Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows

We analyze a simplified Ericksen-Leslie model for nematic liquid crystal flows firstly introduced in [18] with non-autonomous forcing bulk term and boundary conditions on the order parameter field. We obtain existence of weak solutions in the two- and three-dimensional cases. We prove uniqueness, continuous dependence on initial conditions, forcing and boundary terms and also existence of strong solutions in the 2D case. Focusing on the 2D case, we then study the long term behavior of solutions by obtaining existence of global attractors for normal forcing terms (according to [21]). Finally, we prove the existence of exponential attractors for quasi-periodic forcing terms in the 2D model.

1. Introduction. In this paper we study the simplified Ericksen-Leslie model for nematic liquid crystal flows given by (see [18]): in Ω × (0, ∞); where u is the eulerian velocity field, p is the pressure field and d is the order parameter field which describes the local orientation of the liquid crystal molecules.
We recall that f (d) = 1 2 (|d| 2 − 1)d is obtained from the Ginzburg-Landau potential, which relaxes the physical constraint |d| = 1 (see [13], [17] or [25] for some interesting results without this relaxation). In this paper we consider two nonautonomous terms: one volume force on the linear momentum equation g(t) and the non-homogeneous Dirichlet boundary condition for the order parameter field h(t) (see [20] for some results with other boundary conditions).
Only some results are known on the well-posedness of the full classic model (see [5], [27] or [32] for a complete overview of the physical background and [16] for some analytical results) and much of the most recent results refer to the simplified 408 STEFANO BOSIA system (1) which was first introduced in [18] by Lin e Liu (see, however, [34] or [28] for a more complete model).
In section 2 we obtain well-posedness results for system (1). In particular, we prove the existence of weak solutions both in the two-and three-dimensional cases (theorem 2.2), uniqueness and continuous dependence of solutions on the initial data, forcing and boundary terms (theorem 2.4) for the 2D case and existence of strong solutions always in the two-dimensional case (theorem 2.6). See [11], [14] and [15] for some additional regularity results.
We then focus on the long term behavior of solutions (see [34] and [33] for some similar results). In section 3 we prove the existence of global attractors for the flow under very general assumptions on the non-autonomous forcing terms. In particular, we consider the newly class of normal functions introduced by Lu and Wu in 2005 (see [21] and [22]). We note that normal functions are obviously not necessary translation compact whilst all translation compact functions are also normal. This new class is a proper subset of the set of all translation bounded functions (i.e. weakly translation compact functions, see [2]).
Finally in section 4 we prove the existence of exponential attractors for the flow (see [23] for a review of this subject) considering quasi-periodic forcing terms.
2. Well-posedness. We start by studying the well-posedness of the just introduced simplified Ericksen-Leslie model (1). We will essentially follow the proof in [4] for the system without external driving force. We observe that, in [4], the regularity assumptions on the time dependent lifting function (denoted here by d, see below equation (22)) seems in general not to be verified. We therefore give in appendix A a fully revised proof of the existence of weak solutions in the 2D and 3D cases. We will then complete the analysis in the 2D case by obtaining uniqueness of weak solutions and continuous dependence on the data. Moreover, always in the 2D case, we will be able to prove existence of strong solutions.
2.1. Existence of weak solutions. We start by introducing the functional spaces we will use in the analysis of (1). With L 2 we will denote the standard function space made up by vector valued L 2 (Ω) functions. Analogously, H 1 will be the usual vector Sobolev space constituted by componentwise H 1 (Ω) functions. Moreover, let be the usual divergenceless spaces used in the analysis of Navier-Stokes equations (see, e.g., [31]). The family {w n } n will be the Hilbert basis of V given by the eigenfunctions of Stokes' problem: where, thanks to the spectral theorem, the sequence of λ i is monotonically increasing. From the spectral theory for compact operators we know that the functions w i form a complete orthonormal basis in H which is also orthogonal in H 1 . For convenience we will write V m = w 1 , w 2 , . . . , w m for the finite dimensional subspace of V spanned by the first m eigenfunction of Stokes' problem. From the regularity results for this problem (see [31]) and thanks to the finite-dimensionality of V m , we know that the canonical embedding V m → H 2 (Ω) is compact. Finally, we will denote by V * the dual space of V.
Notation. We will write |w| p to indicate the L p norm of w and |w| H s when referring to its H s norm. Sometimes, as in the definition of the nonlinear potential f just after problem (1), we will write |f (x, t)| or shortly |f | when referring to the usual Euclidean norm of vectors in R n . Therefore, while at fixed time |d| 2 will be a real number, |d| will be a real valued function on Ω.
We now give the definition of weak solutions for system (1).

Remark 1.
We observe that this definition of weak solutions involves the following natural compatibility requirement on the initial and boundary conditions for the order parameter field: d 0 = h on ∂Ω×{0}. This compatibility condition will involve some care in the definition of the phase space when studying the long term behavior of our system (see section 3.1 below).
We give here only a brief sketch of the proof leaving the details to appendix A. We start by studying the regularity of the solution of a lifting problem for the non-autonomous boundary conditions on the order parameter. We then give a semi-Galerkin formulation of problem (1) by considering the discretized problem for the velocity field leaving the other equations in the lifted form. Next we prove local existence of solutions for the approximating problem through a fixed point argument. However, the lifespan of these solutions depends in a critical way on the dimension of the approximating subspace V m . Thus, before passing to the limit, we need to extend the approximating solutions. This is achieved by proving the following energy estimate which also holds for the approximating solutions (see page 439 below). Lemma 2.3. Let the assumptions of theorem 2.2 hold. Then any weak solution of (1) satisfies for all t > 0 the estimate: Finally we end the proof by passing to the limit by means of standard arguments.
, then the size of the time interval [t, t + T ] on which the local solutions obtained by theorem 2.2 are defined is independent of t. Moreover, estimate (3) is uniform in t and therefore any local solution can be extended by successive steps up to ∞.

2.2.
Uniqueness and continuous dependence on initial conditions in the 2D case. The following result holds: Theorem 2.4 (Uniqueness and continuous dependence). Under the same assumptions of theorem 2.2, if n = 2, the weak solution of problem (1) is unique. Moreover, it continuously depends on the initial conditions d 0 , u 0 and on the forcing terms g and h, and the following estimate holds: where Remark 3. We observe also, as a simple corollary of estimate (4), that (u, d) ∈ C(L 2 × H 1 ).
Proof of theorem 2.4. As usual, let (u 1 , d 1 ) and (u 2 , d 2 ) be two solutions of system (1) respectively with forcing terms g 1 and g 2 and boundary conditions h 1 and h 2 . We will use δu . = u 1 − u 2 and δd . = d 1 − d 2 to denote the difference between these two solutions and δg . = g 1 − g 2 , δh . = h 1 − h 2 for the difference between the non-autonomous terms. By considering the difference of the equations solved by (u 1 , d 1 ) and (u 2 , d 2 ) and testing against δu and −∆δd we have: Summing up, recalling identity (34), using the estimate of lemma A.1, Hölder's and Sobolev's inequalities we get: We observe that, when uniqueness estimates are of concern, thanks to the Poincaré inequality, the H 2 norm of δd can be replaced by the L 2 norm of ∆δd. In the general case we are treating now, the H 2 norm can be easily estimated as follows: After repeatedly using Young's inequality, reordering and neglecting the positive terms on the left hand side, we finally deduce: d dt . Applying Gronwall's inequality to this last estimate we eventually obtain (4).

2.3.
Strong solution in the 2D case. Having proved existence and uniqueness of weak solutions for system (1), we are now ready to investigate existence and regularity of strong solutions. In this section we will use the following assumptions: (5) the domain Ω will be an open bounded subset of R n , with n = 2, 3 of class C 2,1 ; (6) the non-autonomous forcing term g will be an element of L 2 (0, T ; H); (7) the non-autonomous boundary term h will belong to L 2 (0, T ; H 5/2 (∂Ω)) and be such that ∂ t h ∈ L 2 (0, T ; H 1/2 (∂Ω)).
We introduce the notion of strong solution for system (1).
We now prove the following existence result.
Proof. We start by considering the same lifting (22) and the same lifted problem (26) used in the proof of theorem 2.2. By using −∆u as a test function in the equation for the velocity field u and recalling that in the 2D case (u · ∇)u, ∆u = 0, we obtain: where δ will be determined later.
To get regularity estimates for the order parameter d we can take the duality of the second equation in (26) with ∆(∆ d − f (d)). We note that, since u| ∂Ω = 0 and d| ∂Ω = 0, we have (∆ d − f (d))| ∂Ω = 0 and therefore it satisfies Poincaré's inequality. Integrating by parts and observing that boundary terms vanish, after a few calculations we have: We now have to find bounds for every term on the right hand side of this last equation. We start by observing that Recalling that |∇ d f (d)| ∞ ≤ C because |d| < 1 by the maximum principle (see lemma A.1) and using the lifted equation, we obtain: Using the regularity assumptions of this section we have d ∈ L ∞ (0, T ; H 2 ). Moreover, from the weak regularity estimates of the previous sections, u ∈ L ∞ (0, T ; L 2 ) and d ∈ L ∞ (0, T ; H 1 ) hold. Therefore, by proceeding as usual, we get: Likewise we can bound the second term in the right hand side of equation (6): Arguing similarly for the third and last term, we have: where δ > 0 will be determined in a few passages. We now gather all the results of this section. Summing up estimates (5) and (6) and using the last three inequalities, after reordering all terms, we get: from which we can easily prove that u ∈ L 2 (0, T ; H 2 ) ∩ L ∞ (0, T ; H 1 ) and d ∈ L 2 (0, T ; H 3 ) ∩ L ∞ (0, T ; H 2 ) for all T > 0 as claimed.
3. Global attractors in the two-dimensional case. In this section we will study the existence of a global attractor for system (1). We will follow the approach of Chepyzhov and Vishik (see [2,Part 2]) as developed for less regular forcing terms by Lu et al. in [21] and [22]. We briefly recall the fundamental definitions we will use. We will suppose that the time dependency can be completely described through a finite set of functions that we shall denote by σ(t). We will call σ(t) the time symbol or simply the symbol of the non-autonomous evolution equation. The set of all symbols of interest in a particular case will be called symbol space and will usually be denoted by Σ.
We need some notions of dissipativeness for our evolution operator (see [2, Part 2]).
Definition 3.1. A set B 0 ⊂ X is said to be uniformly (with respect to σ ∈ Σ) absorbing for the family of processes {U σ (t, τ )}, σ ∈ Σ if for any τ ∈ R and every B ∈ B(X) there exists an absorbtion time attracting for the family of processes {U σ (t, τ )}, σ ∈ Σ if it satisfies, for any fixed τ ∈ R and B ∈ B(X), the following relation: lim is the usual Hausdorff semi-distance between subsets of a metric space (X, d X ).
• A Σ is contained in every other closed uniformly attracting set (minimality property).
Before stating the main result we will apply in this section, we still have to recall an additional definition (see [21]).
is bounded for every t and lim t→∞ α(B t ) = 0 where α is the Kuratowski measure of noncompactness defined by: . = inf{δ > 0 | B admits a finite cover by sets of diameter ≤ δ}.
We can now report for the reader's convenience the main result we will use in this section to prove the existence of a global attractor for system (1) (see [

21, Section 2.3]).
Theorem 3.5. Let {U σ (t, τ )}, σ ∈ Σ be a uniformly (w.r.t. σ ∈ Σ) ω-limit compact and (X × Σ, X)-weakly continuous family of processes acting in X, let B 0 be a weakly compact (i.e., bounded) uniformly (w.r.t. σ ∈ Σ) weakly attracting set for {U σ (t, τ )}, let Σ be a weakly compact subset of some Banach space and let {T (t)} be a weakly continuous invariant (T (t)Σ = Σ) semigroup on Σ satisfying the translation identity: Then the extended semigroup {S(t)} defined on X × Σ possesses the compact attrac- • the global attractor satisfies: • the uniform attractor satisfies: where K σ (0) is the section at time t = 0 of the kernel K σ of the process {U σ (t, τ )}, that is: In addition we also recall a useful criterion to prove the uniform ω-limit compactness for a given process (see [21,Theorem 2.3]). Proposition 1. Let X be a uniformly convex Banach space (any L p space with p = 1, ∞ will be suitable). Then the family of processes {U σ (t, τ )}, σ ∈ Σ is uniformly (w.r.t. σ ∈ Σ) ω-limit compact if and only if for any fixed τ ∈ R, B ∈ B(X) and > 0 there exists t 0 = t 0 (τ, B, ) ≥ τ and a finite-dimensional subspace X 1 of X such that: 3.1. Bounded absorbing sets for system (1). In order to apply the just introduced abstract theory to system (1), we need some preliminary estimates. Our first goal will be to find some absorbing sets for the trajectories of our system in various function spaces. Most of the results of this section are simple consequences of a suitable interpretation of the estimates obtained in the existence proofs.
In our case the symbol space will be generated by the two non-autonomous terms g and h by continuous time-shifts. Before introducing the symbol spaces we will use, we need to recall the following definition given first in [21].
With L 2 n (R; E) we shall indicate the space of all normal functions taking values in E.

In both cases H(f ) is the (weak) hull of f
On account of the existence results of the previous section and, in particular, due to the compatibility condition d 0 = h(0) on ∂Ω, some care is needed in the definition of the phase space for our system. Indeed, we observe that we can decompose the state of our system as: whereď andd are defined as in (31). Therefore we can consider the following natural phase space: We also observe that a more geometric description of this phase space is possible. The situation for the velocity field being standard, we describe, for simplicity, only the phase space framework for the order parameter. We introduce the map Φ : ) → x +d(0) which parametrizes a subset X of H 1 . Since Σ is closed and regular, we observe that X is also closed. Moreover, from the translation invariance of Σ, we deduce that X is itself invariant under the action of the family of processes U σ (t, τ ), σ ∈ Σ. The attraction properties, which we are going to prove in the present and subsequent sections, can therefore be interpreted as taking place on the parametrization of the phase space given by H 1 0 × Σ in the metric locally induced by the map Φ −1! .
Thanks to theorem 2.2 and to the dissipation result which follows, we can also define the process associated with the solution operator of equation (1) acting on the phase space X indexed by a symbol σ ∈ Σ 0 (or σ ∈ Σ 1 ).
Theorem 3.7. Under the regularity assumptions of theorem 2.
Moreover, for t ≥ t 0 (B), we also have: Consider estimate (35). We only need to show that the two integrals are bounded if (g, h) ∈ Σ 0 . Actually this can be easily shown under more general assumptions, namely it is enough that g and h are L 2 translation bounded (see [2,Section V.4]). Indeed we have: Recalling a standard elliptic regularity estimate (cp. (31)) |∇d| 2 ≤ |h| H 1/2 and that, under the present hypothesis, h is continuous and bounded with values in H 1/2 (∂Ω), we obtain the absorbing set B 0 as claimed. We will denote by t 0 (B) the absorbtion time of the bounded set B in B 0 . In particular we observe that the absorbtion time t 0 can be obtained from the inequality: which gives the claimed result.
In order to prove the second part of theorem 3.7, we only need to integrate equation (33) from t to t + 1 with t sufficiently large (it is enough to consider t ≥ t 0 ). This immediately gives the estimate we claimed.
Analogously, starting from the result of section 2.3, we can prove the existence of absorbing sets bounded in the stronger topology of V × H 2 .
Theorem 3.8. Under the regularity assumptions of theorem 2.6, system (1) admits a uniformly (w.r.t. σ ∈ Σ 1 ) absorbing set B 2 ⊂ V × H 2 : where ρ 2 and ρ 3 depend only on ν, , Ω, for all T > 0. By choosing = 1 and by recalling the regularity estimate (24) for the lifting solution d, we then easily deduce the existence of the strong absorbing set B 2 .
As in the proof of theorem 3.7, in order to obtain the second part of the theorem we only need to integrate estimate (8) from t to t + 1 with t sufficiently large (again it is enough to suppose t greater than the absorbing time in B 2 ).
We end this section introducing another absorbing set which will prove to be useful when dealing with exponential attractors in the next section.

A smooth attractor.
We can now apply theorem 3.5 to system (1) under strong regularity assumptions. More precisely we will prove the following result. Theorem 3.9. Suppose that (5) and (9) hold, then the process {U (g,h) (t, τ )} generated by the solution operator of problem (1) possesses a compact uniform (w.r.t.
attracts the bounded sets in X in the product norm of X. Moreover we have: Remark 4. From this result we deduce that all the solutions to (1) belonging to the kernel of the solution process are strong and globally bounded. We therefore deduce that system (1) holds a.e. on the kernel.
Proof. We begin our proof by observing that these regularity assumptions on the boundary term h imply that h ∈ L ∞ (0, ∞; H 3/2 (∂Ω)).
Actually, with the above assumptions, we only have to prove ω-limit compactness and weak continuity of the process defined by the solution operator in order to apply theorem 3.5 and prove our claim.
We start with ω-limit compactness. We will consider again the lifted system (32) (forgetting all m's). First of all, however, we prove that the lifted term ∂ t ∇d is bounded and normal. From problem (31) tested against a function v ∈ H 1 (Ω), a simple integration by parts yields: By choosing v =d we eventually get: We then easily obtain: If we apply this last estimate to ∂ td instead of d with ∂ t h substituted to h, we have then proved that ∂ t ∇d ∈ L 2 n (R, L 2 ) since ∂ t h ∈ L 2 n (R, H 1/2 ). We now recall proposition 1, which gives a straightforward way to check ω-limit compactness for the process. Thanks to the Hilbert setting which provides a natural norm-reducing projection onto any linear subspace and on account of the absorbing sets previously identified, the first assumption of proposition 1 has already been verified. We still have to control for the "dissipativeness" of the higher modes. As subspaces we will consider V n for the velocity field and the space D m spanned by the first m eigenfunctions of the laplacian with Dirichlet homogeneous boundary conditions in Ω. Let {λ n } and {µ m } be the ascending sequences of eigenvalues respectively for Stokes's problem and Laplace's problem on Ω and let P n and Q m be the projections on V n and D m respectively. In what follows, we set! u 1 . = P n u, Consider again the equation for the velocity field in (32) and take its scalar product in L 2 with −∆u 2 . Using the orthogonality of the chosen base (notice, for example, that (∆u, ∆u 2 ) = |∆u 2 | 2 2 ), we obtain: , ∆u 2 ) . (11) As usual, we have to estimate all terms on the right hand side of this last expression. In order to obtain the desired estimates we recall a useful interpolation (see [1]).
where the constant C depends only on the domain Ω.
We start by analyzing the well-known trilinear term of Navier-Stokes equations. We have: . Recalling the absorbing sets identified in the previous sections and noticing that |∆u 2 | 2 2 ≤ λ n+1 |∇u 2 | 2 2 , we finally obtain: |((u · ∇)u, ∆u 2 )| ≤ Cρ 2 (1 + ln λ n+1 ) 1/2 |∆u 2 | 2 + Cρ The other nonlinear term can be estimated analogously as follows: The last term on the right hand side of equation (11) is easily dealt with: Putting everything together, we get the first half of the desired estimate: We now turn our attention to the equation for the order parameter. Testing the second equation in (32) with ∆∆ď 2 , integrating by parts and using the orthogonality of the eigenbasis of the laplacian, we get: As with the equation for the velocity field, we now have to bound all terms on the right hand side of this last equality.
The second term is dealt with in a similar way. We only recall that usual elliptic regularity results for problem (31) give |d| 2 Finally the last two bulk terms can be estimated as follows: In order to control the boundary term, we can write: Adding everything together, we eventually get: By recalling estimate (12) and adding the last inequality we have obtained, we find the desired bound on the higher modes of our solution: From Poincaré's inequality in V and H 1 0 we have |∆u 2 | 2 2 ≥ λ n+1 |∇u 2 | 2 2 and similarly |∇∆ď 2 | 2 2 ≥ µ m+1 |∆ď 2 | 2 2 . By setting κ = min{νλ n+1 , µ m+1 } and using Gronwall's inequality we finally get: All terms on the right hand side of last inequality can be made arbitrarily small by choosing n and m sufficiently large such that νλ n ≈ µ m and recalling estimate (10) for translation bounded functions and the normality assumption (see [21]). This proves ω-limit compactness.
To complete the proof of the existence of the global attractor we still have to control weak continuity of the process with respect to initial data and to the symbol. Our argument follows [24, Lemma 2.1] with the obvious changes.
Since we already know that {d n } is bounded in L 2 (τ, T − a; H 3 ), using the same lemma as before, we conclude that {d n } is precompact in L 2 (τ, T − a; H 2 ).
From the boundedness and compactness of the sequences just proved, by means of a diagonal extraction process, we can find a subsequence of {u n , d n } that converges weakly* in L ∞ (τ, ∞; V × H 2 ), weakly in L 2 loc ([τ, ∞); H 2 × H 3 ), strongly in L 2 loc ([τ, ∞); V×H 2 ) to (u, d) and such that {d n } converges weakly* in L ∞ (τ, ∞; L ∞ ). We note that (u, d) solves equation (1) (the passage to the limit in the equation is analogous to that treated in appendix A). From the strong convergence we have (u n (t), d n (t)) → (u(t), d(t)) strongly in V × H 2 for a.e. t ≥ τ . We therefore have: for almost every t ≥ τ and any regular pair of functions (v, w). We note that, from the previous estimates, (∇u n (t), v) and (∆d n (t), w) are equibounded and equicontinuous as functions of t. Therefore the convergences hold for all t ≥ τ , i.e. we have obtained weak continuity for the solution process we are studying.

3.3.
A less regular attractor. We now want to investigate what happens when we consider less regular forcing terms of critical regularity. The main result we shall obtain is the following.
Theorem 3.11. Suppose that (1) and (8) hold, then the process {U (g,h) (t, τ )} associated to the solution of problem (1) possesses a compact uniform (w.r.t. (g, h) ∈ Σ 0 ) attractor A Σ0 in X which uniformly (w.r.t. (g, h) ∈ Σ 0 ) attracts the bounded sets in X in the norm of X. Moreover we have: where K (g,h) is the kernel of the process {U (g,h) (t, τ )} and where K (g,h) is nonempty for all (g, h) ∈ Σ 0 .
Proof. As in the proof of theorem 3.9, we only have to check ω-limit compactness and weak continuity in order to apply theorem 3.5. The argument leading to weak continuity can be carried over to the current setting with no significant changes. However, checking ω-limit compactness involves a slightly more subtle estimate. Indeed, due to the structure of the nonlinear terms and in particular to the convective term (u · ∇)u, the direct approach adopted in the last section does not lead to any useful estimate in this case. We look for a bound on the time derivative of the solution fields in the natural weak norms. From the equation we have: where the nontrivial terms on the right hand side can be bounded as follows: We observe that |d| 2 H 2 ≤ C|∆ď| 2 2 + |h| 2 H 3/2 and therefore we have (∂ t u, ∂ t d) ∈ L 2 loc ([τ, ∞); V * × L 2 ). Moreover, the norm of (∂ t u, ∂ t d) is uniformly bounded in [t, t + δt] w.r.t. t > τ . Thanks to [26,Corollary 4], we deduce that From the precompactness of B [t,t+δt] we deduce that there exists a finite number of pairs (u 1 , d 1 ), . . . , (u N , d N ) such that for any (u, d) ∈ B [t,t+δt] there exists an i that verifies Therefore there exists a timet ∈ [t, t + δt] such that We now use the continuous dependence estimate (4) and get: where all constants depend only on ρ 0 , ρ 1 and are bounded w.r.t. δt ≤ 1. Using the normality assumption (2) and the precompactness of the trajectories, we can bound the left hand side of the last inequality by a fixed constant times . We have thus proven that B T = U T,τ B 0 is compact for sufficiently big T − τ , uniformly w.r.t. τ ∈ R. Since B 0 is absorbing, this also proves the ω-limit compactness for the process.
4. Exponential Attractors. Global attractors are not necessarily the only description of a dissipative dynamical system. A feature one usually wants to guarantee is an exponential attracting rate of trajectories to the attractor preserving the finite dimension. Moreover, some sort of continuous dependence of the attractor on the data is hoped for. This means, e.g., that small changes in the form of the forcing terms should not cause any relevant modification of the attracting sets (we refer to [23] for a complete overview of the subject). One possible solution that guarantees these properties is given by exponential attractors which were proposed in the 90's by Eden et al. (see [7]). Although some important drawbacks (notably the lack of uniqueness) still remain, this theory can easily be applied to the general setting of Banach spaces (see [8]) by using rather natural estimates on the solutions. We review briefly the standard theory recalling the results we shall use later. In order to prove the existence of such absorbing sets we introduce the following dissipativity notion for a semigroup. Definition 4.2. Let E and E 1 be Banach spaces with E 1 compactly embedded in E, let X be a bounded subset of E 1 and let S : E → E. Then S enjoys the smoothing property on X if: Definition 4.3. Let E and E 1 be Banach spaces with E 1 compactly embedded in E and let X be a bounded subset of E 1 . Given positive constants δ and K, a (nonlinear) operator S : is a neighborhood of X of radius δ in the topology of E 1 ; • S enjoys the smoothing property on O δ (X), that is: We can now state the main results we will use in this paper (see [8] for a proof).
Theorem 4.4. For every S ∈ S δ,K (X), there exists an exponential attractor M S in the topology of E 1 , that is: Moreover, the map S → M S can be chosen such that it is Hölder continuous in the following sense: Finally, α, κ and all other constants which appear in the preceding estimates depend only on X, δ and K, but they are otherwise independent of the particular semigroup S ∈ S δ,K (X).

4.1.
A discrete exponential attractor for system (1). This section and the next are devoted to prove the existence of an exponential attractor for system (1). We start by defining the symbol space we will use in this section.
We recall that the translation hull of a (Lipschitz) quasi-periodic function can be identified with the k-dimensional real torus T k . In particular, setting We shall indicate with QP (Ξ) the set of all (Lipschitz) quasi-periodic functions with values in Ξ. Our main assumption will be: (10) g ∈ QP (L 2 ), h ∈ QP (H 5/2 (∂Ω)) and ∂ t h ∈ QP (H 1/2 (∂Ω)).
The main result of this section is: (5) and (10) hold and let {S(t)} be the extended semigroup associated to the solution operator of problem (1) acting on the extended phase space X × T k (here k is equal to the sum of the different irrationally independent periods of h and g) 1 . Then there exists a finite time t * such that the discrete-time semigroup generated by S(t * ) possesses a uniform (w.r.t. the initial phase θ ∈!T k ) exponential attractor.
Proof. Thanks to theorem 4.4, setting X = H × H 1 and X 1 = V × H 2 , we will only need to prove that the extended semigroup S : X × T k → X × T k belongs to the class of operators S δ,K (X) for suitable δ, K and X. It is therefore sufficient to show that there exists an absorbing bounded set X ⊂ X 1 × T k and that the smoothing property holds (cf. definition 4.3).
Since T k is invariant under the action of the extended semigroup, the existence of a suitable absorbing set is a simple consequence of theorem 3.8. Therefore, in what follows, we choose X = B 2 × T k .
We now need to show the smoothing property. Since the proof of this result is quite lengthy, we give here a short overview of our (standard) argument. The main estimate we shall prove will be made up of three major contribution: the first arising from the difference equation for the velocity field obtained from (26) (without ms), the second coming from the difference lifted order parameter equation deduced again from (26) while the third and last derives from the difference timedependent lifted problem got from (22). In any of the three cases, our aim will be to obtain inequalities of the form: where E 2 will be a Banach space (compactly) embedded in E 1 and where δ is any difference of solutions. Using then the uniform Gronwall inequality we get a time dependent bound on |δ(t)| E1 : The smoothing property can then be obtained through the results of section 3.1. We start by considering the equation for the velocity field in (26). Let (u 1 , d 1 ) and (u 2 , d 2 ) be two solutions to (26). Taking the difference of the equations for the velocity field, multiplying by −∆w and integrating by parts, we obtain: 1 2 d dt |∇w| 2 2 − ((w · ∇)u 1 , ∆w) − ((u 2 · ∇)w, ∆w) + ν|∆w| 2 2 = (∇e) t ∆d 1 , ∆w + (∇d 2 ) t ∆e, ∆w + (g 1 − g 2 , ∆w) 1 We observe that here we have implicitly substituted the natural extension T l ⊕ T m of the phase space with the algebraically and geometrically equivalent space T l+m . This equivalence can immediately be proven by considering the standard coordinate description of a k-dimensional torus through vectors of R k with the proper identification. for a.e. t ∈ R, where we have set w = u 1 −u 2 and e = d 1 −d 2 (note that w| ∂Ω = 0).
As in the previous sections, we now bound all the non-linear terms. For the first part of the trilinear term coming from the Navier-Stokes equation we have: where the last estimate follows from theorem 3.8. For the second term arising from the convective contribution in Navier-Stokes equations, we can write: We now consider the two contributions coming from the nonlinear coupling with the order parameter equation. We have: where β is a positive real number that will be determined later. Similarly we get: (∇d 2 ) t ∆e, ∆w ≤ |∇d 2 | 4 |∆e| 4 |∆w| 2 To obtain the first estimate we need to control the non-autonomous forcing term: By putting all these estimates together, we deduce the following inequality: We now consider the lifted equation for the order parameter field (see (26)). By taking the difference of the equations satisfied by the same two solutions (u 1 , d 1 ) and (u 2 , d 2 ) as above, multiplying by ∆(∆ e − f (d 1 ) + f (d 2 )) and integrating by parts we have: where we observe that e| ∂Ω = 0 and that ∆ e − f (d 1 ) + f (d 2 )| ∂Ω = 0. Completing the first term on the left hand side of this relation we obtain: As usual, we have to estimate all the nonlinear terms appearing on the right hand side. We start with the four transport terms. We have: and, analogously, we get: We can also proceed in a similar way for the following two terms, obtaining: and deducing: We now have to consider the last term in (16). We start by observing that the following identity holds: where with ∇ d we denote the gradient with respect to d. Before going on, we recall that the tensor norm we use throughout this work (Frobenius' norm) is compatible with the standard Euclidean norm of vectors. Therefore we have: By using these results we obtain: We observe that, thanks to lemma A.1, we have: In order to finish this part of our argument, we have to find an appropriate estimate for |∂ t e| 2 . From system 25, we obtain: We can now obtain the desired estimate for the last term in (16). By using all the results above, recalling corollary 2 and |e| ∞ ≤ C|e| H 2 , we get: We can now substitute these estimates in (16) obtaining: where (u(s), d(s)) is any solution of (1). We begin with ∂ t u. From the equation for the velocity field in (1) we get: H 3 . By squaring and integrating between τ and t, recalling the results of section 3.1, we easily find that on the absorbing set (that is on a neighborhood of the exponential attractor) |∂ t u| L 2 (τ,t;H) is bounded.
An analogous estimate can be obtained by taking the gradient of the equation for the order parameter. In particular we have: Again simple calculations give a uniform bound on |∂ t d| L 2 (τ,t;H 1 ) on a neighborhood of the exponential attractor.
Remark 6. We observe that in [9], starting from the theory of pullback attractors, a slightly different notion of exponential attractor is introduced. Although here the setting of the problem is not exactly the same, the results now obtained (i.e. absorbing sets and smoothing property) are sufficient to prove the existence of a time dependent exponential attractor when the forcing terms are quasi-periodic. Moreover, thanks to the abstract results of [9], this attractor is continuous w.r.t. the forcing term considered. Since in the context of pullback attractors one considers a fixed non-autonomous forcing term, we also observe that the estimates of this section can easily be adapted to the more general case of arbitrary translation-bounded forcing terms. This leads to the existence of a positively invariant time-dependent exponential attractor even under these more general assumptions. It is interesting to notice that this approach does not require a somewhat involve! d definition of the phase space due to the compatibility conditions on initial and boundary data. These features will however be studied in more detail in a forthcoming paper of which this remark represents only a preliminary account. solution u m ∈ C 1 (0, T ; V m ) and d m ∈ L 2 (0, T ; H 2 ) ∩ L ∞ (0, T ; H 1 ) such that: on ∂Ω × (0, T ) (25) holds, where the linear operator P m : H → V m is the orthogonal (in L 2 ) projection on V m . Actually also the following lifted problem will be important in the proof: = d m − d and d is the solution of the lifting problem (22).
Local time existence of solutions. We will now apply a fixed point argument to prove existence of (at least) a solution on the time interval [0, T m ] for the approximating problem (25). We start by introducing the following splitting. 1. Let u m ∈ C(0, T ; V m ) be a given velocity field. We look after the order parameter field d m ∈ L 2 (0, T ; H 2 ) ∩ L ∞ (0, T ; H 1 ) which solves: 2. Let d m ∈ L 2 (0, T ; H 2 ) ∩ L ∞ (0, T ; H 1 ) be the order parameter field just determined. The second part of the splitting consists in finding a velocity field u m ∈ H 1 (0, T ; V m ) such that the following equation is satisfied: We stress that in this problem the order parameter field d m is given.
Remark 7. The just introduced splitting and the fixed point argument of the following pages can be considered as possible starting points in the design of numerical scheme to solve problem (1). One possible advantage of this strategy is that one can use only existing programs that already efficiently solve Navier-Stokes equations and simple transport-diffusion equations without the need of implementing from scratch a whole new numerical algorithm.
Existence and uniqueness for problem (27a). The existence of a solution which satisfies equation (27a) can be obtained through a standard fixed point argument.
We recall here only the main results which will be important in the sequel observing that here no assumptions on ∂ t h are necessary.
Lemma A.5 (Weak maximum principle, [3]). Suppose that the hypotheses (1), (3) and (4)  We now use the lifting function given by (22). Testing the lifted problem with −∆ d m , routine calculations give the following estimate.
Lemma A.6. Under the same assumptions of lemma A.5, let M ∈ R be a constant such that sup t∈[0,T ] |∇u m | 2 ≤ M . Then the following estimate holds: Moreover, it is easy to prove that the solution is unique and continuously depends on initial data so that the solution operator S Existence and uniqueness for problem (27b). The well-posedness of the problem for the velocity field (27b) can easily be obtained through the standard Galerkin approach. Before stating the main result and estimate, we only observe that since ∇d m (t) ∈ L 6 , ∆d m (t) ∈ L 2 a.e. t ∈ [0, T ], the d m -dependent forcing term can indeed be read as a scalar product in L 2 instead of being a duality. Moreover it is useful to recall the following vector identity: ∇ · (∇d m ∇d m ) = (∇d m ) t ∆d m + 1 2 ∇|∇d m | 2 .
Lemma A.7. Suppose that the hypotheses (1) and (2) We observe that the constant C m , which explicitly depends on m, appears because we made use of the inverse inequality |∇u m | ∞ ≤ C m |∇u m | 2 which holds since V m is finite dimensional.
As before, we can also easily prove that the solution operator of problem (27b): Let now M > 0 such that |u 0 | 2 2 + 2 ν |g| 2 L 2 (0,T ;V * ) ≤ M 2 . Thanks to estimates (28) and (30), if |u m (t)| 2 2 ≤ M for all t ∈ [0, T m ], we have: that is |u m (t)| 2 2 ≤ M for all t ∈ [0, T m ] where 0 < T m ≤ T is sufficiently small so that the norms of d m are suitably bounded. We can therefore apply Schauder's Theorem and obtain existence of a solution u m ∈ H 1 (0, T m ; V m ), d m ∈ L ∞ (0, T m ; H 1 ) ∩ L 2 (0, T m ; H 2 ) to problem 25. Uniqueness can be proven in a standard way and we leave out the straightforward details.
Extending approximating solutions. Since T m → 0 when m → ∞, before passing to the limit on m, we still need to extend these approximating solutions. The a priori estimates needed to accomplish this step can be obtained by considering a different lifting problem for system (1): From the standard theory for elliptic partial differential equations we know the following existence and regularity result.
Lemma A.8. Under assumptions (1) and (4) the lifting problem (31) has a unique solutiond in H 1 (0, T ; L 2 ) ∩ L ∞ (0, T ; H 1 ) ∩ L 2 (0, T ; H 2 ) and the following estimates hold for a.e. t > 0: In the following, we will writeď m . = d m −d to refer to this differently lifted function. We immediately observe thatď m is a solution of the following problem: Setting C 0 = min{ ν C P , 1 C P } = min{νµ 1 , µ 1 } with C P Poincaré constant for the domain Ω and with µ 1 first eigenvalue of the homogeneous Laplace-Dirichlet operator on Ω we eventually have the following estimate: We note that, since |∇d| 2 ≤ C|h| H 1/2 (∂Ω) ≤ C|h| L 2 (H 3/2 ) + C|∂ t h| L 2 (H −1/2 ) , this inequality implies lemma 2.3 for the approximating solutions (u m , d m ). We have, indeed, shown that the L 2 norm of u m and ∇d m are uniformly bounded in m and that we can extend all approximating solutions up to any fixed time T .
Passing to the limit. We can now pass to the limit in (25). We recall that the sequence {(u m , d m )} is bounded in L ∞ (0, T ; H × H 1 ) and in L 2 (0, T ; V × H 2 ) and is such that {d m } is bounded in L ∞ (0, T ; L ∞ ). Using identity (29), we can deduce directly from equation (25) that ∂ t u m is bounded in L p (0, T ; V * ) and that ∂ tď m is bounded in L p (0, T ; L 2 ), with p = 2 when n = 2 and p = 4/3 when n = 3. Actually, for the convective term, the following estimates hold (see [31]): |((u · ∇)v, w)| ≤ C |u|