)| ≤ C1|u|1/2|Au|1/2kvk|w| ∀ u∈D(A), v ∈V, w ∈H. (2) Now, we establish some suitable spaces in order to deal with the delay term, and some appropriate assumptions on the term in (1) containing the delay. Let us denote CH=C([−h, 0]; H), with the norm |ϕ|CH= maxs∈[−h,0] |ϕ(s)|, and L2 X=L2(−h, 0; X) for X=H,V. On the delay operator from (1), we consider that is well defined as g:R×CH→(L2(Ω))2, and it satisfies the following assumptions: 4 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL (I) for all ξ∈CH, the function R3t7→ g(t, ξ)∈(L2(Ω))2is measurable, (II) g(t, 0) = 0, for all t∈R, (III) there exists Lg>0 such that for all t∈R, and for all ξ,η∈CH, |g(t, ξ)−g(t, η)| ≤ Lg|ξ−η|CH, (IV) there exists Cg>0 such that for all τ≤t, and for all u,v∈C([τ−h, t]; H), Zt τ |g(s, us)−g(s, vs)|2ds ≤C2 gZt τ−h |u(s)−v(s)|2ds. Examples of fixed, variable, and distributed delay operators can be found, for instance, in [3, Section 3], [5, Sections 3.5 and 3.6], and [10, Section 3], and we omit them here just for the sake of brevity. Observe that (I)−(III) imply that given T > τ and u∈C([τ−h, T]; H), the function gu: [τ, T]→(L2(Ω))2defined by gu(t) = g(t, ut) for all t∈[τ, T ], is measurable and, in fact, belongs to L∞(τ, T; (L2(Ω))2). Then, thanks to (IV), the mapping G:u∈C([τ−h, T]; H)→gu∈L2(τ, T; (L2(Ω))2) has a unique extension to a mapping e Gwhich is uniformly continuous from L2(τ− h, T;H) into L2(τ, T; (L2(Ω))2). From now on, we will denote g(t, ut) = e G(u)(t) for each u∈L2(τ−h, T ;H), and thus property (IV) will also hold for all u, v∈L2(τ−h, T;H). Assume that uτ∈H,φ∈L2 H, and f∈L2 loc(R;V0). Definition 1. A weak solution of (1) is a function uthat belongs to L2(τ−h, T;H) ∩L2(τ, T;V)∩L∞(τ, T;H) for all T > τ, with u(τ) = uτand u(t) = φ(t−τ) a.e. t∈(τ−h, τ), and such that for all v∈V, d dt(u(t), v) + νhAu(t), vi+b(u(t), u(t), v) = hf(t), vi+ (g(t, ut), v),(3) where the equation must be understood in the sense of D0(τ, ∞). Remark 1. If uis a weak solution of (1), then from (3) we deduce that for any T > τ, one has u0∈L2(τ, T;V0), and so u∈C([τ, ∞); H), whence the initial datum u(τ) = uτhas full sense. Moreover, in this case the following energy equality holds: |u(t)|2+2νZt s ku(r)k2dr=|u(s)|2+2Zt shf(r), u(r)i+(g(r, ur), u(r))dr ∀τ≤s≤t. A notion of more regular solution is also suitable for problem (1). Definition 2. A strong solution of (1) is a weak solution uof (1) such that u∈ L2(τ, T;D(A)) ∩L∞(τ, T;V) for all T > τ. Remark 2. If f∈L2 loc(R; (L2(Ω))2) and uis a strong solution of (1), then u0∈ L2(τ, T;H) for all T > τ, and so u∈C([τ, ∞); V). In this case the following energy equality holds: ku(t)k2+ 2νZt s |Au(r)|2dr + 2 Zt s b(u(r), u(r), Au(r)) dr =ku(s)k2+ 2 Zt s (f(r) + g(r, ur), Au(r)) dr ∀τ≤s≤t. (4) REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 5 Concerning the existence and uniqueness of weak and strong solutions for (1), we have the following result which can be proved similarly as [3, Theorem 2.1] or [4, Theorem 2.5] (see also [10, Theorem 2.3] for a more general case). Theorem 1. Let us consider uτ∈H,φ∈L2 H,f∈L2 loc(R;V0), and g:R×CH→ (L2(Ω))2satisfying (I)–(IV). Then, for each τ∈R, there exists a unique weak solution u=u(·;τ, uτ, φ)of (1). Moreover, if f∈L2 loc(R; (L2(Ω))2), then (a) u∈C([τ+ε, T]; V)∩L2(τ+ε, T;D(A)) for all T > τ +ε > τ. (b) If uτ∈V,uis in fact a strong solution of (1). Before establishing the original results about the regularity of pullback attractors, we recall the main existence results studied in [5,17,19]. Firstly, in order to do that, we remember briefly the abstract theory on pullback attractors in the next section. 2. Abstract results on minimal pullback attractors. Now, we present a summary of some results from [8] about the existence of minimal pullback attractors (see also [1,2,18]). In particular, we assume that the process Uis closed (see Definition 3below). Consider given a metric space (X, dX), and let us denote R2 d={(t, τ)∈R2:τ≤t}. A process Uon Xis a mapping R2 d×X3(t, τ, x)7→ U(t, τ)x∈Xsuch that U(τ, τ)x=xfor any (τ, x)∈R×X, and U(t, r)(U(r, τ)x) = U(t, τ)xfor any τ≤r≤tand all x∈X. Definition 3. Let Ube a process on X. (a) Uis said to be continuous if for any pair τ≤t, the mapping U(t, τ) : X→X is continuous. (b) Uis said to be closed if for any τ≤t, and any sequence {xn} ⊂ X, if xn→x∈Xand U(t, τ)xn→y∈X, then U(t, τ)x=y. Remark 3. It is clear that every continuous process is closed. Let us denote by P(X) the family of all nonempty subsets of X, and consider a family of nonempty sets b D0={D0(t) : t∈R}⊂P(X). Definition 4. We say that a process Uon Xis pullback b D0-asymptotically compact if for any t∈Rand any sequences {τn} ⊂ (−∞, t] and {xn} ⊂ Xsatisfying τn→ −∞ and xn∈D0(τn) for all n, the sequence {U(t, τn)xn}is relatively compact in X. Denote Λ( b D0, t) = \ s≤t[ τ≤s U(t, τ)D0(τ) X ∀t∈R, where {· · · }Xis the closure in X. Given two subsets of X,O1and O2, we denote by distX(O1,O2) the Hausdorff semi-distance in Xbetween them, defined as distX(O1,O2) = sup x∈O1 inf y∈O2 dX(x, y). Let be given Da nonempty class of families parameterized in time b D={D(t) : t∈R}⊂P(X). The class Dwill be called a universe in P(X). 6 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL Definition 5. A process Uon Xis said to be pullback D-asymptotically compact if it is pullback b D-asymptotically compact for any b D∈ D. It is said that b D0={D0(t) : t∈R}⊂P(X) is pullback D-absorbing for the process Uon Xif for any t∈Rand any b D∈ D, there exists a τ0(t, b D)≤tsuch that U(t, τ)D(τ)⊂D0(t)∀τ≤τ0(t, b D). With the above definitions, we may establish the main result of this section (cf. [8, Theorem 3.11]). Theorem 2. Consider a closed process U:R2 d×X→X, a universe Din P(X), and a family b D0={D0(t) : t∈R}⊂P(X)which is pullback D-absorbing for U, and assume also that Uis pullback b D0-asymptotically compact. Then, the family AD={AD(t) : t∈R}defined by AD(t) = Sb D∈D Λ( b D, t) X , has the following properties: (a) for any t∈R, the set AD(t)is a nonempty compact subset of X, and AD(t)⊂ Λ( b D0, t), (b) ADis pullback D-attracting, i.e., limτ→−∞ distX(U(t, τ)D(τ),AD(t)) = 0 for all b D∈ D, and any t∈R, (c) ADis invariant, i.e., U(t, τ)AD(τ) = AD(t)for all (t, τ)∈R2 d, (d) if b D0∈ D, then AD(t) = Λ( b D0, t)⊂D0(t)Xfor all t∈R. The family ADis minimal in the sense that if b C={C(t) : t∈R} ⊂ P(X)is a family of closed sets such that for any b D={D(t) : t∈R}∈D,lim τ→−∞ distX(U(t, τ)D(τ), C(t)) = 0, then AD(t)⊂C(t). Remark 4. Under the assumptions of Theorem 2, the family ADis called the minimal pullback D-attractor for the process U. If AD∈ D, then it is the unique family of closed subsets in Dthat satisfies (b)–(c). A sufficient condition for AD∈ D is to have that b D0∈ D, the set 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). We will denote by DF(X) the universe of fixed nonempty bounded subsets of X, i.e., the class of all families b Dof the form b D={D(t) = D:t∈R}with Da fixed nonempty bounded subset of X. Now, it is easy to conclude the following result. Corollary 1. Under the assumptions of Theorem 2, if the universe Dcontains the universe DF(X), then both attractors, ADF(X)and AD, exist, and ADF(X)(t)⊂ AD(t)for all t∈R. Remark 5. It can be proved (see [18]) that, under the assumptions of the preceding corollary, if for some T∈R, the set ∪t≤TD0(t) is a bounded subset of X, then ADF(X)(t) = AD(t) for all t≤T. Now, and since it will be useful below, we establish an abstract result (cf. [8, Theorem 3.15]) that allows us to compare two attractors for a process under appropriate assumptions. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 7 Theorem 3. Let {(Xi, dXi)}i=1,2be two metric spaces such that X1⊂X2with continuous injection, and for i= 1,2, let Dibe a universe in P(Xi), with D1⊂ D2. Assume that we have a map Uthat acts as a process in both cases, i.e., U:R2 d×Xi→ Xifor i= 1,2is a process. For each t∈R, let us denote Ai(t) = [ b Di∈Di Λi(b Di, t) Xi i= 1,2, where the subscript iin the symbol of the omega-limit set Λiis used to denote the dependence of the respective topology. Then, A1(t)⊂ A2(t)for all t∈R. Suppose moreover that the two following conditions are satisfied: (i) A1(t)is a compact subset of X1for all t∈R, (ii) for any b D2∈ D2and any t∈R, there exist a family b D1∈ D1and a t∗ b D1≤t(both possibly depending on tand b D2), such that Uis pullback b D1asymptotically compact, and for any s≤t∗ b D1there exists a τs≤ssuch that U(s, τ)D2(τ)⊂D1(s)for all τ≤τs. Then, under all the conditions above, A1(t) = A2(t)for all t∈R. Remark 6. In the preceding theorem, if instead of assumption (ii) we consider the following condition: (ii’) for any b D2∈ D2and any sequence τn→ −∞, there exist another family b D1∈ D1and another sequence τ0 n→ −∞ with τ0 n≥τnfor all n, such that U is pullback b D1-asymptotically compact, and U(τ0 n, τn)D2(τn)⊂D1(τ0 n)∀n, then, with a similar proof, one can obtain that the equality A1(t) = A2(t) also holds for all t∈R. Observe that a sufficient condition for (ii’) is that there exists T > 0 such that for any b D2∈ D2, there exists a b D1∈ D1satisfying that Uis pullback b D1-asymptotically compact, and U(τ+T, τ)D2(τ)⊂D1(τ+T) for all τ∈R. 3. Previous results on processes and pullback attractors in H.In this section we recall some known results (cf. [5,17,19]) on the existence of minimal pullback attractors in the Hnorm for suitable processes associated to problem (1). In order to apply the theory of the above section, and following [5,17,19], we may consider the Banach space CH, and the Hilbert space M2 H=H×L2 Hwith associated norm k(uτ, φ)k2 M2 H=|uτ|2+Z0 −h |φ(s)|2ds for (uτ, φ)∈M2 H. We can define two processes for problem (1). Proposition 1. Assume that f∈L2 loc(R;V0), and g:R×CH→(L2(Ω))2satisfies (I)–(IV). Then, the bi-parametric families of mappings U(t, τ) : CH→CHand S(t, τ) : M2 H→M2 Hgiven respectively by U(t, τ)φ=ut(·;τ, φ(0), φ)for φ∈CH, τ ≤t, (5) and S(t, τ)(uτ, φ)=(u(t;τ, uτ, φ), ut(·;τ, uτ, φ)) for (uτ, φ)∈M2 H, τ ≤t, (6) 8 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL where uis the unique weak solution of (1), are well defined continuous processes on CHand M2 Hrespectively. Proof. The result follows from Theorem 1above, and from [5, Theorem 9]. Now, in order to establish asymptotic estimates for the solutions of (1), we impose a fifth assumption on gand f. Denote by λ1the first eigenvalue of the Stokes operator A. (V) Assume that νλ1> Cg, and that there exists a value η∈(0,2(νλ1−Cg)) such that for every u∈L2(τ−h, t;H), Zt τ eηs|g(s, us)|2ds ≤C2 gZt τ−h eηs|u(s)|2ds for any τ≤t, and Z0 −∞ eηskf(s)k2 ∗ds < ∞. Lemma 1. Suppose that f∈L2 loc(R;V0), and that fand g:R×CH→(L2(Ω))2 satisfy (I)–(V). Then, for any (uτ, φ)∈M2 H, the following estimate holds for the solution uto (1) for all t≥τ: |u(t)|2≤e−η(t−τ)max{1, Cg}k(uτ, φ)k2 M2 H+β−1e−ηt Zt τ eηskf(s)k2 ∗ds, (7) where β= 2ν−(η+ 2Cg)λ−1 1.(8) Proof. By the energy equality (see Remark 1), and Young’s inequality, we have d dt|u(t)|2+ 2νku(t)k2 ≤βku(t)k2+β−1kf(t)k2 ∗+Cg|u(t)|2+C−1 g|g(t, ut)|2,a.e. t > τ. Thus, d dteηt|u(t)|2+eηt2ν−β−(η+Cg)λ−1 1ku(t)k2 ≤eηtβ−1kf(t)k2 ∗+eηtC−1 g|g(t, ut)|2,a.e. t > τ, and therefore, integrating above and using property (V), we obtain eηt|u(t)|2+2ν−β−(η+Cg)λ−1 1Zt τ eηsku(s)k2ds ≤eητ |uτ|2+β−1Zt τ eηskf(s)k2 ∗ds +CgZt τ−h eηs|u(s)|2ds ≤eητ max{1, Cg}k(uτ, φ)k2 M2 H+β−1Zt τ eηskf(s)k2 ∗ds +CgZt τ eηs|u(s)|2ds, for all t≥τ, and from this last inequality and (8), in particular we deduce (7). After the above result, it turns out appropriate the introduction of the following tempered universes. Definition 6. For any η > 0, we will denote by Dη(CH) the class of all families of nonempty subsets b D={D(t) : t∈R}⊂P(CH) such that lim τ→−∞ eητ sup ϕ∈D(τ) |ϕ|2 CH= 0. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 9 Analogously, we will denote by Dη(M2 H) the class of all families of nonempty subsets b D={D(t) : t∈R}⊂P(M2 H) such that lim τ→−∞ eητ sup (w,ϕ)∈D(τ) k(w, ϕ)k2 M2 H= 0. Furthermore, accordingly to the notation introduced in the previous section, DF(CH) and DF(M2 H) will denote the universes of fixed bounded sets in CHand M2 Hrespectively. Remark 7. (i) The choices of the above universes are right and convenient to keep, in the sense that, on the one hand, M2 His more general as phase space for the initial data of problem (1). On the other hand, the regularity of the solution to (1) (cf. Theorem 1) makes that, after an elapsed time h, every solution is continuous with values on H. Indeed, as it was observed in [5], in the case of the universes of fixed bounded sets, pullback attractors in both spaces do exist, and they are intrinsically related through the canonical embedding j:CH→M2 Hdefined by j(ϕ) = (ϕ(0), ϕ) (see Theorem 4below). (ii) The universes Dη(CH) and Dη(M2 H),which are inclusion-closed, contain respectively the universes DF(CH) and DF(M2 H). Now, we obtain pullback absorbing families for U:R2 d×CH→CHand S: R2 d×M2 H→M2 H. Corollary 2. Under the assumptions of Lemma 1, the family b D1,η ={D1,η(t) : t∈ R} ⊂ P(CH)defined by D1,η(t) = BCH(0, rη(t)), the closed ball in CHof center zero and radius rη(t), where r2 η(t) = 1 + β−1e−η(t−h)Zt −∞ eηskf(s)k2 ∗ds, with βgiven by (8), is pullback Dη(CH)-absorbing for the process Uon CHdefined by (5) (and therefore pullback DF(CH)-absorbing too), and b D1,η belongs to Dη(CH). Besides, the family b D2,η ={D2,η(t) : t∈R} ⊂ P(M2 H)defined by D2,η(t) = BM2 H(0, Rη(t)), the closed ball in M2 Hof center zero and radius Rη(t), with R2 η(t) = 1 + β−1(1 + heηh)e−ηt Zt −∞ eηskf(s)k2 ∗ds, is pullback Dη(M2 H)-absorbing for the process Son M2 Hgiven by (6) (and thus also pullback DF(M2 H)-absorbing), and b D2,η belongs to Dη(M2 H). Since it will be useful in order to compare the pullback attractors defined in the spaces CHand M2 H, we consider the bi-parametric family of mappings e U(t, τ) : M2 H→L2 Hdefined as e U(t, τ)(uτ, φ) = ut(·;τ, uτ, φ) for (uτ, φ)∈M2 H, τ ≤t. Remark 8. Observe that e U(t, τ) maps M2 Hinto CHif t≥τ+h, and therefore we can write S(t, τ)(uτ, φ) = j(e U(t, τ)(uτ, φ)) for (uτ, φ)∈M2 H, t ≥τ+h, where S(·,·) is given by (6). Moreover, it is clear that U(t, τ)φ=e U(t, τ)j(φ) for φ∈CH, t ≥τ, 16 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL Then, since all Jnare non-increasing, we deduce that for all n≥n(kδ) Jn(tn)−J(t∗)≤Jn(˜ tkδ)−J(t∗) ≤ |Jn(˜ tkδ)−J(t∗)| ≤ |Jn(˜ tkδ)−J(˜ tkδ)|+|J(˜ tkδ)−J(t∗)|< δ. Therefore, as δ > 0 is arbitrary, we obtain that lim sup n→∞ Jn(tn)≤J(t∗), and consequently, again by (15) and (16), lim sup n→∞ kun(tn)k≤ku(t∗)k, which combined with (20) and (17) allows us to claim that un(tn)→u(t∗) strongly in V, in contradiction with (19). Thus, (18) is proved as desired. As an immediate consequence of the previous lemma, we have the following result. Corollary 3. Under the assumptions of Lemma 4, it holds: (a) For any ˜ h∈[0, h], the process U:R2 d×C˜ h,V H→C˜ h,V His pullback D˜ h,V η(CH)- asymptotically compact. (b) The process S:R2 d×M2 V→M2 Vis pullback DV η(M2 H)-asymptotically compact. We establish now the following result about the existence of minimal pullback attractors for the process Uon C˜ h,V H, which can be proved in a same way as [9, Theorem 5.1]. Theorem 5. Assume that f∈L2 loc(R; (L2(Ω))2), and that fand g:R×CH→ (L2(Ω))2satisfy (I)–(V). Then, for any ˜ h∈[0, h], the process Uon C˜ h,V Hpossesses a minimal pullback D˜ h,V η(CH)-attractor AD˜ h,V η(CH), a minimal pullback D˜ h,V F(CH)- attractor AD˜ h,V F(CH), and a minimal pullback DF(C˜ h,V H)-attractor ADF(C˜ h,V H). Besides, the following relations hold: ADF(C˜ h,V H)(t)⊂ AD˜ h,V F(CH)(t) ⊂ ADF(CH)(t) ⊂ AD˜ h,V η(CH)(t) = ADη(CH)(t) ⊂CV∀t∈R,(21) and for any family b D∈ Dη(CH), lim τ→−∞ distCV(U(t, τ)D(τ),ADη(CH)(t)) = 0 ∀t∈R. Finally, if moreover fsatisfies sup s≤0e−ηs Zs −∞ eηθ|f(θ)|2dθ<∞,(22) then all attractors in (21) coincide, and this family is tempered in CV, in the sense that lim t→−∞ eηt sup v∈ADη(CH)(t) kvk2 CV= 0, where kvkCV= maxs∈[−h,0] kv(s)kfor any v∈CV. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 17 Remark 13. Observe that, under the assumptions of Theorem 5, one has that AD˜ h,V η(CH)≡ ADh,V η(CH)for any ˜ h∈[0, h], i.e., the pullback attractor AD˜ h,V η(CH)is independent of ˜ h. Actually, if falso satisfies (22), then AD˜ h,V F(CH)≡ ADh,V F(CH), and ADF(C˜ h,V H)≡ ADF(Ch,V H). Remark 14. Under the assumptions of Theorem 5, since b D1,η,h belongs to Dh,V η(CH), and the set D1,η,h(t) is closed in Ch,V Hfor all t∈R, from Remarks 4and 10, and the equality in (21), we deduce that ADη(CH)belongs to Dh,V η(CH). In fact, if in addition fsatisfies (22), then, for each T∈R,the set {ADη(CH)(t) : t≤T}is bounded in Ch,V H. We are also able to obtain the existence of minimal pullback attractors for the process Son M2 V. Theorem 6. Suppose that f∈L2 loc(R; (L2(Ω))2), and that fand g:R×CH→ (L2(Ω))2satisfy (I)–(V). Then, there exist the minimal pullback DF(M2 V)-attractor ADF(M2 V), and the minimal pullback DV η(M2 H)-attractor ADV η(M2 H)for the process S on M2 V, and the following relations hold: ADF(M2 V)(t)⊂ ADF(M2 H)(t)⊂ ADη(M2 H)(t) = ADV η(M2 H)(t)∀t∈R.(23) In particular, for any family b D∈ Dη(M2 H), lim τ→−∞ distM2 V(S(t, τ)D(τ),ADη(M2 H)(t)) = 0 ∀t∈R.(24) Finally, if falso satisfies (22), then ADF(M2 V)(t) = ADF(M2 H)(t) = ADη(M2 H)(t) = ADV η(M2 H)(t)∀t∈R, and this family is tempered in M2 V, i.e., lim t→−∞ eηt sup (w,ϕ)∈ADη(M2 H)(t) k(w, ϕ)k2 M2 V= 0.(25) Proof. The existence of ADF(M2 V)and ADV η(M2 H)is a direct consequence of Theorem 2, Corollary 1, Proposition 2, Proposition 3, and Corollary 3. In (23), the inclusions follow from Corollary 1, Theorem 3, and Remark 11. The equality holds by applying Theorem 3and Remark 6, using Theorem 1, Lemma 2, Remark 11, and Corollary 3. The pullback attraction result (24) comes from Remark 8, Lemma 2, and the fact that by the regularity property (a) in Theorem 1, for any b D∈ Dη(M2 H) and any τ < t −h−1, distM2 V(S(t, τ)D(τ),ADη(M2 H)(t)) = distM2 V(S(t, τ +h+ 1)(S(τ+h+ 1, τ)D(τ)),ADη(M2 H)(t)) = distM2 V(S(t, τ +h+ 1)(j(e U(τ+h+ 1, τ)D(τ))),ADη(M2 H)(t)) = distM2 V(S(t, τ +h+ 1)(j(D(h+1)(τ))),ADV η(M2 H)(t)), since it is clear that the family {j(D(h+1)(τ)) : τ∈R}belongs to DV η(M2 H). If moreover fsatisfies (22), the equality ADF(M2 H)(t) = ADη(M2 H)(t) follows from Remark 5, and the equality ADF(M2 V)(t) = ADF(M2 H)(t) is again a consequence of 18 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL Theorem 3, by using the second estimate in (13), Remark 11, and Corollary 3, since (22) is equivalent to sup s≤0Zs s−1 |f(θ)|2dθ < ∞.(26) Lastly, the tempered property (25) comes from (22) (and therefore (26)) and the tempered character of ρ2(t) defined in Lemma 3. Remark 15. Under the assumptions of Theorem 6, reasoning analogously as in Remark 14, one has that ADη(M2 H)belongs to DV η(M2 H). To conclude, we relate the minimal pullback attractors obtained in C˜ h,V Hand M2 V through the canonical injection j. Theorem 7. Assume that f∈L2 loc(R; (L2(Ω))2), and that fand g:R×CH→ (L2(Ω))2satisfy (I)–(V). Then, the following relations hold: j(ADF(Ch,V H)(t)) ⊂ ADF(M2 V)(t)∀t∈R,and (27) j(AD˜ h,V η(CH)(t)) = ADV η(M2 H)(t)∀˜ h∈[0, h], t ∈R.(28) Actually, if falso satisfies (22), then, for any ˜ h∈[0, h], j(ADF(C˜ h,V H)(t)) = j(AD˜ h,V F(CH)(t)) = ADF(M2 V)(t)∀t∈R.(29) Proof. In order to prove the inclusion in (27) we proceed similarly as in [19, Theorem 5], taking into account that the map jis continuous from Ch,V Hinto M2 V, and that j(DF(Ch,V H)) ⊂ DF(M2 V). The equality in (28) is a consequence of property (11) in Theorem 4, using the equalities (21) and (23). Finally, the equalities in (29) follow from (28) and the known facts that, under the additional assumption (22), all attractors in (21) and (23) coincide. Acknowledgments. While finishing this paper among other projects, our coauthor Prof. Jos´e Real deceased. This work is dedicated to his memory, with our deepest and most sincere admiration, gratitude, and love. He was P.M.-R. and J.G.-L.’s PhD-advisor, clever mathematician with a sharp view on problems, generous and wonderful colleague, and better friend. He passed away too soon, being only 60 years old. We miss him deeply, but he will stay forever in our hearts. 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