Attractors for a Double Time-Delayed 2D-Navier-Stokes Model

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume XX, Number 0xx, XXXXXX 20xx pp. – ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL Julia Garc´ ıa-Luengo1, Pedro Mar´ ın-Rubio1& Gabriela Planas2 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080–Sevilla, Spain 2Departamento de Matem´atica Instituto de Matem´atica, Estat´ıstica e Computa¸c˜ao Cient´ıfica Universidade Estadual de Campinas 13083-859 Campinas, SP, Brazil (Communicated by XXXXX) Dedicated to the memory of Professor Jos´e Real with our deepest admiration, gratitude, and love. Abstract. In this paper, a double time-delayed 2D-Navier-Stokes model is considered. It includes delays in the convective and the forcing terms. Existence and uniqueness results and suitable dynamical systems are established. We also analyze the existence of pullback attractors for the model in several phase-spaces and the relationship among them. 1. Introduction and statement of the problem. The importance of physical models for fluid mechanics 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. [19] for a wind-tunnel model). The study of Navier-Stokes models including delay terms –existence, uniqueness, stationary solutions, exponential decay, existence of attractors, etcetera– was initiated in the references [5,6,7], 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. [28,11,21,23,9] among others). While the theory of linear viscoelasticity in fluid mechanics has often considered the inclusion of delay effects in the viscous part of the model (e.g. cf. [26]), the inclusion in other parts has not been investigated so often. In the recent paper [18] a time-delayed term in the Burgers’ equation was considered. Such a kind of delay in the trajectory that a particle should follow could present some obstacles to a rigorous physical interpretation. However, as many other simplified and/or approximative models in fluid mechanics (with truncations, as the globally modified Navier-Stokes equations, e.g. cf. [4,15,16,27,20]), this kind of effect may be interesting to study from the mathematical point of view. 2010 Mathematics Subject Classification. Primary: 35Q35, 35Q30, 35B40, 37L30. Key words and phrases. 2D Navier-Stokes equations; delay terms; pullback attractors. 1 2 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS Consider a bounded domain Ω ⊂R2, τ ∈R,and the non-autonomous functional Navier-Stokes model              ∂u ∂t −ν∆u+ (u(t−ρ(t)) · ∇)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) in Ω, u(x, τ +s) = φ(x, s) in Ω ×(−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, utdenotes –as usual– the delay function ut(s) = u(t+s) where it has sense. The delay function ρin the convective term is assumed to belong to C1(R; [0, h]) with ρ0(t)≤ρ∗<1 for all t∈R,where h > 0 is fixed, and uτand φare the initial data in τand (τ−h, τ) respectively. Existence, uniqueness, some regularity features for this model, and some partial long-time estimates were studied in [25] in dimension two (see [12] for the case in dimension three). The interesting point of the model in dimension two is that the natural estimate of u0is in L4/3(V0) (see below for the proper definitions), as the Navier-Stokes equations in three dimensions without delay does. This means that, without any additional assumption on the phase-space, the appearance of a delay –however small it be– in the nonlinear term has an important influence. Therefore, the study of existence of attractor (or pullback attractor) is more involved for problem (1), leading to the same kind of (lack of uniqueness or lack of continuity in time) troubles (e.g. cf. [1,24,14,13] for multi-valued approaches). Our approach in this paper is to modify the phase-space improving slightly the initial conditions, such that existence and uniqueness of solution hold. For the associated single-valued process, we will study the existence of pullback attractors in different universes and the relation among them. The structure of the paper is as follows. In the rest of this section we recall the abstract setting of the problem with the standard functional spaces, and the definition of a weak solution to problem (1). In Section 2we recall for completeness the proof of existence of weak solutions, and the uniqueness under additional assumptions in the phase-space (that are closely related to the existence of an energy equality). Continuity with respect to the initial data are also given. Section 3provides a very briefly summary on the theory of minimal pullback attractors, that will be used in the last part of the paper. Our main results are given in Section 4, where estimates on the solutions, absorption, and asymptotic compactness are proved, leading to the existence of several minimal pullback attractors, in different phase-spaces. We also establish some relations among these families. Finally, Section 5is devoted to expose the above results in the autonomous framework. This allows to simplify the statements and concentrate in the problem of a delay perturbation in the convective term. Existence of global attractors and relationship among them are so deduced. We will consider the usual functional spaces to deal with the problem in an abstract setting (e.g. cf. [17,29]). Let be V={u∈(C∞ 0(Ω))2: div u= 0}; ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 3 His 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; Vis 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. Define the operator A:V→V0as hAu, vi:= ((u, v)) ∀u, v ∈V. Let us denote 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. For suitable uand v(for instance in V) it is also useful to denote B(u, v) the operator of V0given by hB(u, v), wi=b(u, v, w) for any w∈V. On other hand, let us recall that the operator bsatisfies b(u, v, v) = 0 ∀u∈V, v ∈(H1 0(Ω))2,(2) and since we are in dimension two there exists a constant C > 0,depending only on Ω,such that |b(u, v, w)| ≤ C|u|1/2kuk1/2kvk|w|1/2kwk1/2∀u, v, w ∈V. (3) Before continuing, for short, we introduce the notation Lp X=Lp(−h, 0; X),which will be used in the sequel for suitable choices of pand X. The norm in these spaces will be denoted by k·kLp X.On other hand, CH=C([−h, 0]; H) will also be used, and the sup norm in CHwill be denoted by |·|CH.Finally, BE(0, α) will denote the closed ball in a metric space Eof center zero and radius α. The second delay operator is g:R×CH→(L2(Ω))2,and we assume that it satisfies the following assumptions: (H1) for all ξ∈CH,the function R3t7→ g(t, ξ)∈(L2(Ω))2is measurable, (H2) g(t, 0) = 0,for all t∈R, (H3) 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, (H4) there exists Cg>0 such that for all τ≤tand for all u, v ∈C([τ−h, t]; H) Zt τ |g(r, ur)−g(r, vr)|2dr ≤C2 gZt τ−h |u(r)−v(r)|2dr. 4 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS Examples of fixed, variable, and distributed delay operators can be found, for instance, in [5, Section 3], [7, Sections 3.5 and 3.6], and [11, Section 3], and we omit them here just for the sake of brevity. Remark 1. From (H1)–(H3), for T > τ and u∈C([τ−h, T]; H),the function gu: [τ, T ]→(L2(Ω))2given by gu(t) = g(t, ut) is measurable and belongs to L∞(τ, T ; (L2(Ω))2).By using (H4), the mapping C([τ−h, T]; H)3u7→ G(u) := 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).We will still denote by g(t, ut) = e G(u)(t) for each u∈L2(τ−h, T;H),and therefore assumption (H4) will hold for all u, v ∈L2(τ− h, T;H). Concerning the goal of finding solutions to problem (1), different choices are possible for the initial data. Let us consider that uτ∈H, φ ∈L2 V,and f∈L2 loc(R;V0). Definition 1. A weak solution to (1) is a function u∈L∞(τ, T ;H)∩L2(τ−h, T;V) for all T > τ, such that u(τ) = uτ, uτ=φ, and satisfies d dt(u(t), v) + νhAu(t), vi+b(u(t−ρ(t)), u(t), v) = hf(t), vi+ (g(t, ut), v)∀v∈V, where the equation must be understood in the sense of D0(τ, ∞). Remark 2. Let us observe that if uis a weak solution to (1), from (3), in particular we have that there exists a constant e C > 0 such that for any v∈V, |b(u(t−ρ(t)), u(t), v)| ≤ e Cku(t−ρ(t))kku(t)k1/2|u(t)|1/2kvk, where we have used the continuous embedding of Vinto H. Then, by Young inequality, we conclude that B(u(·−ρ(·)), u(·)) ∈L4/3(τ, T;V0). Therefore, u0∈L4/3(τ, T ;V0) too. So, u∈C([τ, T]; V0) and in particular (e.g. cf. [29]) u∈Cw([τ, T ]; H) for all T > τ (whence to impose an initial datum uτ∈H is meaningful). Although the above choice of phase-space will lead to an existence result (see Theorem 1below), the well-posedness of the problem in the sense of Hadamard will require more regularity on the initial data, pointing out that the above was an unnatural choice (compare with Remark 3and Theorem 2below). 2. Existence of solutions, uniqueness, and continuity results. We have the following result concerning existence of weak solutions. It is also worth mentioning that the delay in the convective term, even if his small, does matter in the sense that uniqueness of solution to (1) is unknown (compare Remark 2–essentially as the case without delay in dimension three– with Remark 3and Theorem 2below, where this difficulty is sorted out). Theorem 1. Consider uτ∈H, φ ∈L2 V, f ∈L2 loc(R;V0),and g:R×CH→ (L2(Ω))2satisfying assumptions (H1)–(H4). Then, there exists at least one weak solution u(·;τ, uτ, φ)to (1). ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 5 Proof. The existence of weak solution can be proved as in [25, Theorem 2.1], and we include its proof here just for the sake of clarity. Consider a special basis of Hformed by normalized eigenfunctions of the Stokes operator, {wj}j≥1,with corresponding eigenvalues {λj}j≥1being 0 < λ1≤λ2≤. . . with limj→∞ λj=∞.Pose the approximate problems (for each k≥1) of finding uk∈Vk:= span[w1, . . . , wk] with uk(t) = Pk j=1 γjk(t)wjsuch that d dt(uk(t), wj) + νhAuk(t), wji+b(uk(t−ρ(t)), uk(t), wj) =hf(t), wji+ (g(t, uk t), wj), a.e. t > τ, ∀1≤j≤k, (4) fulfilled with the initial conditions uk(τ) = Pkuτand uk(τ+s) = Pkφ(s) in s∈(−h, 0), where Pkis the orthogonal projector from Honto Vk. It is well known (e.g. cf. [5]) that the above system of ordinary functional differential equations (the unknowns are {γjk}k j=1) is well-posed in some local interval [τ, tk).We fix a value T > τ and will provide uniform estimates that will imply that actually it holds that tk=Tand pass to the limit via compactness arguments, whence existence of a weak solution on (τ, T ) will be ensured. Indeed, multiplying each equation in (4) by γjk(t) and summing from j= 1 to k, we obtain 1 2 d dt|uk(t)|2+νkuk(t)k2=hf(t), uk(t)i+ (g(t, uk t), uk(t)), a.e. t ∈(τ, tk), where we have used (2) to remove the nonlinear term b. By integrating in time, from H¨older and Young inequalities, and the assumptions on the delay operator g, we obtain that |uk(t)|2+ 2νZt τ kuk(s)k2ds ≤ |uτ|2+1 νZt τ kf(s)k2 ∗ds +νZt τ kuk(s)k2ds +Zt τ |g(s, uk s)|2ds +Zt τ |uk(s)|2ds ≤ |uτ|2+C2 gZ0 −h |φ(s)|2ds +1 νZt τ kf(s)k2 ∗ds +νZt τ kuk(s)k2ds +(1 + C2 g)Zt τ |uk(s)|2ds for all t∈[τ, tk). So, we deduce that |uk(t)|2+νZt τ kuk(s)k2ds ≤ |uτ|2+C2 gZ0 −h |φ(s)|2ds +1 νZt τ kf(s)k2 ∗ds + (1 + C2 g)Zt τ |uk(s)|2ds for all t∈[τ, tk). Now, from Gronwall lemma, we conclude that tk=T, and that {uk}is bounded in L∞(τ, T ;H)∩L2(τ−h, T;V).Moreover, from (3) (see also Remark 2) we have that {duk/dt}is bounded in L4/3(τ, T ;V0),whence by compactness results, the Dominated Convergence Theorem, assumption (H4), and Remark 1, we may extract 6 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS a subsequence (relabelled the same) and ensure the existence of a function u∈ L2(τ−h, T;V) with du/dt ∈L4/3(τ, T;V0) with uτ=φ, such that uk→ustrongly in L2(τ−h, T;H), uk* u weakly in L2(τ−h, T ;V), duk/dt * du/dt weakly in L4/3(τ, T;V0), uk(· − ρ(·)) →u(· − ρ(·)) strongly in L2(τ, T ;H), g(·, uk ·)→g(·, u·) strongly in L2(τ, T ;H). (5) It is standard to pass to the limit in (4). Just for clarity, we point out how to deal with the delayed convective term, which is the novelty here. Indeed, it holds that b(uk(· − ρ(·)), uk(·), wj)→b(u(· − ρ(·)), u(·), wj) in L1(τ, T ), since |b(uk(t−ρ(t)), uk(t), wj)−b(u(t−ρ(t)), u(t), wj)| =|b(uk(t−ρ(t)), wj, uk(t)) −b(u(t−ρ(t)), wj, u(t)) ±b(u(t−ρ(t)), wj, uk(t))| ≤ |b(uk(t−ρ(t)) −u(t−ρ(t)), wj, uk(t))|+|b(u(t−ρ(t)), wj, uk(t)−u(t))|. We will prove that the first addend in the right hand-side goes to zero in the L1(τ, T) norm (the second addend follows analogously). Using (3), H¨older inequality, and the fact that wjis an eigenfunction of the Stokes operator, we have that ZT τ |b(uk(t−ρ(t)) −u(t−ρ(t)), wj, uk(t))|dt ≤Cλ1/2 jkuk(· − ρ(·)) −u(· − ρ(·))k1/2 L2(τ,T ;H)kuk(· − ρ(·)) −u(· − ρ(·))k1/2 L2(τ,T ;V) ×kuk(·)k1/2 L2(τ,T ;H)kuk(·)k1/2 L2(τ,T ;V). From (5) the above goes to zero, and the claim is proved. Thus, we conclude that uis a weak solution to (1) in the interval (τ, T ). By concatenation of solutions, it is clear that we obtain at least one global (defined on (τ, ∞)) weak solution to (1). If we modify slightly the initial data we may improve the above result in the sense that we gain an energy equality (and therefore uniqueness of solution and continuity of the solutions with respect to initial data). So we will be in a good position to study the associated dynamical system (which will be continuous). Roughly speaking, what we do now is to impose on the initial data the same regularity as we expect for the weak solutions. Remark 3. Suppose that uτ∈Hand φ∈L2 V∩L∞ H.Then, we may improve the regularity for the operator B(u(· − ρ(·)), u(·)) obtained in Remark 2. Indeed, from (3) we have that for any v∈V, |b(u(t−ρ(t)), u(t), v)| ≤ C|u(t−ρ(t))|1/2ku(t−ρ(t))k1/2kvk|u(t)|1/2ku(t)k1/2.(6) Therefore, we can conclude now that B(u(·−ρ(·)), u(·)) ∈L2(τ, T;V0) for all T > τ, and so u0∈L2(τ, T ;V0) and u∈C([τ, T]; H) for all T > τ. Now, the following energy equality holds for any solution to (1), |u(t)|2+ 2νZt s ku(r)k2dr =|u(s)|2+ 2 Zt s hf(r), u(r)idr + 2 Zt s (g(r, ur), u(r))dr ∀τ≤s≤t. (7) ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 7 Next, we establish a uniqueness result for problem (1). Theorem 2. Consider uτ∈H, φ ∈L2 V∩L∞ H, f ∈L2 loc(R;V0),and g:R× CH→(L2(Ω))2satisfying assumptions (H1)–(H4). Then, there exists a unique weak solution to (1), u(·;τ, uτ, φ)∈C([τ, ∞); H),which satisfies the energy equality (7). Moreover, if for short we denote by u(·)and v(·)the corresponding solutions to (1) with respective initial data (uτ, φ)and (vτ, ψ),then ess sup s∈(t−h,t) |u(s)−v(s)|2≤kφ−ψk2 L∞ H+ (λ−1 1+ν 2)Z0 −h kφ(s)−ψ(s)k2ds ×exp b CZt τ (ku(s)k2+ 1)ds,(8) νZt τ ku(s)−v(s)k2ds ≤kφ−ψk2 L∞ H+ (λ−1 1+ν 2)Z0 −h kφ(s)−ψ(s)k2ds(9) ×1 + b Cexp b CZt τ (ku(s)k2+1)dsZt τ (ku(s)k2+ 1)ds for all t≥τ, where b C=C2ν−1(1 −ρ∗)−1/2+C2 g+ 1. Proof. The existence of at least one weak solution was already proved in Theorem 1. The energy equality (7) was given in Remark 3for any solution to (1). So, it only remains to check uniqueness, and estimates (8) and (9). Actually, we will obtain uniqueness as a by-product of (8). Indeed, consider two solutions u(·) and v(·) to (1) with corresponding initial data (uτ, φ) and (vτ, ψ) respectively, and denote w=u−v. Then, from (2) and the energy equality for w, we obtain that 1 2 d dt|w(t)|2+νkw(t)k2+b(w(t−ρ(t)), u(t), w(t)) = (g(t, ut)−g(t, vt), w(t)), a.e. t > τ. (10) Now, as φ, ψ ∈L2 V∩L∞ H,by (3) we have the following estimate for the trilinear term b, |b(w(t−ρ(t)), u(t), w(t))| ≤C|w(t−ρ(t))|1/2kw(t−ρ(t))k1/2ku(t)k|w(t)|1/2kw(t)k1/2 ≤Cess sup r∈(t−h,t) |w(r)|ku(t)kkw(t−ρ(t))k1/2kw(t)k1/2. Integrating in time (10) and using the above estimate, the assumptions on g, and Young and H¨older inequalities with a suitable constant (to be fixed later on), we deduce that |w(t)|2+ 2νZt τ kw(s)k2ds ≤ |w(τ)|2+C2 εZt τ ess sup r∈(s−h,s) |w(r)|2ku(s)k2ds +εZt τ kw(s−ρ(s))kkw(s)kds +2 Zt τ (g(s, us)−g(s, vs), w(s))ds 8 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS ≤ |w(τ)|2+C2 εZt τ ess sup r∈(s−h,s) |w(r)|2ku(s)k2ds +ν 2Zt τ kw(s)k2ds +ε2 2νZt τ kw(s−ρ(s))k2ds +Zt τ−h |w(s)|2ds +C2 gZt τ |w(s)|2ds ∀t≥τ. In particular, after a change of variable in the integral of w(s−ρ(s)),thanks to the upper bound on ρ0,and choosing ε2=ν2(1 −ρ∗),we arrive at |w(t)|2+νZt τ kw(s)k2ds ≤ |w(τ)|2+ (λ−1 1+ν/2) Zτ τ−h kw(s)k2ds +C2 εZt τ ess sup r∈(s−h,s) |w(r)|2ku(s)k2ds + (1 + C2 g)Zt τ |w(s)|2ds ∀t≥τ.(11) Thus, neglecting the integral term in the left hand side above, putting s∈(t−h, t) instead of t, and taking the essential supremum in the resulting left hand side, we conclude that ess sup s∈(t−h,t) |w(s)|2≤ kwτk2 L∞ H+ (λ−1 1+ν/2) Zτ τ−h kw(s)k2ds +C2 ε+C2 g+ 1Zt τ (ku(s)k2+ 1) ess sup r∈(s−h,s) |w(r)|2ds for all t≥τ, whence (8) holds by applying Gronwall lemma. Finally, (9) is a consequence of (11) by using (8). Remark 4. It is worth mentioning that even with φ∈L2 Valone, the regularisation of the equation means that after an elapsed time hthe weak solution obtained in Theorem 1becomes well-posed and continuous. The problem is that before that elapsed time we cannot guarantee uniqueness of solution. So a possible dynamical system in such phase-space H×L2 Vwould be eventually multi-valued, which means that all the study of the asymptotic behaviour would be an open question (among many conditional results for this type of problems, we recall the seminal paper by J. M. Ball [1]). 3. Existence and comparison of minimal pullback attractors. We give a brief summary of some well-known abstract results on existence and comparison of minimal pullback attractors for dynamical systems (e.g. cf. [2,3,22,8]). 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. A process Uis said to be continuous if for any pair τ≤t, the mapping U(t, τ) : X→Xis continuous. On other hand, a process 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. 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). ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 9 Definition 2. 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). Definition 3. 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). Next result was proved in [8, Theorem 3.11]. Theorem 3. 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) = S b 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 5. Under the assumptions of Theorem 3, 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). 16 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS Indeed, in order to prove (24), fix an arbitrary value t∈R,and observe that ADF(L2 V∩CH)(t) = [ B⊂L2 V∩CH Bbounded ΛL2 V∩CH(B, t) L2 V∩CH , where the symbol ΛL2 V∩CHdenotes the omega-limit construction with respect to the topology of the space L2 V∩CH. Analogously, we have that ADF(H×(L2 V∩L∞ H))(t) = [ B⊂H×(L2 V∩L∞ H) Bbounded ΛH×(L2 V∩L∞ H)(B, t) H×(L2 V∩L∞ H) , where the symbol ΛH×(L2 V∩L∞ H)denotes the omega-limit construction with respect to the topology of the space H×(L2 V∩L∞ H). Now, observe that for any bounded set B⊂L2 V∩CH,since the operator jis clearly linear and continuous, then j(B) is also bounded in H×(L2 V∩L∞ H). If x∈ΛL2 V∩CH(B, t),then there exist sequences {τn},with τn≤tfor all n, and limnτn=−∞,and {xn} ⊂ B, such that x= lim τn→−∞ U(t, τn)xnin L2 V∩CH. But this implies that (x(0), x) = lim τn→−∞ S(t, τn)(xn(0), xn) in H×(L2 V∩L∞ H), whence we deduce that j(ΛL2 V∩CH(B, t)) ⊂ΛH×(L2 V∩L∞ H)(j(B), t) for all bounded set B⊂L2 V∩CH.Thus, (24) follows. The inclusion to the right in (25) can be proved analogously. Let us now prove the inclusion to the left in (25). Indeed, for any t∈Rand b D∈ DH,L2 H η(H×(L2 V∩L∞ H)), we have that for any τ < t −h distH×(L2 V∩L∞ H)(S(t, τ)D(τ), j(ADCH η(L2 V∩CH)(t))) = distH×(L2 V∩L∞ H)(S(t, τ +h)(S(τ+h, τ)D(τ)), j(ADCH η(L2 V∩CH)(t))) = distH×(L2 V∩L∞ H)(j(U(t, τ +h)D(h)(τ)), j(ADCH η(L2 V∩CH)(t))) ≤C(j)distL2 V∩CH(U(t, τ +h)D(h)(τ),ADCH η(L2 V∩CH)(t)), where we have used the notation introduced in Lemma 3for the family b D(h),which belongs to DCH η(L2 V∩CH),and once more the fact that jis a linear and continuous operator from L2 V∩CHto H×(L2 V∩L∞ H).Thus, we have that the right-hand side of the above inequality goes to zero when τgoes to −∞,and so the left-hand side also does. Therefore, the inclusion ADH,L2 H η(H×(L2 V∩L∞ H))(t)⊂j(ADCH η(L2 V∩CH)(t)) follows since ADH,L2 H η(H×(L2 V∩L∞ H))(t) is the minimal closed set in H×(L2 V∩L∞ H) that attracts any family b D∈ DH,L2 H η(H×(L2 V∩L∞ H)) at time tin a pullback sense. ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 17 Last claim about the equalities in (23) and (24) when falso satisfies (22) follows again from Corollary 1since then it holds that supt≤TRH(t) and supt≤TRV(t) are bounded for any T∈R.This gives immediately the equality in (23). Then, combining this with the equality in (25) and the equality in (21), we conclude that (24) becomes an equality too, for all t∈R. Remark 6. Under the assumptions of the above theorem, if besides fsatisfies (22), then, for each T∈R,the sets {ADCH η(L2 V∩CH)(t)}t≤Tand {ADH,L2 H η(H×(L2 V∩L∞ H))(t)}t≤T are bounded in L2 V∩CHand H×(L2 V∩L∞ H) respectively. 5. The autonomous case. In this section we translate and adapt the previous results to the framework of time-independent forces. Observe that without an explicit dependence on time, the dynamical system then becomes autonomous, which means that only the elapsed time is important, rather than the pair of initial and final times. Actually, the autonomous results are just a particular case of all the previous exposition, but for some readers it might be a more clear exposition of the nature of the problem itself without the interferences of non-autonomous modifications. In particular, we will be able to state the existence of the global attractor for the cited (autonomous) dynamical system under suitable conditions. Consider the functional Navier-Stokes model              ∂u ∂t −ν∆u+ (u(t−h)· ∇)u+∇p=f+g(ut) in Ω ×(0,∞), div u= 0 in Ω ×(0,∞), u= 0 on ∂Ω×(0,∞), u(x, 0) = u0(x) in Ω, u(x, s) = φ(x, s) in Ω ×(−h, 0), (26) where all the unknowns were already explained in the introduction of the paper (h > 0 is fixed and now ρ≡h). Observe too that f, the non-delayed external force field, and g, the external force with some hereditary characteristics, are timeindependent. Let us also observe that in contrast to (1), here τ= 0 (actually, since the problem is autonomous, the initial time is not relevant). For the delay operator gwe assume that g:CH→(L2(Ω))2satisfies (observe that the assumption (H1) holds trivially in this framework): (H2’) g(0) = 0, (H3’) there exists Lg>0 such that for all ξ, η ∈CH, |g(ξ)−g(η)| ≤ Lg|ξ−η|CH, (H4’) there exists Cg>0 such that for all 0 ≤τ≤t, and for all u, v ∈C([−h, t]; H) Zt τ |g(ur)−g(vr)|2dr ≤C2 gZt τ−h |u(r)−v(r)|2dr. Then, the immediate translation of the first existence result (cf. Theorem 1) is the following Theorem 6. Consider u0∈H, φ ∈L2 V, f ∈V0,and g:CH→(L2(Ω))2satisfying assumptions (H2’)–(H4’). Then, there exists at least one weak solution u(·; 0, u0, φ) to (26). 18 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS Of course, the concept of weak solution given in Definition 1to problem (1) just needs to substitute τ= 0 to be referred to problem (26). The trouble of this result arises from the fact that (cf. Remark 2)B(u(· − h), u(·)) ∈L4/3(0, T;V0),which does not allow to apply an energy equality, and therefore uniqueness is unknown. However, if we improve slightly the initial data (cf. Remark 3) we gain B(u(· − h), u(·)) ∈L2(0, T ;V0).We state these results precisely in the following Theorem 7. Consider u0∈H, φ ∈L2 V∩L∞ H, f ∈V0,and g:CH→(L2(Ω))2 satisfying assumptions (H2’)–(H4’). Then, there exists a unique weak solution to (26), which additionally satisfies u(·; 0, u0, φ)∈C([0,∞); H)and the energy equality (7) for all 0≤s≤t. Moreover, if for short we denote by u(·)and v(·)the corresponding solutions to (26) with respective initial data (u0, φ)and (v0, ψ),then the estimates (8) and (9) (with τ= 0) hold for all t≥0,where b C=C2ν−1+C2 g+ 1. Once that a solution operator to problem (26) is suitably given, since continuous dependence with respect to initial data holds (by the previous theorem), and concatenation of solutions is clearly a solution too, one may use the standard results of (autonomous) dynamical systems (see e.g. [29] for a detailed exposition on concepts and results). Let us for the sake of brevity just include the very essential elements we need for our analysis. Definition 6. A semi flow Son a metric space (X, dX) is a mapping R+×X3 (t, x)7→ S(t)x∈Xsuch that S(0) =IdX, and S(t)S(s)x=S(t+s)xfor any t, s ≥0 and all x∈X. It is said that the semi flow Sis continuous if for any t∈R+, the mapping S(t) : X→Xis continuous. The semi flow Sis said to be asymptotically compact if for any bounded sequence {xn} ⊂ Xand {tn} ⊂ R+with limntn=∞,the sequence {S(tn)xn}is relatively compact in X. A subset B0⊂Xis said to be absorbing for the semi flow Sif for any bounded subset B⊂Xthere exists a time T(B)≥0 such that S(t)B:= ∪b∈BS(t)b⊂B0 for all t≥T(B). A subset A ⊂ Xis said to be a global attractor for the semi flow Son Xif it is compact, invariant (i.e. S(t)A=Afor all t∈R+), and it attracts bounded sets of X, i.e. limt→∞ distX(S(t)B, A) = 0 for all B⊂Xbounded. Observe that from the definition of a global attractor for a semi flow, if it exists, it is unique. Moreover, it is the minimal closed set with the property of attracting all bounded sets, and the maximal compact invariant set. With the above concepts, the basic result on existence of global attractor is the following. Theorem 8. (cf. [29]) Consider a semi flow Sdefined on a metric space (X, dX), which is continuous. Then, there exists the global attractor Afor Sif and only if the semi flow is asymptotically compact and it has a bounded absorbing set B0⊂X. Moreover, then A=ω(B0) := \ t≥0[ s≥t S(s)B0 X . ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 19 To apply the above result, as in Section 4we consider the Banach space X= H×(L2 V∩L∞ H),wit the norm k(ζ, φ)kX=|ζ|+kφkL2 V+kφkL∞ Hfor a pair (ζ, φ)∈X. After Corollary 2, but within the assumptions of Theorem 7, the family of mappings that form a continuous semi flow is now given by S(t) = S(t, 0) from into X, and for any t∈R+,i.e. S(t) : H×(L2 V∩L∞ H)→H×(L2 V∩L∞ H) given by S(t)(u0, φ) = (u(t), ut) where uis the weak solution to (26). In order to obtain asymptotic estimates, we impose this new condition: (H5’) Assume that νλ1> Cg, and that there exists a value η∈(0,2(νλ1−Cg)) such that for any 0 ≤τ≤tand for every u∈L2(τ−h, t;H), Zt τ eηs|g(us)|2ds ≤C2 gZt τ−h eηs|u(s)|2ds ∀t≥τ≥0. The analogous result to Lemma 1is the following Lemma 4. Consider given f∈V0and g:CH→(L2(Ω))2satisfying conditions (H2’)–(H5’). Then, for any (u0, φ)∈H×(L2 V∩L∞ H), the following inequalities hold for the solution uto (26) for all t≥s≥0: |u(t)|2≤e−ηt(|u0|2+Cgkφk2 L2 H) + 1 βη kfk2 ∗, νZt s ku(r)k2dr ≤ |u(s)|2+Cgkusk2 L2 H+1 νkfk2 ∗(t−s)+2CgZt s |u(r)|2dr, where βis given by (14). Since condition (15) now is fulfilled trivially for a constant f∈V0,as a particular case of Corollary 3we have the first ingredient for applying Theorem 8: the existence of an absorbing set. Corollary 4. Under the assumptions of Lemma 4, the set b B0:= BH(0,b RH)×(BL2 V(0,b RV)∩BL∞ H(0,b RH)) ⊂H×(L2 V∩L∞ H), where b R2 H= 1 + (βη)−1kfk2 ∗,b R2 V=ν−1[(1 + 3Cgh)b R2 H+ν−1hkfk2 ∗], is absorbing for the semi flow Son H×(L2 V∩L∞ H). Second ingredient for applying Theorem 8is the asymptotic compactness of S, but this is again a consequence of the previously proved result in Section 4(see Lemma 2; observe that the autonomous or non-autonomous formulation is not really a matter for the application of the energy method). Lemma 5. Under the assumptions of Lemma 4, the semi flow Sis asymptotically compact. Main result of this section is the following Theorem 9. Assume that f∈V0,and g:CH→(L2(Ω))2satisfies conditions (H2’)–(H5’). Then, there exists the global attractor ADF(H×(L2 V∩L∞ H)) for Son H×(L2 V∩L∞ H). Since the adaptation of Section 4.1 is also obvious (we omit the details here just for the sake of brevity), and we may consider a natural semi flow e S:R+×L2 V∩CH→ L2 V∩CHgiven by e S(t) = U(t, 0),and the continuity of this semi flow and absorption and asymptotic compactness properties are not difficult to obtain (inherited from 20 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND G. PLANAS the proofs in that section), we conclude the following result (compare with Theorem 5). Theorem 10. Assume that f∈V0,and g:CH→(L2(Ω))2satisfies conditions (H2’)–(H5’). Then, there exists the global attractor ADF(L2 V∩CH)for e Son L2 V∩CH. Moreover, the following relation holds: j(ADF(L2 V∩CH)) = ADF(H×(L2 V∩L∞ H)) where j:L2 V∩CH→H×(L2 V∩L∞ H)is defined by j(ϕ) = (ϕ(0), ϕ). Acknowledgments. This work is dedicated to the memory of Prof. Jos´e Real, with our deepest 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. The authors thank the referees for a careful reading that led to some improvements in the presentation of the paper. This work has been partially supported by Ministerio de Educaci´on–DGPU (Spain) project PHB2010-0002-PC, Ministerio de Educaci´on y Ciencia (MEC, Spain), grant MTM2011-22411, Junta de Andaluc´ıa grant P07-FQM-02468, and CapesDGU (Brazil) 238/11. G.P. also thanks to CNPq - Brazil, grant 303302/2009-7. J.G.-L. is a fellow of Programa de FPU del Ministerio de Educaci´on (SPAIN). REFERENCES [1] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the NavierStokes equations, J. Nonlinear Sci. 7(1997), 475–502. [2] T. Caraballo, G. Lukaszewicz, and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006), 484–498. [3] T. Caraballo, G. Lukaszewicz, and J. Real, Pullback attractors for non-autonomous 2DNavier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris 342 (2006), 263–268. [4] T. Caraballo, P. E. Kloeden, and J. Real, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud. 6(2006), 411–436. [5] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), 2441–2453. [6] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), 3181–3194. [7] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations 205 (2004), 271–297. [8] J. Garc´ıa-Luengo, P. Mar´ın-Rubio, and J. Real, Pullback attractors in Vfor non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations 252 (2012), 4333–4356. [9] J. Garc´ıa-Luengo, P. Mar´ın-Rubio, and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud. 13 (2013), 331–357. [10] J. Garc´ıa-Luengo, P. Mar´ın-Rubio, and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal. To appear. [11] M. J. Garrido-Atienza and P. Mar´ın-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal. 64 (2006), 1100–1118. [12] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B 16 (2011), 225–238. [13] O. V. Kapustyan, P. O. Kasyanov, and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl. 373 (2011), 535–547. [14] A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations 240 (2007), 249–278. ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL 21 [15] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real, and J. Valero, The three-dimensional globally modified Navier-Stokes equations, Advances in Nonlinear Analysis: Theory Methods and Applications, Math. Probl. Eng. Aerosp. Sci. 311–22, Cambridge Scientific Publishers, Cambridge, 2009. [16] P. E. Kloeden, P. Mar´ın-Rubio, and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal. 8(2009), 785–802. [17] J. L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod, Paris, 1969. [18] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers’ equation, Discrete Contin. Dyn. Syst. Ser. B 2(2002), 47–56. [19] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation, IEEE Trans. Automat. Control 29 (1984), 1058–1068. [20] P. Mar´ın-Rubio, A. M. M´arquez-Dur´an, and J. Real, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud. 11 (2011), 917–927. [21] P. Mar´ın-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal. 67 (2007), 2784–2799. [22] P. Mar´ın-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal. 71 (2009), 3956–3963. [23] P. Mar´ın-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst. 26 (2010), 989–1006. [24] P. Mar´ın-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stoch. Dyn. 3(2003), 279–297. [25] G. Planas and E. Hern´andez, Asymptotic behaviour of two-dimensional time-delayed NavierStokes equations, Discrete Contin. Dyn. Syst. 21 (2008), 1245–1258. [26] M. Renardy, A class of quasilinear parabolic equations with infinite delay and application to a problem of viscoelasticity, J. Differential Equations 48 (1983), 280–292. [27] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud. 9(2009), 425–427. [28] T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst. 12 (2005), 997–1018. [29] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1988. E-mail address, J. Garc´ıa-Luengo: [email protected] E-mail address, P. Mar´ın-Rubio: [email protected] E-mail address, G. Planas: [email protected]