ATTRACTORS FOR A DOUBLE TIME-DELAYED 2D-NAVIER-STOKES MODEL

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.
H is the closure of V in (L 2 (Ω)) 2 with the norm | · |, and inner product (·, ·), where for u, v ∈ (L 2 (Ω)) 2 , V is the closure of V in (H 1 0 (Ω)) 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 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 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. For suitable u and v (for instance in V ) it is also useful to denote B(u, v) the operator of V given by B(u, v), w = b(u, v, w) for any w ∈ V.
Before continuing, for short, we introduce the notation L p X = L p (−h, 0; X), which will be used in the sequel for suitable choices of p and X. The norm in these spaces will be denoted by · L p X . On other hand, C H = C([−h, 0]; H) will also be used, and the sup norm in C H will be denoted by | · | C H . Finally, B E (0, α) will denote the closed ball in a metric space E of center zero and radius α.
Concerning the goal of finding solutions to problem (1), different choices are possible for the initial data.

Let us consider that
where the equation must be understood in the sense of D (τ, ∞).
Although the above choice of phase-space will lead to an existence result (see Theorem 1 below), 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 3 and Theorem 2 below).
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 h is 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 3 and Theorem 2 below, where this difficulty is sorted out).
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 H formed by normalized eigenfunctions of the Stokes operator, {w j } j≥1 , with corresponding eigenvalues {λ j } j≥1 being 0 < λ 1 ≤ λ 2 ≤ . . . with lim j→∞ λ j = ∞. Pose the approximate problems (for each k ≥ 1) of finding fulfilled with the initial conditions where P k is the orthogonal projector from H onto V k . It is well known (e.g. cf. [5]) that the above system of ordinary functional differential equations (the unknowns are {γ jk } k j=1 ) is well-posed in some local interval [τ, t k ). We fix a value T > τ and will provide uniform estimates that will imply that actually it holds that t k = T and 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 γ jk (t) and summing from j = 1 to k, we obtain where we have used (2) to remove the nonlinear term b. By integrating in time, from Hölder and Young inequalities, and the assumptions on the delay operator g, we obtain that for all t ∈ [τ, t k ). Now, from Gronwall lemma, we conclude that t k = T, and that {u k } is bounded in L ∞ (τ, T ; H) ∩ L 2 (τ − h, T ; V ). Moreover, from (3) (see also Remark 2) we have that {du k /dt} is bounded in L 4/3 (τ, T ; V ), whence by compactness results, the Dominated Convergence Theorem, assumption (H4), and Remark 1, we may extract a subsequence (relabelled the same) and ensure the existence of a function u ∈ 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 We will prove that the first addend in the right hand-side goes to zero in the L 1 (τ, T ) norm (the second addend follows analogously). Using (3), Hölder inequality, and the fact that w j is an eigenfunction of the Stokes operator, we have that . From (5) the above goes to zero, and the claim is proved.
Thus, we conclude that u is 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.
Moreover, if for short we denote by u(·) and v(·) the corresponding solutions to (1) with respective initial data (u τ , φ) and (v τ , ψ), then ess sup Proof. The existence of at least one weak solution was already proved in Theorem 1. The energy equality (7) was given in Remark 3 for 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 (u τ , φ) and (v τ , ψ) respectively, and denote w = u − v. Then, from (2) and the energy equality for w, we obtain that Now, as φ, ψ ∈ L 2 V ∩ L ∞ H , by (3) we have the following estimate for the trilinear term b, Integrating in time (10) and using the above estimate, the assumptions on g, and Young and Hölder inequalities with a suitable constant (to be fixed later on), we deduce that In particular, after a change of variable in the integral of w(s − ρ(s)), thanks to the upper bound on ρ , and choosing ε 2 = ν 2 (1 − ρ * ), we arrive at 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 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 ∈ L 2 V alone, the regularisation of the equation means that after an elapsed time h the weak solution obtained in Theorem 1 becomes 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 × L 2 V would 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, d X ), and let us denote A process U is said to be continuous if for any pair τ ≤ t, the mapping U(t, τ ) : X → X is continuous.
On other hand, a process 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. 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 2. 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 where {· · · } X is the closure in X. Given two subsets of X, O 1 and O 2 , we denote by dist X (O 1 , O 2 ) the Hausdorff semi-distance in X between them, defined as Let be given D 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 3.
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 Next result was proved in [8, Theorem 3.11].
Theorem 3. 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: (a) for any t ∈ R, the set A D (t) is a nonempty compact subset of X, and A D (t) ⊂ Λ( D 0 , t), (b) A D is pullback D-attracting, i.e., lim τ →−∞ dist X (U(t, τ )D(τ ), A D (t)) = 0 for all D ∈ D, and any t ∈ R, The family A D is minimal in the sense that if C = {C(t) : t ∈ R} ⊂ P(X) is a family of closed sets such that for any Remark 5. Under the assumptions of Theorem 3, 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 (b)-(c).
A sufficient condition for A D ∈ D is to have that D 0 ∈ D, the set D 0 (t) is closed for all t ∈ R, and the family D is inclusion-closed (i.e., if D ∈ D, and D = {D (t) : t ∈ R} ⊂ P(X) with D (t) ⊂ D(t) for all t, then D ∈ D).
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 3, if the universe D contains the universe D F (X), then both attractors, A D F (X) and A D , exist, and

4.
Dynamical system associated to (1) and long-time behaviour. In view of Theorems 1 and 2, we will apply the above abstract results in the phase-space The first consequence after the Theorems 1 and 2 is the following Corollary 2. Consider given f ∈ L 2 loc (R; V ) and g : R×C H → (L 2 (Ω)) 2 satisfying (H1)-(H4). Then, the biparametric family of mappings S(t, τ ) : where u is the weak solution to (1), defines a continuous process. Now we introduce an additional assumption in order to obtain some energy estimates. (H5) Assume that νλ 1 > C g , and that there exists a value η ∈ (0, 2(νλ 1 − C g )) such that for every u ∈ L 2 (τ − h, t; H), We have the following result (cf. [10]), which proof is included only for the sake of completeness. Lemma 1. Consider given f ∈ L 2 loc (R; V ) and g : R × C H → (L 2 (Ω)) 2 satisfying conditions (H1)-(H5). Then, for any (u τ , φ) ∈ H × (L 2 V ∩ L ∞ H ), the following inequalities hold for the solution u to (1) for all t ≥ s ≥ τ : Proof. By the energy equality (7) and Young inequality, we have d dt ≤ e ηt β −1 f (t) 2 * + e ηt C −1 g |g(t, u t )| 2 , a.e. t > τ, and therefore, integrating in time above and using property (H5), we obtain for all t ≥ τ , and from this last inequality and (14), in particular we deduce (12). Finally, observing that a.e. t > τ, and integrating in [s, t], by (H4) we conclude (13).
At the light of the previous result, we will now define an appropriate concept of (tempered) universe for problem (1).

Definition 4. We will denote by
Observe that the above definition does not make the most use of the natural norm of (ζ, ϕ) in H × (L 2 V ∩ L ∞ H ), but just in H × L 2 H . Another immediate observation is that the above universe is inclusion-closed.
We will denote by D F (H × (L 2 V ∩ L ∞ H )) the universe of fixed bounded sets in H × (L 2 V ∩ L ∞ H ). As a consequence of Lemma 1 we have the following and g : R × C H → (L 2 (Ω)) 2 fulfills conditions (H1)-(H5). Then, the family Proof. Fix t ∈ R. From (12) we deduce that for any D ∈ D In particular, we deduce that u t (13) and using the above estimate, an immediate computation leads to u t ) follows from the definition of R H and (15) (cf. Definition 4). The proof is finished.
Analogously to Corollary 3, from Lemma 1 we have that there exists τ ( D, t) < t − 4h − 1 such that the subsequence {u n : , and thanks to (H4) and (6), . Therefore, by the Aubin-Lions compactness lemma (e.g., cf. [17]), there exists u ∈ L ∞ (t − 4h − 1, t; H) ∩ L 2 (t − 3h − 1, t; V ) with u ∈ L 2 (t − 2h − 1, t; V ) such that, for a subsequence (relabelled the same), the following convergences hold, From (H4) we also have that g(·, u n · ) → g(·, u · ) strongly in L 2 (t − h − 1, t; H). In particular, observe that thanks to the above convergences u ∈ C([t − 2h − 1, t]; H) is a weak solution to (1) We also deduce from (16) where we have used (17) to identify the weak limit. If not, there would exist ε > 0, a value t * ∈ [t−h, t], and subsequences (relabelled the same) {u n } and {t n } ⊂ [t − h, t], with lim n t n = t * , such that Moreover, from (18) we have that From the energy equality (7) for u n and for u, we deduce that the following functions are non-increasing in [t − h − 1, t] : (g(r, u n r ), u n (r))dr, Moreover, J and J n are continuous, and by the above convergences, we have that Consider an arbitrary value δ > 0. By the continuity of J, there exists k δ such that Now, let us take n(k δ ) such that for all n ≥ n(k δ ) it holds t n ≥t k δ and |J n (t k δ ) − J(t k δ )| < δ/2.
Then, since all J n are non-increasing, we deduce that for all n ≥ n(k δ ) Therefore, as δ > 0 is arbitrary, we obtain that lim sup n→∞ J n (t n ) ≤ J(t * ), and consequently, by (16), whence, jointly with (20) and (18), gives the strong convergence u n (t n ) → u(t * ) in H, in contradiction with (19). Thus, Claim 1 is proved.
Claim 2: u n → u strongly in L 2 (t − h, t; V ). Indeed, by using again the energy equality (7) satisfied by u and u n , all the convergences in (16), and Claim 1, we conclude that u n t L 2 V . This convergence of the norms, jointly with the weak convergence already proved in (16), concludes this Claim 2.
The proof follows from Claims 1 and 2.
From the above results, we may establish the main result of the paper.
The goal of this section is to compare these two families of attractors with others associated to this problem, related to the space L 2 V ∩ C H . In order to do so, we need to introduce some additional notation. Analogously to Definition 4, let us introduce a new universe. We will also denote by D F (L 2 V ∩ C H ) the universe of fixed bounded sets in L 2 Let us also introduce the biparametric family of mappings It is clear that U is also a process on L 2 V ∩ C H , and that under the assumptions of Theorem 2, U is continuous.
Before establishing the main result of this section, we need an auxiliar lemma.
Proof. By Theorem 2 it is clear that D (r) (τ ) ⊂ L 2 V ∩ C H . Fix an arbitrary value τ ∈ R and denote by u, for short, the solution to (1) with (arbitrary) initial data (u τ , φ) ∈ D(τ ). From (12) we can deduce that Taking into account that f satisfies (15), the proof is finished.
Now we may establish the following result.
Theorem 5. Assume that f ∈ L 2 loc (R; V ) satisfies (15) and g : R×C H → (L 2 (Ω)) 2 fulfills conditions (H1)-(H5). Then, there exist the minimal pullback attractors for the universes of fixed bounded sets and for those with tempered growth in L 2 V ∩ C H . Both pullback attractors belong to D C H η (L 2 V ∩ C H ) and the following relations hold: where j : . Finally, if f satisfies (22), then, the inclusions in (23) and (24) are in fact equalities for all t ∈ R.
Proof. From Corollary 3, we have that the family , and its time sections are closed.
From Lemma 2 we have that U is pullback D C H η (L 2 V ∩ C H )−asymptotically compact.
Therefore, we may apply again Theorem 3 and Corollary 1 to conclude the existence of the minimal pullback attractors in the statement and the inclusion relation (23).
Relations (24) and (25) through the canonical embedding j from L 2 can be obtained by the construction of the attractors, arguments of minimality of minimal pullback attractors, and estimates after a time-shift of length h, in the same manner as in [23,Theorem 5] or [10,Theorem 23].
Indeed, in order to prove (24), fix an arbitrary value t ∈ R, and observe that where the symbol Λ L 2 V ∩C H denotes the omega-limit construction with respect to the topology of the space L 2 V ∩ C H . Analogously, we have that where the symbol Λ H×(L 2 V ∩L ∞ H ) denotes the omega-limit construction with respect to the topology of the space H × (L 2 V ∩ L ∞ H ). Now, observe that for any bounded set B ⊂ L 2 V ∩ C H , since the operator j is clearly linear and continuous, then , then there exist sequences {τ n }, with τ n ≤ t for all n, and lim n τ n = −∞, and {x n } ⊂ B, such that But this implies that whence we deduce that (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 ∈ R and D ∈ D , where we have used the notation introduced in Lemma 3 for the family D (h) , which belongs to D C H η (L 2 V ∩ C H ), and once more the fact that j is a linear and continuous operator from .Thus, we have that the right-hand side of the above inequality goes to zero when τ goes to −∞, and so the left-hand side also does. Therefore, the inclusion Last claim about the equalities in (23) and (24) when f also satisfies (22) follows again from Corollary 1 since then it holds that sup t≤T R H (t) and sup t≤T R V (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 f satisfies (22), then, for each T ∈ R, the sets 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 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 g we assume that g : C H → (L 2 (Ω)) 2 satisfies (observe that the assumption (H1) holds trivially in this framework): (H2') g(0) = 0, (H3') there exists L g > 0 such that for all ξ, η ∈ C H , Then, the immediate translation of the first existence result (cf. Theorem 1) is the following Theorem 6. Consider u 0 ∈ H, φ ∈ L 2 V , f ∈ V , and g : C H → (L 2 (Ω)) 2 satisfying assumptions (H2')-(H4'). Then, there exists at least one weak solution u(·; 0, u 0 , φ) to (26).
Of course, the concept of weak solution given in Definition 1 to problem (1) just needs to substitute τ = 0 to be referred to problem (26).
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 S on a metric space (X, d X ) is a mapping R + × X (t, x) → S(t)x ∈ X such that S(0) =Id X , and S(t)S(s)x = S(t + s)x for any t, s ≥ 0 and all x ∈ X.
It is said that the semi flow S is continuous if for any t ∈ R + , the mapping S(t) : X → X is continuous.
The semi flow S is said to be asymptotically compact if for any bounded sequence {x n } ⊂ X and {t n } ⊂ R + with lim n t n = ∞, the sequence {S(t n )x n } is relatively compact in X.
A subset B 0 ⊂ X is said to be absorbing for the semi flow S if for any bounded subset B ⊂ X there exists a time T (B) ≥ 0 such that S(t)B := ∪ b∈B S(t)b ⊂ B 0 for all t ≥ T (B).
A subset A ⊂ X is said to be a global attractor for the semi flow S on X if it is compact, invariant (i.e. S(t)A = A for all t ∈ R + ), and it attracts bounded sets of X, i.e. lim t→∞ dist X (S(t)B, A) = 0 for all B ⊂ X bounded.
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 S defined on a metric space (X, d X ), which is continuous. Then, there exists the global attractor A for S if and only if the semi flow is asymptotically compact and it has a bounded absorbing set B 0 ⊂ X. Moreover, then the proofs in that section), we conclude the following result (compare with Theorem 5).