PULLBACK ATTRACTORS FOR 2D NAVIER–STOKES EQUATIONS WITH DELAYS AND THE FLATTENING PROPERTY

. This paper treats the existence 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 carry out our study under less restrictive assumptions than in the previous reference. More pre- cisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms. Here we show that the asymptotic compactness of the corresponding processes required to establish the existence of pullback attractors, obtained in [Garc´ıa-Luengo, Mar´ın-Rubio and Real, Adv. Nonlinear Stud. 13 (2013), 331–357] by using an energy method, can be also proved by verifying the ﬂattening property – also known as “Condition (C)”. We deal with dynamical systems in suitable phase spaces within two metrics, the L 2 norm and the H 1 norm. Moreover, we provide results on the existence of pullback attractors for two possible choices of the attracted universes, namely, the standard one of ﬁxed bounded sets, and sec-ondly, one given by a tempered condition.

1. Introduction. 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. [24] 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 dimension for modified terms) have been addressed (e.g., cf. [17,28,32,30,25,31,18,26,27] among others).
Generally, in all finite delay frameworks the assumptions for the delay terms used to involve estimates in L 2 spaces, which in turn means some restrictive conditions on the operators and on the function driving the delayed time. However, as long as the solution for the problem (without delay) in dimension two is continuous in time, it seemed natural to develop a theory just considering a phase space only requiring continuity in time. In this sense, in [14] we treated a relaxation on the assumptions for the delay operator, removing conditions related to the control of the L 2 norm of the delay terms (e.g., cf. conditions (IV) and (V) in [3,4,5,17]). 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 or C 1 with bounded derivative, as usually in the literature.
In this more general framework for the delay operators, in [14] we also established the existence of minimal pullback for the associated processes in both L 2 and H 1 norms, by proving the asymptotic compactness via an energy method (see also [15] for related results). In this sense, to verify asymptotic compactness one can either proceed directly, or make use of a splitting of the solutions into high and low components. Such a splitting is a very common technique in the study of the qualitative behaviour of solutions for PDE problems, in particular when considering the long-time behaviour of dynamics, as in the construction of invariant manifolds [8,19] and inertial manifolds [12,9], the squeezing property [10,35], the notion of 'determining modes' [11,20], and the theory of attractors [23]. In the context of proofs of the existence of attractors it was formalised by Ma, Wang, and Zhong [23] as their 'Condition (C)'. A more descriptive terminology, 'the flattening property', was coined by Kloeden and Langa [21], and we adopt this terminology here. However, it is worth making the observation that this is not so much a 'property' as a (powerful) technique for obtaining the asymptotic compactness of a flow, be it autonomous or non-autonomous.
In this paper we are able to prove that the processes satisfy the flattening property in L 2 and H 1 , and as a consequence we obtain the asymptotic compactness. It is worth pointing out that, while in the case of L 2 a direct proof of asymptotic compactness is no harder than a proof of the flattening property, in H 1 a proof of asymptotic compactness via the flattening property is significantly shorter than as provided in [14], that is based on an energy method. This is due to the fact that there are stronger estimates available for the nonlinear term in H 1 than in L 2 (see estimates (3) and (4) below).
The structure of the paper is as follows. Section 2 contains some preliminaries, including the functional setting of the problem. Section 3 is devoted to recalling standard results from the theory of pullback attractors (within the framework of time-dependent universes of sets), such as existence and comparison, and the flattening property in C([−h, 0]; Z), the space of continuous functions from [−h, 0] into Z, with Z a Banach space. The analysis in the space L 2 is carried out in Section 4, where we conclude the existence of minimal pullback attractors for a universe not only of fixed bounded sets but also for a set of tempered universes. Finally, in Section 5 we establish additional attraction, namely, in the H 1 norm instead of L 2 as in Section 4. Different families of universes (tempered and non-tempered) are introduced and we obtain the flattening property in H 1 , which implies the asymptotic compactness of the corresponding process in this norm. We finish with the existence of minimal pullback attractors and comparison among them under suitable additional assumptions.
2. Statement of the problem. Let Ω ⊂ R 2 be an open bounded set with smooth enough boundary ∂Ω, and consider an arbitrary initial time τ ∈ R, and the following functional Navier-Stokes problem: where we assume that ν > 0 is the kinematic viscosity, u = (u 1 , u 2 ) is the velocity field of the fluid, p is the pressure, f is a non-delayed external force field, g is another external force containing some hereditary characteristics, and φ( To set our problem in the abstract framework, we consider the usual spaces in the variational theory of the Navier-Stokes equations: 2 with the norm · associated to the inner product ((·, ·)), where for u, v ∈ (H 1 0 (Ω)) 2 , We will use · * for the norm in V and ·, · for the duality product between V and V . We consider every element h ∈ H as an element of V , given by the equality h, v = (h, v) for all v ∈ V . It follows that V ⊂ H ⊂ V , where the injections are dense and continuous, and, in fact, compact.
Define the operator A : V → V as Au, v = ((u, v)) for all u, v ∈ V . Let us denote D(A) = {u ∈ V : Au ∈ H}. By the regularity of ∂Ω, one has D(A) = (H 2 (Ω)) 2 ∩ V , and Au = −P ∆u for all u ∈ D(A) is the Stokes operator (P is the ortho-projector from (L 2 (Ω)) 2 onto H). On D(A) we consider the norm | · | D(A) defined by |u| D(A) = |Au|. Observe that on D(A) the norms · (H 2 (Ω)) 2 and |·| D(A) are equivalent (see [7] or [34]), and D(A) is compactly and densely injected in V . By standard spectral theory, since it will be useful in the paper, let us denote by {w j } j≥1 a Hilbert basis of H formed by ortho-normalized eigenfunctions of the Stokes operator A with corresponding eigenvalues {λ j } j≥1 being 0 < λ 1 ≤ λ 2 ≤ . . . and lim j→∞ λ j = ∞.
Let us define for every functions u, v, w : Ω → R 2 for which the right-hand side is well defined. In particular, b has sense for all u, v, w ∈ V , and is a continuous trilinear form on V × V × V . Some useful properties concerning b that we will use in the next sections are the following (see [33] or [35] and there exists a constant C 1 > 0, only dependent on Ω, such that Now, we establish some appropriate assumptions on the term in (1) containing the delay.
It is worth pointing out that no condition involving L 2 norms of the memory term in g is assumed (in opposition to conditions (IV) and (V) in [3,4,5,17]).
Assume that φ ∈ C H , and f ∈ L 2 loc (R; V ).
where the equation must be understood in the sense of D (τ, ∞).

Remark 1.
If u is a weak solution of (1), then from (5) we deduce that for any T > τ, one has u ∈ L 2 (τ, T ; V ), and the following energy equality holds: A notion of more regular solution is also suitable for problem (1).

3.
Abstract results on minimal pullback attractors. Pullback D 0 -flattening property. Now, we present a summary of some results from [13] about the existence of minimal pullback attractors (see also [1,2,29]). In particular, we assume that the process U is closed (see Definition 3 below).
Consider given a metric space (X, d X ), and let us denote Definition 3. Let U be a process on X.
(a) U is said to be continuous if for any pair τ ≤ t, the mapping U (t, τ ) : X → X is continuous.
(b) U is said to be closed if for any τ ≤ t, and any sequence {x n } ⊂ X, if x n → x ∈ X and U (t, τ )x n → 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 D 0 = {D 0 (t) : t ∈ R} ⊂ P(X).
Definition 4. We say that a process U on X is pullback D 0 -asymptotically compact if for any t ∈ R and any sequences {τ n } ⊂ (−∞, t] and {x n } ⊂ X satisfying τ n → −∞ and x n ∈ D 0 (τ n ) for all n, the sequence Let D be a nonempty class of families parameterized in time D = {D(t) : t ∈ R} ⊂ P(X). The class D will be called a universe in P(X).

Definition 5.
A process U on X is said to be pullback D-asymptotically compact if it is pullback D-asymptotically compact for any D ∈ D.
It is said that D 0 = {D 0 (t) : t ∈ R} ⊂ P(X) is pullback D-absorbing for the process U on X if for any t ∈ R and any D ∈ D, there exists a τ 0 (t, D) ≤ t such that for the process U on X, and U is pullback D 0 -asymptotically compact, then U is also D-asymptotically compact.
With the above definitions, we may establish the main result of this section (cf. [13,Theorem 3.11]). Theorem 2. Consider a closed process U : R 2 d × X → X, a universe D in P(X), and a family D 0 = {D 0 (t) : t ∈ R} ⊂ P(X) which is pullback D-absorbing for U , and assume also that U is pullback D 0 -asymptotically compact.
Then, the family has the following properties: The family A D is minimal in the sense that if C = {C(t) : t ∈ R} ⊂ P(X) is another family of closed sets such that for any Remark 5. Under the assumptions of Theorem 2, the family A D is called the minimal pullback D-attractor for the process U . If A D ∈ D, then it is the unique family of closed subsets in D that satisfies We will denote by D F (X) the universe of fixed nonempty bounded subsets of X, i.e., the class of all families D of the form D = {D(t) = D : t ∈ R} with D a 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 D contains the universe D F (X), then both attractors, A D F (X) and A D , exist, and the following relation holds: Remark 6. It can be proved (see [29]) that, under the assumptions of the preceding corollary, if for some T ∈ R, the set ∪ t≤T D 0 (t) is a bounded subset of X, then Now, we introduce a notion which is a slight modification of Ma, Wang, and Zhong's "Condition (C)" [23] (renamed the "flattening property" by Kloeden and Langa [21]), after Definition 2.24 in the book by Carvalho, Langa, and Robinson [6], where P ε need not be a projection operator.
Definition 6. Assume that Z is a Banach space and that a process U on C Z is well-defined. Let be D 0 = {D 0 (t) : t ∈ R} ⊂ P(C Z ) a given family. We will say that the process U on C Z satisfies the pullback D 0 -flattening property if for any t ∈ R, and ε > 0, there exist τ ε < t, a finite dimensional subspace Z ε of Z, and a continuous mapping P ε : Z → Z ε , all depending on D 0 , t and ε, such that Similarly to the results in [23], [21] and [16] (see also [6]), we will see that in order to show that a process U is pullback D 0 -asymptotically compact, it is enough to verify the pullback D 0 -flattening property given in the definition above.
Proposition 1. Assume that Z is a Banach space, U is a well-defined process on C Z , and D 0 = {D 0 (t) : t ∈ R} ⊂ P(C Z ) is a given family such that U satisfies the pullback D 0 -flattening property. Then, the process U is pullback D 0 -asymptotically compact.
Proof. Let be fixed t ∈ R, a sequence {τ n } ⊂ (−∞, t] such that τ n → −∞, and a sequence {φ n } ⊂ C Z such that φ n ∈ D 0 (τ n ) for all n, be fixed. We must show that For a fixed integer k ≥ 1, by the pullback D 0 -flattening property, there exist N k ≥ 1, a finite dimensional subspace Z k of Z, and a continuous mapping P k : Z → Z k , such that {P k U (t, τ n )φ n : n ≥ N k } is a bounded and equi-continuous subset of C Z k , and therefore a relatively compact subset of C Z , and |(I−P k )U (t, τ n )φ n | C Z ≤ 1/(2k) for all n ≥ N k . Thus, {U (t, τ n )φ n : n ≥ 1} can be covered by a finite number of balls in C Z of radius 1/k. As k is arbitrary, it is not difficult to check that {U (t, τ n )φ n : n ≥ 1} possesses a Cauchy subsequence in C Z . Since C Z is complete, this subsequence is convergent, whence {U (t, τ n )φ n : n ≥ 1} is relatively compact in C Z .
Finally, we recall an abstract result that allows us to compare two attractors for a process under appropriate assumptions (see [13,Theorem 3.15]). 2 be two metric spaces such that X 1 ⊂ X 2 with continuous injection, and for i = 1, 2, let D i be a universe in P(X i ), with D 1 ⊂ D 2 . Assume that we have a map U that acts as a process in both cases, i.e., U : R 2 d ×X i → X i for i = 1, 2 is a process.
For each t ∈ R, let us denote where the subscript i in the symbol of the omega-limit set Λ i is used to denote the dependence of the respective topology. Then, If in addition is a compact subset of X 1 for all t ∈ R, and (ii) for any D 2 ∈ D 2 and any t ∈ R, there exist a family D 1 ∈ D 1 and a t * D1 ≤ t (both possibly depending on t and D 2 ), such that U is pullback D 1asymptotically compact, and for any s ≤ t * D1 there exists a τ s ≤ s such that 4. Existence of minimal pullback attractors in H norm. The goal of this section is to establish the existence of minimal pullback attractors for a suitable process U on C H associated to problem (1). This result was obtained in [14] by applying an energy method. Now, we will make use of the pullback flattening property. As pointed out in the Introduction, in the phase space C H , the proofs of the flattening property and the asymptotic compactness of this process are in fact very similar and no extra benefit is appreciated, in contrast with Section 5 as shown below for the C V -norm.
We first define the following process U on C H associated to (1) (cf. [14, Proposition 4.1]).
Concerning the existence of a pullback absorbing family for the process U on C H defined above, we have the following results which were also established in [14]. We recall them for the sake of clarity. Lemma 1. Under the assumptions of Proposition 2, consider µ ∈ (0, 2νλ 1 ) and φ ∈ C H . Then the following estimates hold for the solution to (1) for all t ≥ τ From now on we will assume that there exists 0 < µ < 2νλ 1 such that 2e µh L g < µ, Remark 7. The above conditions will be the key for the uniform asymptotic estimates in what follows. Observe that when L g << 1 these are essentially the same assumptions as in the case without delay (i.e. g ≡ 0). In the current case when g exists and therefore L g > 0, the above relation among the elements h, λ 1 , L g , ν and the integrability condition on f indicate some balance such that the resulting system will be dissipative.
Once the estimate (8) has been obtained, we introduce the following tempered universe in P(C H ). Observe that for any σ > 0, D F (C H ) ⊂ D σ (C H ) and that the universe D σ (C H ) is inclusion-closed.
Corollary 2. Under the assumptions of Proposition 2, if moreover conditions (10) and (11) are satisfied, then the family D 0,µ = {D 0,µ (t) : t ∈ R}, with D 0,µ (t) = B C H (0, ρ µ (t)), the closed ball in C H of center zero and radius ρ µ (t), where is pullback D σµ (C H )-absorbing for the process U defined by (7). Moreover, D 0,µ ∈ D σµ (C H ). Now, we establish several estimates for the process U when the initial time is sufficiently shifted in a pullback sense, that will be used in order to prove the pullback flattening property.

Lemma 2.
Under the assumptions of Corollary 2, for any t ∈ R and D ∈ D σµ (C H ), there exist τ 1 ( D, t, h) < t − 2h and functions {ρ i } 3 i=1 depending on t and h, such that for any τ ≤ τ 1 ( D, t, h) and any φ τ ∈ D(τ ), where Proof. The first two estimates (13) and (14) follow directly from (8) and (9). Now, from (2), (3), (5), and the fact that A is an isometric isomorphism, we obtain which combined with properties (II) and (III) of g, implies that Thus, from this last inequality we can conclude the third estimate (15).
In order to prove the pullback flattening property for the process U on C H , we need the following auxiliary result, which can be obtained from [14, Proposition 4.2], taking into account Remark 4.
Remark 8. While in the non-delayed case (see [16]), since the process is defined on the Hilbert space H, the pullback asymptotic compactness guarantees the pullback flattening property (it is known that both are equivalent in any uniformly convex Banach space, see [6]), it does not happen in the delayed case, where the phase space is C H .
We also need the next corollary of Lemma 3, that can be proved similarly to [16,Corollary 20].
In particular, Proof. Consider t ∈ R and D ∈ D σµ (C H ). For simplicity we split the proof in two parts.
Step 2. The general statements (17) and (16) hold. Namely we just extend to the whole interval [t − h, t] the previously proved inequalities (19) and (18) using again the estimate (13) and uniform continuity arguments and the compactness of [t − h, t].
We will also use the following result, whose proof is analogous to that of [22,Lemma 12].
In particular, the process U on C H satisfies the pullback D-flattening property for any D ∈ D σµ (C H ).
Indeed, using the energy equality, for each m ≥ 1 one has = f (r), q m (r) + (g(r, u r ), q m (r)) where we have used Young's inequality and the assumptions (II) and (III) of g.
Actually, the other three terms in the RHS in (21) can be also controlled by ε/4 by using Lemma 4, and since |q m (t − h)| 2 ≤ |u(t − h)| 2 ≤ ρ 2 1 (t) and λ m → ∞ as m → ∞, so it suffices to take eventually a bigger value m. So, we conclude |q m (t)| < ε.
Finally, observe that (20) holds by applying the above arguments uniformly in the interval [t−h, t]. Indeed, all the estimates for (21) with t replaced by ξ ∈ [t−h, t] hold in the same uniform manner, since δ in Corollary 3 can be chosen in such a way for the whole interval [t − h, t].
From the previous results, we obtain the existence of minimal pullback attractors for the process U on C H (see [14,Theorem 4.1]).
Theorem 4. Assume that f ∈ L 2 loc (R; V ), and g : R × C H → (L 2 (Ω)) 2 satisfying the assumptions (I)-(III), (10) and (11), are given. Then, there exist the minimal pullback D F (C H )-attractor A D F (C H ) , and the minimal pullback D σµ (C H )attractor A Dσ µ (C H ) , for the process U defined by (7). The family A Dσ µ (C H ) belongs to D σµ (C H ), and the following relation holds: Remark 9. If, additionally, we assume that where σ µ is given by (12), then, taking into account Remark 6, we deduce that 5. Regularity of pullback attractors and attraction in V norm. Now, we will make use again of the pullback flattening property in order to establish the existence of minimal pullback attractors in the C V norm, improving the results of the previous Section 4. In this case, a proof of the pullback flattening property allows us to obtain the pullback asymptotic compactness of the process in a shorter way than by using an energy method as in [14].
First, we define some new phase spaces. For anyh ∈ [0, h], let us denote Observe that the space Proof. It follows from Theorem 1 above and from [14,Proposition 5.2].
The following result is similar to Lemma 2 in Section 4, but now we are able to establish estimates for the process U in higher norms, thanks to the regularity result (a) in Theorem 1 and the energy equality (6). It can be obtained analogously to [14,Lemma 5.2].
Lemma 5. Under the assumptions of Proposition 4, if moreover conditions (23) and (24) are satisfied, then for any t ∈ R and D ∈ D σµ (C H ), there exist τ 2 ( D, t, h) < t − 2h − 1 and non-negative functions {R i } 4 i=1 depending on t and h, such that for any τ ≤ τ 2 ( D, t, h) and any φ τ ∈ D(τ ), it holds  Finally, we will denote by D F (Ch ,V H ) the class of families D = {D(t) = D : t ∈ R} with D a fixed nonempty bounded subset of Ch ,V H . Remark 11. The following chain of inclusions for the universes in the above definition and the universes introduced in Section 4 it holds Moreover, D 0,µ,h belongs to Dh ,V σµ (C H ). Now, we will prove some sort of pullback flattening property for the family of processes U : h] (note that we are not considering the process U restricted to R 2 d × C V ). Nevertheless, we will be able to obtain the pullback asymptotic compactness reasoning as in the proof of Proposition 1.
Analogously to Lemma 4, we have the following result.
Since {w j } j≥1 is a special basis, P m is non-expansive in V . From this and the second estimate in Lemma 5, we deduce the boundedness in C V of the set {P m U (t, τ )D(τ ) : τ ≤ τ 2 ( D, t, h)}, for all m ≥ 1.

Remark 12.
(i) Observe that, thanks to the regularity property (a) in Theorem 1, we neither need to restrict the process U to R 2 d ×C V nor to define a universe in P(C V ) in order to obtain the asymptotic compactness of the process U : (ii) It is worth pointing out that in this way the proof of the asymptotic compactness is much shorter than [14,Lemma 5.3].
As an immediate consequence of Proposition 1 and Proposition 6 we have the following Combining all the above statements, we obtain the existence of minimal pullback attractors for the process U : R 2 d × Ch ,V H → Ch ,V H , for anyh ∈ [0, h]. Moreover, we improve the attraction of the pullback attractor A Dσ µ (C H ) , so that it actually attracts in the C V norm (see [14,Theorem 5.1]).