Some regularity results for a double time-delayed 2d-Navier-Stokes model

DISCRETE AND CONTINUOUS doi: DYNAMICAL SYSTEMS SERIES B Volume , Number 0, pp. 000–000 SOME REGULARITY RESULTS FOR A DOUBLE1 TIME-DELAYED 2D-NAVIER-STOKES MODEL2 Julia Garc ´ ıa-Luengo Departamento de Matem´atica Aplicada a las TIC Universidad Polit´ecnica de Madrid C/ Nikola Tesla s/n, 28031, Madrid, Spain Pedro Mar ´ ın-Rubio Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla C/ Tarfia s/n, 41012, Sevilla, Spain Gabriela Planas Departamento de Matem´atica Instituto de Matem´atica, Estat´ıstica e Computa¸c˜ao Cient´ıfica Universidade Estadual de Campinas Rua Sergio Buarque de Holanda 651, 13083-859 Campinas, SP, Brazil Dedicated to Professor Peter E. Kloeden on occasion of his Seventieth Birthday Abstract. In this paper we analyze some regularity properties of a double time-delayed 2D-Navier-Stokes model, that includes not only a delay force but also a delay in the convective term. The interesting feature of the model -from the mathematical point of viewis that being in dimension two, it behaves similarly as a 3D-model without delay, and extra conditions in order to have uniqueness were required for well-posedness. This model was previously studied in several papers, being the existence of attractor in the L2-framework obtained by the authors [Discrete Contin. Dyn. Syst. 34 (2014), 4085–4105]. Here regularization properties of the solutions and existence of (regular) attractors for several associated dynamical systems are established. Moreover, relationships among these objects are also provided. 1. Introduction, statement of the problem, and previous results. Consider3 a bounded domain Ω ⊂R2,τ∈R,and the following non-autonomous functional4 Navier-Stokes model5             ∂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) 2010 Mathematics Subject Classification. Primary: 35Q30, 35Q35, 35B65, 35B41, 37L30. Key words and phrases. Navier-Stokes equations; nonlinear delay terms; well-posed/ill-posed Navier-Stokes; delayed convective term; regular attractors; tempered universes for non-autonomous dynamical systems. 1 2 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO AND G. PLANAS where ν > 0 is the kinematic viscosity, u= (u1, u2) is the velocity field of the fluid,1 pis the pressure, fis a non-delayed external force field, gis another external force2 with some hereditary characteristics with memory length bounded by h > 0, ut 3 denotes the delay function ut(s) = u(t+s) where it has sense. The delay function4 ρin the convective term is assumed to belong to C1(R; [0, h]) with ρ0(t)≤ρ∗<15 for all t∈R,and uτand φare the initial data in τand (τ−h, τ) respectively.6 7 The study of Navier-Stokes models including delay terms –existence, uniqueness,8 stationary solutions, exponential decay, existence of attractors, and other issues– is9 initiated by Caraballo and Real [1,2,3], and after that, different questions have10 been addressed (e.g.,[19,9,13,15,16,6,7]).11 12 In particular, the inclusion of a delay term in the convective part is firstly con-13 sidered in [12] for a Burgers’ equation. Then Planas and collaborators [17,10,11]14 treat problem (1), the analysis of well-posedness (including uniqueness) and an15 unbounded delay case too. The asymptotic behavior in the sense of attractors in16 L2-norm is carried out in [4]. It is worth also to mention that in [20] the inclusion of17 a delay is used as an approximation to a 3D Navier-Stokes model when the length18 of the delay vanishes. Nevertheless, in the 2D case our interest in the problem is19 just mathematical, due to the difficulties arising in controlling the norm of the de-20 rivatives as cited in the abstract (see also [4]). Our goal in this paper is to improve21 the results in [4] providing regularity for both solutions and attractors.22 23 Let us first introduce some notation. As usual we will denote by Hand Vthe24 Hilbert spaces that are the closure of V(infinitely differentiable functions in Ω with25 compact support and free divergence) in the L2and H1norm respectively, and26 denote their inner products and norms by (·,·) and |·| in H, and by ((·,·)) (product27 of gradients, thanks to the Poincar´e inequality) and k·k in V(see [4] for more28 details). We do the identification of the Hilbert space Hwith its dual, so we have29 the chain of dense and compact inclusions V⊂H≡H0⊂V0.The duality between30 elements in V0and Vwill be denoted by h·,·i.For short, we introduce the notation31 Lp X=Lp(−h, 0; X),for several choices of pand X, and its norm will be denoted by32 k·kLp X.In the same sense, we denote CX=C([−h, 0]; X).33 34 Recall that A:V→V0given by hAv, wi= ((v, w)),satisfies that Au =−P∆u for all u∈D(A) (the Stokes operator), where Pis the Leray-Helmholtz projector from (L2(Ω))2onto H. The trilinear operator bassociated to Navier-Stokes model is given by 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, bis well defined on V×V×V, and therefore we can consider the associated bilinear form B:V×V→V0 given by hB(u, v), wi:= b(u, v, w). It is well-known (e.g., cf. [18]) that bsatisfies35 the following inequalities (recall we are in dimension two) for a certain constant36 REGULARITY RESULTS FOR A 2D-NAVIER-STOKES MODEL WITH DELAYS 3 C > 0 depending only on Ω,1 |b(u, v, w)| ≤ C|u|1/2kuk1/2kvk|w|1/2kwk1/2∀u, v, w ∈V, (2) |b(u, v, w)| ≤ C|u|1/2kuk1/2kvk1/2|Av|1/2|w| ∀ u∈V, v ∈D(A), w ∈H. (3) The delay operator in the right-hand side is g:R×CH→(L2(Ω))2,and we2 assume that it satisfies the following assumptions:3 (H1) for all ξ∈CH,the function R3t7→ g(t, ξ)∈(L2(Ω))2is measurable,4 (H2) g(t, 0) = 0,for all t∈R,5 (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. Examples of several types of delay operators can be found in [1, Section 3], [3,6 Sections 3.5 and 3.6] and [9, Section 3].7 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−8 h, T;H) into L2(τ, T; (L2(Ω))2).We will still denote by g(t, ut) = e G(u)(t) for each9 u∈L2(τ−h, T;H),and therefore assumption (H4) will hold for all u, v ∈L2(τ−10 h, T;H).11 12 Let us consider that uτ∈H, φ ∈L2 V,and f∈L2 loc(R;V0).13 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, ∞).14 If uis a weak solution to (1), since 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 that b(u, v, w) = −b(u, w, v) for all u, v, w ∈V, (2) and15 the continuous embedding of Vinto H), we conclude that B(u(· − ρ(·)), u(·)) ∈16 L4/3(τ, T;V0) and u0∈L4/3(τ, T;V0) too. Thus, u∈C([τ, ∞); V0)∩Cw([τ, ∞); H).17 This continuity in time in V0and weakly in Hdoes not seem enough to apply18 an energy method to gain asymptotic compactness (in the study of the long time19 behavior). Somehow this indicates the border between an ill-posed and well-posed20 problems, as we recall now in a new phase-space.21 When the initial (memory) data is more regular, namely φ∈L2 V∩L∞ H,we can22 improve the above estimates again using (2).23 Some existence results given in those settings (cf. [17, Theorem 2.1] and [4,24 Theorems 1 and 2]) are summarized in the following result.25 4 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO AND G. PLANAS Theorem 1. Consider f∈L2 loc(R;V0),and g:R×CH→(L2(Ω))2satisfying1 assumptions (H1)–(H4). Then, the following statements hold:2 (a) If uτ∈H, φ ∈L2 V, there exists at least one weak solution u(·;τ, uτ, φ)to (1).3 (b) If uτ∈Hand φ∈L2 V∩L∞ H, then there exists a unique weak solution to (1), u(·;τ, uτ, φ)∈C([τ, ∞); H),with u0∈L2(τ, T;V0)for all T > τ, and satisfying the energy equality |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. (4) Remark 1. As a by-product of the proof of the above result, it is not difficult to check that the following estimates hold for the weak solution obtained in statement (a) (unique under the assumptions of statement (b)) and also for suitable Galerkin approximations (see (6) below) for any T > τ : kuk2 L∞(τ,T ;H)≤(|uτ|2+C2 gkφk2 L2 H+ν−1kfk2 L2(τ,T ;V0))exp((1 + C2 g)(T−τ)) =: CL∞(H)(τ, T, uτ, φ), kuk2 L2(τ,T ;V)≤ν−1(|uτ|2+C2 gkφk2 L2 H+ν−1kfk2 L2(τ,T ;V0)) +ν−1(1 + C2 g)(T−τ)CL∞(H)(τ, T, uτ, φ) =: CL2(V)(τ, T, uτ, φ). Our goal in this work is to improve some previous results obtained in [4,17],4 addressing to the existence of strong solutions and attractors in a higher norm.5 The structure of the paper is the following: in Section 2we establish the exist-6 ence of strong solutions and the regularization effect in 2D. Estimates leading to7 continuity of involved processes and absorbing properties are obtained in Section 3.8 Section 4is devoted to prove the asymptotic compactness via an energy method.9 In order to do that, previous uniform estimates in several spaces are deduced. The10 required computations for these results are more involved than in the non-delayed11 case due mainly to the extra difficulty of the delay in the convective term. Finally,12 in Section 5all the previous results allow us to ensure the existence of several famil-13 ies of pullback attractors in higher norms. Several relationships between them (and14 also compared with those obtained in [4]) will be pointed out too.15 2. Regularization effect and strong solution. One can also expect to introduce16 a concept of strong solution for problem (1).17 Definition 2. A strong solution to (1) is a weak solution that also satisfies u∈18 L∞(τ, T;V)∩L2(τ, T;D(A)) for all T > τ.19 Remark 2. Observe that if φ∈L2 V∩L∞ Hand uis a strong solution for (1), from (3) it yields that B(u(· − ρ(·)), u(·)) ∈L2(τ, T;H) for all T > τ. Therefore, if f∈L2 loc(R;H),then u0∈L2(τ, T;H) for all T > τ,u∈C([τ, ∞); V), and the following energy equality holds, ku(t)k2+ 2 Zt s [ν|Au(r)|2+b(u(r−ρ(r), u(r), Au(r))]dr =ku(s)k2+ 2 Zt s (f(r) + g(r, ur), Au(r))dr ∀τ≤s≤t. (5) The following regularizing effect holds for the problem.20 REGULARITY RESULTS FOR A 2D-NAVIER-STOKES MODEL WITH DELAYS 5 Theorem 2. Consider uτ∈H,φ∈L2 V∩L∞ H,f∈L2 loc(R;H),and g:R×1 CH→(L2(Ω))2satisfying assumptions (H1)–(H4). Then, the weak solution uto2 (1) regularizes to a strong solution in the sense that u∈L∞(τ+ε, T ;V)∩L2(τ+3 ε, T;D(A)) ∩C((τ, ∞); V)for all T > τ +ε>τ.4 Moreover, if uτ∈V, then uis indeed a strong solution to (1), so u∈C([τ, ∞); V)∩5 L2 loc(τ, ∞;D(A)), and satisfies the energy equality (5).6 Proof. 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) =(f(t), wj)+(g(t, uk t), wj), a.e. t > τ, ∀1≤j≤k, (6) fulfilled with the initial conditions7 uk(τ) = Pkuτand uk(τ+s) = Pkφ(s)a.e. s ∈(−h, 0),(7) where Pkis the orthogonal projector from Honto Vk.8 Multiplying each equation by λjγjk(t),summing from j= 1 to k, and taking into account (3), we obtain 1 2 d dtkuk(t)k2+ν|Auk(t)|2≤C|uk(t−ρ(t))|1/2kuk(t−ρ(t))k1/2kuk(t)k1/2|Auk(t)|3/2 + (f(t) + g(t, uk t), Auk(t)) a.e. t > τ. Using the Cauchy-Schwartz and Young inequalities d dtkuk(t)k2+ν 2|Auk(t)|2≤e C|uk(t−ρ(t))|2kuk(t−ρ(t))k2kuk(t)k2 +2 ν|f(t)|2+2 ν|g(t, uk t)|2a.e. t > τ, (8) where e C=27C4 2ν3.Integrating in time, using (H4) and Remark 1, we deduce kuk(t)k2+ν 2Zt s |Auk(r)|2dr ≤kuk(s)k2+b CZt s kuk(r−ρ(r))k2kuk(r)k2dr +2 νZt s |f(r)|2dr +2 νC2 gZt τ−h |uk(r)|2dr (9) for all τ≤s < t ≤T, where b C=e Cmax{kφk2 L∞ H, CL∞(H)(τ, T, uτ, φ)}.9 In particular, from the above, integrating again with respecto to s∈[τ, τ +ε] (with τ+ε<T), it yields kuk(t)k2≤ε−1Zτ+ε τ kuk(s)k2ds +b CZt τ kuk(r−ρ(r))k2kuk(r)k2dr +2 νZt τ |f(r)|2dr +2 νC2 gZt τ−h |uk(r)|2dr ∀τ+ε≤t≤T. 6 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO AND G. PLANAS Now, applying the Gronwall lemma we obtain that kuk(t)k2≤ε−1Zτ+ε τ kuk(r)k2dr +2 νZt τ |f(r)|2dr +2 νC2 gZt τ−h |uk(r)|2dr ×exp b CZt τ kuk(r−ρ(r))k2dr∀t∈[τ+ε, T]. From (7) and Remark 1we conclude that {uk}is bounded in L∞(τ+ε, T;V).1 Turning back now to (9) we also obtain that {uk}is bounded in L2(τ+ε, T;D(A)).2 Passing through the limit in k, we conclude the first claim of the theorem.3 Last claim is simpler. If uτ∈V, it only requires integration in (8) in [τ, t] and4 application of the Gronwall lemma. The details are omitted for brevity.5 An immediate consequence of the above is the following result.6 Corollary 1. Assume that f∈L2 loc(R;H)and g:R×CH→(L2(Ω))2fulfills7 (H1)–(H4). Then, for any bounded set B⊂H×(L2 V∩L∞ H)8 (i) The set of weak solutions to (1){u(·;τ, uτ, φ):(uτ, φ)∈B}is bounded in9 L∞(τ+ε, T;V)∩L2(τ+ε, T;D(A)) for any ε > 0and any T > τ +ε.10 (ii) Moreover, if Bis bounded in V×(L2 V∩L∞ H),then {u(·;τ, uτ, φ) : (uτ, φ)∈B}11 is bounded in L∞(τ, T;V)∩L2(τ, T ;D(A)) for any T > τ.12 3. Processes and their continuity and absorbing properties. In this section13 we start recalling the biparametric families of mappings Sand Udefined in [4]14 through the solution to (1) given in Theorem 1, which in fact were proved to be15 continuous processes. After that, and thanks to the improved regularity we have16 several meaningful choices to restrict these mappings to others with higher norms.17 Their continuity and absorbing properties will be established.18 Recall that S(t, τ) : H×(L2 V∩L∞ H)→H×(L2 V∩L∞ H) and U(t, τ) : L2 V∩CH→L2 V∩CH for any t≥τ, given by S(t, τ)(uτ, φ)=(u(t), ut) for any (uτ, φ)∈H×(L2 V∩L∞ H)19 and U(t, τ)φ=ut(·;τ, φ(0), φ) for any φ∈L2 V∩CHare well-defined mappings after20 Theorem 1(cf. [4]).21 Actually, each of these mappings form a process (see definitions in [4, Section 3]), which for short we denote (H×(L2 V∩L∞ H),{S(t, τ)}t≥τ) and (L2 V∩CH,{U(t, τ)}t≥τ) respectively. They are continuous in their corresponding phase-spaces (cf. [4, Co-22 rollary 2]) under the assumptions of Theorem 1.23 Now, after Theorem 2we may consider the corresponding restrictions (the notation for the operators will not be modified, since no confusion arises) (V×(L2 V∩L∞ H),{S(t, τ)}t≥τ),(CV,{U(t, τ)}t≥τ),(C˜ h,V H,{U(t, τ)}t≥τ), (V×(L2 D(A)∩L∞ V),{S(t, τ)}t≥τ),(L2 D(A)∩CV,{U(t, τ)}t≥τ), which are well-defined processes, where C˜ h,V H={ϕ∈CH:ϕ|[−˜ h,0] ∈B([−˜ h, 0]; V)},24 being B([−˜ h, 0]; V) the space of bounded functions from [−˜ h, 0] into V(with ˜ h∈25 [0, h]).26 For the sake of clarity and brevity in the exposition, among all of the above27 processes we restrict ourselves to the most interesting ones, namely (V×(L2 D(A)∩28 L∞ V),{S(t, τ)}t≥τ) and (L2 D(A)∩CV,{U(t, τ)}t≥τ).29 REGULARITY RESULTS FOR A 2D-NAVIER-STOKES MODEL WITH DELAYS 7 In order to develop a more regular theory of attractors we start studying the1 continuity of these new processes. We have the following result.2 Proposition 1. Consider f∈L2 loc(R;H)and g:R×CH→(L2(Ω))2fulfilling (H1)–(H4), (uτ, φ)and (vτ, ψ)∈V×(L2 V∩L∞ H),and denote u=u(·;τ, uτ, φ)and v=v(·;τ, vτ, ψ)the corresponding solutions to (1) with (respective) initial data. Then, ku(s)−v(s)k2≤kuτ−vτk2+e CZ0 −h kφ(r)−ψ(r)k2dr +C4 2ν2Zt τ |u(r−ρ(r))−v(r−ρ(r))|2ku(r)k2|Au(r)|2dr ×exp Zt τe C+27C4 2ν3|v(r−ρ(r))|2kv(r−ρ(r))k2dr,(10) ν 2Zs τ |Au(r)−Av(r)|2dr ≤kuτ−vτk2+2C2 g νZs τ−h |u(r)−v(r)|2dr + 2 Zs τ ku(r−ρ(r)) −v(r−ρ(r))k2dr +C4 2ν2Zs τ |u(r−ρ(r)) −v(r−ρ(r))|2ku(r)k2|Au(r)|2dr +27C4 2ν3Zs τ |v(r−ρ(r))|2kv(r−ρ(r))k2ku(r)−v(r)k2dr (11) for all τ≤s≤t, where e C= 2C2 g/(λ1ν)+2/(1 −ρ∗).3 Proof. Using the second energy equality (5) for w:= u−v, 1 2 d dtkw(t)k2+ν|Aw(t)|2=b(v(t−ρ(t)), v(t), Aw(t)) −b(u(t−ρ(t)), u(t), Aw(t)) + (g(t, ut)−g(t, vt), Aw(t)) a.e.t > τ. Standard manipulations in the trilinear term byields b(v(t−ρ(t)), v(t), Aw(t)) −b(u(t−ρ(t)), u(t), Aw(t)) =−b(w(t−ρ(t)), u(t), Aw(t)) −b(v(t−ρ(t)), w(t), Aw(t)) a.e. t > τ. Now from (3) and the Young inequality, |b(w(t−ρ(t)), u(t), Aw(t))|+|b(v(t−ρ(t)), w(t), Aw(t))| ≤C|w(t−ρ(t))|1/2kw(t−ρ(t))k1/2ku(t)k1/2|Au(t)|1/2|Aw(t)| +C|v(t−ρ(t))|1/2kv(t−ρ(t))k1/2kw(t)k1/2|Aw(t)|3/2 ≤C2 ν|w(t−ρ(t))|kw(t−ρ(t))kku(t)k|Au(t)|+ν 4|Aw(t)|2 +27C4 4ν3|v(t−ρ(t))|2kv(t−ρ(t))k2kw(t)k2+ν 4|Aw(t)|2a.e. t > τ. 8 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO AND G. PLANAS Combining the above with the H¨older inequality and integrating, we obtain kw(s)k2− kw(τ)k2+ν 2Zs τ |Aw(r)|2dr ≤2C2 g νZs τ−h |w(r)|2dr + 2 Zs τ kw(r−ρ(r))k2dr +C4 2ν2Zs τ |w(r−ρ(r))|2ku(r)k2|Au(r)|2dr +27C4 2ν3Zs τ |v(r−ρ(r))|2kv(r−ρ(r))k2kw(r)k2dr ∀τ≤s. (12) Splitting the first integral in the RHS, that on [τ−h, s] in the initial datum part and the evolutive solution on [τ, s],and using the Poincar´e inequality and an eventual change of variables for t−ρ(t), in particular we conclude kw(s)k2≤kw(τ)k2+e CZ0 −h kφ(r)−ψ(r)k2dr+C4 2ν2Zs τ |w(r−ρ(r))|2ku(r)k2|Au(r)|2dr +Zs τe C+27C4 2ν3|v(r−ρ(r))|2kv(r−ρ(r))k2kw(r)k2dr ∀s≥τ. The Gronwall lemma gives (10). Finally, (11) follows immediately from (12).1 From the continuity of (H×(L2 V∩L∞ H),{S(t, τ)}t≥τ) and (L2 V∩CH,{U(t, τ)}t≥τ)2 (cf. [4, Corollary 2]) combined with Corollary 1and Proposition 1, we deduce3 Corollary 2. Assume that f∈L2 loc(R;H)and g:R×CH→(L2(Ω))2satisfies4 (H1)–(H4). Then (V×(L2 D(A)∩L∞ V),{S(t, τ)}t≥τ)and (L2 D(A)∩CV,{U(t, τ)}t≥τ)5 are continuous processes.6 The first of the two key ingredients to ensure the existence of attractors is the7 absorbing property. We recall from [4] several concepts, definitions and extra con-8 dition on fsuch that the absorbing property holds in a natural universe associated9 to problem (1). These results will be improved in the sequel, in order to gain at-10 tractors in higher norms. Namely, we start recalling an additional assumption for11 extra energy estimates.12 (H5) 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 ∀t≥τ. Now we recall the estimates leading to the introduction of one universe for the13 study of pullback attractors (cf. [4, Lemma 1]).14 Lemma 1. Consider given f∈L2 loc(R;V0)and g:R×CH→(L2(Ω))2satisfying conditions (H1)–(H5). Then, for any (uτ, φ)∈H×(L2 V∩L∞ H), the following inequalities hold for the solution uto (1) for all t≥s≥τ: |u(t)|2≤e−η(t−τ)(|uτ|2+Cgkφk2 L2 H) + e−ηt βZt τ eηrkf(r)k2 ∗dr, (13) νZt s ku(r)k2dr ≤|u(s)|2+Cgkusk2 L2 H+1 νZt s kf(r)k2 ∗dr + 2CgZt s |u(r)|2dr, (14) where15 β= 2ν−(η+ 2Cg)λ−1 1>0.(15) REGULARITY RESULTS FOR A 2D-NAVIER-STOKES MODEL WITH DELAYS 9 In the context of pullback attractors, a universe is a family of time-dependent1 sections, i.e. family of subsets in the phase-space, that allows to establish good dy-2 namical properties, acting as basis of attraction. In [4] it was introduced DH,L2 H η(H×3 (L2 V∩L∞ H)),the class of all families of nonempty subsets b D={D(t) : t∈R} ⊂4 P(H×(L2 V∩L∞ H)) such that5 lim τ→−∞ eητ sup (ζ,ϕ)∈D(τ) (|ζ|2+kϕk2 L2 H)= 0.(16) This notion of universe is naturally related to estimate (13), where the rate of growth6 in −∞ is such that the initial data is killed by the dissipativity of the problem. Let7 us also observe that DH,L2 H η(H×(L2 V∩L∞ H)) is inclusion-closed, which allows certain8 advantages in the application of the theory.9 We will denote by DF(H×(L2 V∩L∞ H)) the universe of fixed bounded sets in10 H×(L2 V∩L∞ H).11 Then, after the above comments and having in mind (13), we gain the first12 absorbing family (cf. [4, Corollary 3]).13 Corollary 3. Assume that g:R×CH→(L2(Ω))2fulfills conditions (H1)–(H5)14 and that f∈L2 loc(R;V0)satisfies15 Z0 −∞ eηrkf(r)k2 ∗dr < ∞.(17) Then, the family b D0={D0(t) : t∈R}⊂P(H×(L2 V∩L∞ H)) defined by D0(t) = BH(0, RH(t)) ×(BL2 V(0, RV(t)) ∩BL∞ H(0, RH(t))), where R2 H(t) =1 + β−1e−η(t−2h)Zt −∞ eηrkf(r)k2 ∗dr, R2 V(t) =ν−1h(1 + 3Cgh)R2 H(t) + ν−1kfk2 L2(t−h,t;V0)i, is pullback DH,L2 H η(H×(L2 V∩L∞ H))−absorbing for the process Son H×(L2 V∩L∞ H)16 (and therefore pullback DF(H×(L2 V∩L∞ H))−absorbing too), and b D0belongs to17 DH,L2 H η(H×(L2 V∩L∞ H)).18 As additional universe for the study of the problem in [4, Definition 5] it was19 introduced DCH η(L2 V∩CH),the class of all families of nonempty subsets b D={D(t) :20 t∈R}⊂P(L2 V∩CH) such that21 lim τ→−∞ eητ sup φ∈D(τ) |φ|2 CH!= 0.(18) Remark 3. After Corollary 3it is immediate to realize that b D1={D1(t) : t∈22 R}⊂P(L2 V∩CH) given by D1(t) = BL2 V(0, RV(t)) ∩BCH(0, RH(t)) is pullback23 DCH η(L2 V∩CH)−absorbing for Uon L2 V∩CH.24 Since the pullback estimates necessary for the dissipativity only require the25 tempered character in norms H×L2 Hor CH,we may consider more restricted26 universes, with higher norms but the same tempered condition. So, analogously to27 the previous definitions of DH,L2 H η(H×(L2 V∩L∞ H)) and DCH η(L2 V∩CH) we introduce28 the following classes.29 16 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO AND G. PLANAS Therefore, |uτ+r|2 CH= sup s∈[r−h,r] |u(τ+s)|2 ≤e−η(r−h)(|uτ|2+Cgkφk2 L2 H) + β−1e−η(τ+r−h)Zτ+r τ eηθkf(θ)k2 ∗dθ, whence limτ→−∞ eητ supuτ+r∈D(r)(τ)|uτ+r|2 CH= 0.1 Now we may establish our main result.2 Theorem 3. Assume that g:R×CH→(L2(Ω))2and f∈L2 loc(R;H)satisfy3 (H1)–(H5) and (17) respectively. Then there exist the minimal pullback attractors4 {ADF(V×(L2 D(A)∩L∞ V))(s)}s∈Rand {ADH,L2 H η(V×(L2 D(A)∩L∞ V))(s)}s∈Rfor (V×(L2 D(A)∩5 L∞ V),{S(t, τ)}t≥τ)and {ADF(L2 D(A)∩CV)(s)}s∈Rand {ADCH η(L2 D(A)∩CV)(s)}s∈Rfor6 (L2 D(A)∩CV,{U(t, τ)}t≥τ).7 The following relations hold for any s∈R ADF(V×(L2 D(A)∩L∞ V))(s)⊂ADH,L2 H η(V×(L2 D(A)∩L∞ V))(s)⊂D0(s),(29) ADF(L2 D(A)∩CV)(s)⊂ADCH η(L2 D(A)∩CV)(s)⊂D1(s),(30) ej(ADF(L2 D(A)∩CV)(s)) ⊂ADF(V×(L2 D(A)∩L∞ V))(s),(31) ej(ADCH η(L2 D(A)∩CV)(s)) =ADH,L2 H η(V×(L2 D(A)∩L∞ V))(s),(32) where ej:L2 D(A)∩CV→V×(L2 D(A)∩CV)is defined by ej(φ)=(φ(0), φ).8 Finally, if fsatisfies9 sup s≤0Zs s−1 |f(θ)|2dθ < ∞,(33) then (31) becomes an equality, ADF(V×(L2 D(A)∩L∞ V)) =ADH,L2 H η(V×(L2 D(A)∩L∞ V)) and10 ADF(L2 D(A)∩CV)=ADCH η(L2 D(A)∩CV).11 Proof. The abstract result ensuring existence of pullback attractors for the tempered12 universes (e.g., cf. [5, Theorem 3.11] or [4, Theorem 3]) can be applied for both13 (V×(L2 D(A)∩L∞ V),{S(t, τ)}t≥τ) and (L2 D(A)∩CV,{U(t, τ)}t≥τ) since they are14 continuous processes (cf. Corollary 2), with absorbing families (cf. Remark 4) and15 fulfilling the pullback asymptotic compact property (cf. Corollary 4) w.r.t. the16 universes DH,L2 H η(V×(L2 D(A)∩L∞ V)) and DCH η(L2 D(A)∩CV) respectively.17 The existence of pullback attractors for the case of universes of fixed bounded18 sets is simpler (e.g., cf. [14] or [4, Corollary 1]) since they are contained in the19 tempered universes. This also implies relations (29) and (30).20 Relations (31) and (32) via ejfollows by the characterization of minimal pullback21 attractors and Lemma 4.22 Last statement follows again from [4, Corollary 1] since absorbing families in23 these phase-spaces are uniformly bounded in time (see expressions of ρ2and ρ3in24 Lemma 2).25 Using comparison results for attractors (e.g., cf. [5, Theorem 3.15]) and relations26 among tempered families of different universes after a time-shift (thanks to Lemma27 REGULARITY RESULTS FOR A 2D-NAVIER-STOKES MODEL WITH DELAYS 17 4), it is possible to relate the attractors ADF(H×(L2 V∩L∞ H)) and ADH,L2 H η(H×(L2 V∩L∞ H)) 1 for (H×(L2 V∩L∞ H),{S(t, τ)}t≥τ) and ADF(L2 V∩CH)and ADCH η(L2 V∩CH)for (L2 V∩2 CH),{U(t, τ)}t≥τ) obtained in [4, Theorems 4 and 5] with the ones obtained here3 in Theorem 3. Arguments are analogous to the above proof so it is omitted.4 Theorem 4. Under the assumptions of Theorem 3, the attractors cited previously satisfy the following relations ADF(V×(L2 D(A)∩L∞ V))(s)⊂ ADF(H×(L2 V∩L∞ H))(s),(34) ADF(L2 D(A)∩CV)(s)⊂ ADF(L2 V∩CH)(s),(35) ADH,L2 H η(H×(L2 V∩L∞ H))(s) = ADH,L2 H η(V×(L2 D(A)∩L∞ V))(s), ADCH η(L2 V∩CH)(s) = ADCH η(L2 D(A)∩CV)(s) for any s∈R.5 Moreover, if fsatisfies (33), then (34) and (35) are in fact equalities.6 Acknowledments. The authors wish to congratulate Prof. Kloeden for his 70th7 birthday. Particularly, P.M.-R. thanks Prof. Kloeden for their relationship during8 all these years, both academic and friendship, and for his continuous encouragement9 from the beginning of his scientific career.10 The authors also wish to thank the referees for their careful reading and com-11 ments on a previous version of the manuscript.12 Partially supported by the projects MTM2015-63723-P (MINECO/FEDER, EU)13 and P12-FQM-1492 (Junta de Andaluc´ıa). G. Planas was partially supported by14 CNPq-Brazil, grants 306646/2015-3 and 402388/2016-0.15 REFERENCES16 [1] T. Caraballo and J. Real, Navier-Stokes equations with delays,R. Soc. Lond. Proc. Ser. A17 Math. Phys. Eng. Sci.,457 (2001), 2441–2453.18 [2] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations19 with delays,R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.,459 (2003), 3181–3194.20 [3] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,J. Differential21 Equations,205 (2004), 271–297.22 [4] J. Garc´ıa-Luengo, P. Mar´ın-Rubio and G. Planas, Attractors for a double time-delayed 2D-23 Navier-Stokes model,Discrete Contin. Dyn. Syst.,34 (2014), 4085–4105.24 [5] J. Garc´ıa-Luengo, P. Mar´ın-Rubio and J. Real, Pullback attractors in Vfor non-autonomous25 2D-Navier-Stokes equations and their tempered behaviour,J. Differential Equations,25226 (2012), 4333–4356.27 [6] J. Garc´ıa-Luengo, P. Mar´ın-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes28 equations with delays and their regularity,Adv. Nonlinear Stud.,13 (2013), 331–357.29 [7] J. Garc´ıa-Luengo, P. Mar´ın-Rubio and J. Real, Regularity of pullback attractors and attrac-30 tion in H1in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite31 delay,Discrete Contin. Dyn. Syst.,34 (2014), 181–201.32 [8] J. Garc´ıa-Luengo, P. Mar´ın-Rubio and J. Real, Some new regularity results of pullback at-33 tractors for 2D Navier-Stokes equations with delays,Commun. Pure Appl. Anal.,14 (2015),34 1603–1621.35 [9] M. J. Garrido-Atienza and P. Mar´ın-Rubio, Navier-Stokes equations with delays on unboun-36 ded domains,Nonlinear Anal.,64 (2006), 1100–1118.37 [10] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with38 bounded delay,Discrete Contin. Dyn. Syst. Ser. B,16 (2011), 225–238.39 [11] S. M. Guzzo and G. Planas, Existence of solutions for a class of Navier-Stokes equations with40 infinite delay,Appl. Anal. 94 (2015), 840–855.41 18 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO AND G. PLANAS [12] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers’ equation,Discrete Contin.1 Dyn. Syst. Ser. B,2(2002), 47–56.2 [13] P. Mar´ın-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some3 unbounded domains,Nonlinear Anal.,67 (2007), 2784–2799.4 [14] P. Mar´ın-Rubio and J. Real, On the relation between two different concepts of pullback5 attractors for non-autonomous dynamical systems,Nonlinear Anal.,71 (2009), 3956–3963.6 [15] P. Mar´ın-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays7 in continuous and sub-linear operators,Discrete Contin. Dyn. Syst.,26 (2010), 989–1006.8 [16] P. Mar´ın-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-9 Stokes model in an infinite delay case,Nonlinear Anal.,74 (2011), 2012–2030.10 [17] G. Planas and E. Hern´andez, Asymptotic behaviour of two-dimensional time-delayed Navier-11 Stokes equations,Discrete Contin. Dyn. Syst.,21 (2008), 1245–1258.12 [18] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-13 Verlag, New York, 1997.14 [19] T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external15 force,Discrete Contin. Dyn. Syst.,12 (2005), 997–1018.16 [20] W. Varnhorn, The Navier-Stokes equations with time delay, Applied Mathematical Sciences,17 2(2008), 947–960.18 E-mail address, Julia Garc´ıa-Luengo: [email protected]19 E-mail address, Pedro Mar´ın-Rubio: [email protected]20 E-mail address, Gabriela Planas: [email protected]21