Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

Homogenisation of global $\mathcal{A}^\epsilon$ and exponential $\mathcal{M}^\epsilon$ attractors for the damped semi-linear anisotropic wave equation $\partial_t^2 u^\epsilon +\gamma\partial_t u^\epsilon-{\rm div} \left(a\left( \tfrac{x}{\epsilon} \right)\nabla u^\epsilon \right)+f(u^\epsilon)=g$, on a bounded domain $\Omega \subset \mathbb{R}^3$, is performed. Order-sharp estimates between trajectories $u^\epsilon(t)$ and their homogenised trajectories $u^0(t)$ are established. These estimates are given in terms of the operator-norm difference between resolvents of the elliptic operator ${\rm div}\left(a\left( \tfrac{x}{\epsilon} \right)\nabla \right)$ and its homogenised limit ${\rm div}\left(a^h\nabla \right)$. Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts $\mathcal{A}^0$ and $\mathcal{M}^0$ are established. These results imply error estimates of the form ${\rm dist}_X(\mathcal{A}^\epsilon, \mathcal{A}^0) \le C \epsilon^\varkappa$ and ${\rm dist}^s_X(\mathcal{M}^\epsilon, \mathcal{M}^0) \le C \epsilon^\varkappa$ in the spaces $X =L^2(\Omega)\times H^{-1}(\Omega)$ and $X =(C^\beta(\overline{\Omega}))^2$. In the natural energy space $\mathcal{E} : = H^1_0(\Omega) \times L^2(\Omega)$, error estimates ${\rm dist}_{\mathcal{E}}(\mathcal{A}^\epsilon, {T}_\epsilon \mathcal{A}^0) \le C \sqrt{\epsilon}^\varkappa$ and ${\rm dist}^s_{\mathcal{E}}(\mathcal{M}^\epsilon, {T}_\epsilon \mathcal{M}^0) \le C \sqrt{\epsilon}^\varkappa$ are established where ${T}_\epsilon$ is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.


Introduction
In this article we consider the following damped semi-linear wave equation in a bounded smooth domain Ω ⊂ R with rapidly oscillating coe cients: (0.1) Such equations appear, for example, in the context of non-linear ascoustic oscillations in periodic composite media (see for example [1]). For xed ε > , the long-time behaviour of u ε has been intensively studied in many works under various assumptions on the non-linearity f and force g. In the context of dissipative PDEs the long-time dynamics can be studied in terms of global attractors. Intuitively speaking, the global attractor is a compact subset of the in nite-dimensional phase space which attracts all trajectories that originate from bounded regions of phase space. Therefore, the global attractor is in some sense a 'much smaller' subset of phase space that characterises the long-time dynamics of the system (see for example [2][3][4][5][6][7]).
It is well-known that for suitable assumptions on the non-linearity (cf. [2,7]) that problem (0.1) possesses a global attractor A ε and an important question to ask, from the point of view of applications, is about the asymptotic structure, with respect to ε, of the global attractor A ε in the limit of small ε. Asymptotics for global attractors have been studied, in the context of reaction di usion equations and the damped wave equation, with respect to 'lower-order' rapid spatial oscillations in the dampening, non-linearity and/or forces g (see [8][9][10][11]). Yet surprisingly, to the best knowledge of the authors, little or no work has been performed on the asymptotics of attractors for hyperbolic dissipative systems with 'higher-order' rapid spatial oscillations such as in (0.1). We mention here the works [12] that perform a quantitative analysis of the asymptotics of global attractors in the context of reaction di usion equations. We also mention the works [13,14] that determine the limit-behaviour of global attractors, in the context of reaction-di usion and hyperbolic equations, for a particular choice of rapidly oscillating coe cients that degenerate in the limit of small period. Aside from the very limited amount of work done on the asymptotics of global attractors for dissipative PDEs with rapid oscillations, no work has been done on the asymptotics of exponential attractors. This article is dedicated to performing these studies for problems of the form (0.1).
In this article we aim to study the long-time behaviour of trajectories u ε to (0.1), for small parameter ε, from the point of view of homogenisation theory. In homogenisation theory, the mapping Aε u := − div a( · ε )∇u , for periodic uniformly elliptic and bounded coe cients a(·), is well-known to converge (in an appropriate sense) in the limit of small ε to A u := − div a h ∇u , where a h is the 'e ective' or 'homogenised' constant-coe cient matrix associated to a(·) (see for example [15] and references therein). As such, it is natural to compare the long-time dynamics of u ε to the long-time dynamics of u the solution to homogenised problem    ∂ t u + γ∂ t u − div a h ∇u + f (u ) = g(x), x ∈ Ω, t ≥ , (u , ∂ t u )| t= = ξ , u | ∂Ω = .
(0.2) Homogenisation theory has been studied intensively since the 1970's and amongst the extensive works we focus on works related to quantitative estimates of the form where the mappings have been equipped with appropriate boundary conditions. Such (sharp) order-ε results, that are now standard, has been proved by various authors using various techniques (see the monograph [16] for a review of some of these techniques). We mention here the results of particular interest to our article; in the case of bounded domain with Dirichlet or Neumann boundary conditions the order-sharp estimates were proved for the rst time in [17,18] and utilised the (order-sharp) estimate proved in [19,20] for the whole space (and periodic torus). While some work has been done to provide order-sharp operator estimates for individual trajectories in the parabolic (cf. [15,16]) or hyperbolic settings (for smooth enough initial data) (cf. [21][22][23][24]), no work is done on providing order-sharp operator estimates for attractors in dissipative PDEs.
Our rst main result is the following estimate¹ between the global attractors A ε and A , associated to problem (0.1) and (0.2) respectively, in the energy spaces E − := L (Ω) × H − (Ω) and (C β (Ω)) (see Theorem 4.3 and Corollary 4.1): Brought to you by | University of Durham Authenticated Download Date | 9/18/19 12:35 PM for some κ, θ ∈ ( , ). Upon combining this result with the operator estimate (0.3) gives the desired error estimates between global attractors.
The above inequality is new in the homogenisation theory of attractors. Moreover, this result is important from the general perspective as it establishes the upper semi-continuity of global attractors of the damped wave equation in terms of the elliptic part of the PDE. Indeed, in the proof of this result we do not use the asymptotic structure in ε of Sε(t) in terms of S (t). The arguments are purely operator-theoretic in nature and only require that the elliptic operator is self-adjoint and boundedly invertible (see Section 4). In particular, if Aε and A were positive elliptic operators A = div(a∇) and B = div(b∇) for two di erent matrices a and b, the above continuity result still holds. Additionally, the same can be said for di erent boundary conditions: one can replace Dirichlet boundary conditions with other types of boundary conditions under the sole requirement that A = div(a∇) de nes a self-adjoint operator in L (Ω) (see Section 7 for details).
Let us say a few words on the method of proof of (0.4). This result is essentially proved by establishing the following (sharp) estimate between trajectories u ε (t) and u (t) for initial data in A ε (Theorem 4.2): Then, for such a choice of initial data, we readily establish inequality (0.7) (Theorem 5.2 and Corollary 5.1) and consequently prove (0.6). Such initial data was originally introduced in [25] in the homogenisation (without error estimates) of the linear wave equation. An important question from the point of view of applications is whether or not the estimates (0.4), (0.6) hold in the symmetric Hausdor distance To prove this one would need to show that for su ciently small ε the global attractor A ε is in fact (generically) an exponential attractor with exponent, and set of generic forces, independent of ε. Such a result seems reasonable from the perspective of considering A ε to be an 'appropriate' perturbation of the global attractor A and applying the theory of regular attractors, see for example [2,26]. Such a result has yet to be established and we intend to carry out this study in future work.
That being said, it is known that, in general, global attractors are not continuous (in the symmetric Hausdor distance) under perturbations and that the rate of attraction can be arbitrarily slow. For this reason the theory of exponential attractors was developed; such exponential attractors are known to be stable under perturbations and attract bounded sets exponentially fast in time. Importantly, exponential attractors also occupy 'small' subsets of phase space in the sense that they have nite fractal dimension, cf. [27][28][29][30].
Motivated by the above discussion, and the desire for estimates in the symmetric Hausdor distance, we also study the relationship between exponential attractors associated to problems (0.1) and (0.2). In fact we construct exponential attractors M ε and M whose ( nite) fractal dimension and exponents of attraction are independent of ε, and we determine the following analogues of (0.4) and (0.6) in the symmetric distance (Theorem 6.1, Corollary 6.1 and Theorem 6.3): To establish the last inequality above we developed further (in Theorem 6.4) the known abstract construction of exponential attractors of semi-groups to include the case of semi-groups that admit asymptotic expansions (i.e. 'corrections' such as Tε).
We end the introduction with some words on the structure of this article. In Section 1, we formulate precise assumptions on the non-linearity f and the elliptic part of (0.1), (0.2). Also, we recall relevant known well-posedness results as well as results on the existence of global attractors associated with (0.1), (0.2). For the reader's convenience, details on the corresponding attractor theory is provided in Appendix A. In Section 2, for the dynamical systems generated by problems (0.1), (0.2), we establish existence and smoothness results for an attracting set (which contains the global attractors). These results will be crucial in justifying error estimates between anisotropic and homogenised attractors. In Section 3, we establish the convergence, in the limit of ε → , of the anisotropic global attractor A ε to the homogenised attractor A in the spaces E − and (C β (Ω)) . In Section 4, we derive the central (order-sharp) estimate (0.5) on the di erence between trajectories u ε (t) and u (t) of the corresponding anisotropic and homogenised problems. Then, based on this, we demonstrate the quantitative estimates (0.4) on the distance between global attractors A ε and A . Estimate (0.6) between the global attractor A ε and rst-order correction TεA in the energy space E is proved in Section 5. Section 6 is devoted to exponential attractors M ε , M associated with problems (0.1), (0.2) and consists of two parts. In Subsection 6.1, existence of the exponential attractors is proved and estimates (0.8) in E − and (C β (Ω)) are obtained. The results in this section rely on a variant of a standard abstract result on the construction of exponential attractors; this construction is included in Appendix B. In Subsection 6.2, we compare the distance between the exponential attractor M ε and the rst-order correction TεM in the energy space E. Subsection 6.2 rests on a new abstract theorem, presented in Appendix C, which compares the distance between exponential attractors which admit correction. We discuss, and prove the corresponding results for the cases of Neumann and periodic boundary conditions in Section 7. Some re nements of the

Notations
We document here notations frequently used throughout the article. The L (Ω) inner product is given by (u, v) := Ω u(x)v(x) dx, with norm denoted by u := (u, u) / for u, v ∈ L (Ω). We frequently consider initial data in the energy spaces E − := L (Ω) × H − (Ω), and E := H (Ω) × L (Ω). These spaces are equipped with norms whose squares are given as ξ E − := ξ + ξ H − (Ω) and ξ E := ∇ξ + ξ for admissible pairs² ξ = (ξ , ξ ). For any function z(t) we set ξz(t) to be the pair (z(t), ∂ t z(t)) where ∂ t z denotes the distributional (time) derivative. For a Banach space E, B E ( , r) denotes the ball centered at of radius r in E; the symbol [ · ] E denotes the closure in E; the one-sided and symmetric Hausdor distances between two sets A, B ⊂ E are respectively de ned as dist E (A, A) . The standard Euclidean basis is denoted by {e k } k= .

Preliminaries
Throughout the article, unless stated otherwise, we adopt the convention that M and K denote generic constants whose precise value may vary from line to line. For a given matrix a(·) = {a ij (·)} i,j= we denote by a h = {a h ij } i,j= the homogenised matrix corresponding to a(·) whose constant coe cients are given by the formula Here N i , i ∈ { , , }, is the solution to the so-called cell problem:    − divy a(y)∇y N i (y) = divy a(y)e i , y ∈ Q = [ , ) , (1.1) It is well-known that if a(·) is symmetric, bounded and uniformly elliptic, then so is a h with the exact same bounds (see for example [15,Section 1]). Furthermore, as a h is constant it is clearly periodic. Consequently, both problem (0.1) and (0.2) are problems of the form with the same generic assumptions on coe cients, forces and non-linearity; we collect these assumptions together here: and f ∈ C (R) satisfying 2 Here we adopt the common clash of notation for (·, ·) to mean both an inner product and represent a pair in a product space. It will be clear from the context which meaning is appropriate.
s ∈ R. Also note that the assumption f ( ) = is, in fact, not a restriction since f ( ) always can be included into the forcing term g.
We begin with some basic existence, continuity and dissipative estimate results. Particular attention is paid to the dependence of these results on the matrix a, assuming that the other variables (Ω and f ) are xed. As these results are standard we shall omit the proofs, commenting here that they are easily argued by the techniques employed in Appendix A. Theorem 1.1. Assume (H1). Then, for any initial data ξ ∈ E, problem (1.2) possesses a unique energy solution u with ξu ∈ C(R+; E). Moreover, the following dissipative estimate is valid: .
We now proceed to study the long-time behaviour of solutions u from the point of view of in nite-dimensional dynamical systems. In particular the problem (1.2) de nes a dynamical system (E, S(t)) by where u is a solution to the problem (1.2) with initial data ξ . The limit behaviour of a dissipative dynamical system as time goes to +∞ can be described in terms of a so-called global attractor. Let us brie y recall its de nition (see [2,3,5,7] One can show that if a global attractor exists then it is unique. Also, the following description of the global attractor in terms of bounded trajectories is known (see e. g. [2,3]): (1.6) To prove that a global attractor exists for problem (1.2) we utilise the following classical result ( [2,3,5,7] holds for some positive constants R, M and β that depend only on ν.
The proof of Theorem 1.3 is presented for the reader's convenience in Appendix A and is based on a splitting of trajectory u, into the smooth and contractive parts, that was developed in [31]. Consequently, as E is compact in E we see from Theorem 1.3 that K = B E ( , R) is a compact attracting set and, by Theorem 1.2, there exists a global attractor. That is, the following result holds. Theorem 1.4. Assume (H1). Then, the dynamical system (E, S(t)) given by (1.4

) possesses a global attractor
A ⊂ E such that:

Smoothness of the global attractor
Above we demonstrated that the global attractor A is a bounded subset of E . We shall now establish some additional regularity of A. These results will be used later on to derive homogenisation error estimates.
We are going to show that A is contained in the more regular set E := ξ ∈ E | div(a∇ξ ) + g ∈ H (Ω) and div(a∇ξ ) ∈ L (Ω) , ξ E := div(a∇ξ ) + g H (Ω) + div(a∇ξ ) + div(a∇ξ ) , and that A is bounded in the following sense: To this end, we shall show that B E ( , R) is exponentially attracted, in E, to some 'ball' ⁴ Then by utilising the so-called transitivity property of exponential attraction we establish that B (from (1.6)) is attracted to B E ( , R ) exponentially in E and, therefore, we will show that A is bounded in E . Let us begin with the following theorem which provides a useful dissipative estimate for problem (0.2) with initial data in E (see (1.7)).
Theorem 2.1. Assume (H1). Then for any initial data ξ ∈ E the energy solution u to problem (1.2) is such that ξu ∈ L ∞ (R+; E ), and the following dissipative estimate is valid: for some non-decreasing function M and constant β > that depend only on ν.
Since this result is standard we omit the proof. We only remark here that, by di erentiating the rst equation of (1.2) in time, one rst obtains a dissipative estimate for ξ ∂t u (t) E which readily implies the uniform bound on div(a∇u)(t) .
Remark 2.1. Note that by elliptic regularity we have the inequality for su ciently small α = α(ν) and admissible u. Here C α (Ω) is the Hölder space of order α: Thus, we have a dissipative estimate for u, given by Theorem 2.1, in the C α (Ω) norm.
for some non-decreasing function M that depends only on ν.
Proof. By di erentiating the rst equation of (2.3) in time and by our choice of initial data (G, ) we nd that p := ∂ t w solves (2.5) By the dissipative estimate in E (cf. Theorem 2.1 and Remark 2.1) we nd that This inequality and the conditions on the non-linearity f (see (H1)) imply that Therefore, using the dissipative estimate in E ((1.3)) we conclude Returning back to p = ∂ t w, we rewrite (2.5) to nd div(a∇∂ t w)(t) = − G (t) + γ∂ t p(t) + ∂ t q(t) ≤ M( g ), t ≥ .
Rewriting the rst equation in (2.3), and using cubic growth of f (see Remark 1.1.a) gives Hence, the desired result holds and the proof is complete.
Combining (2.4) and Lemma 2.1 produces the following result.
for some constants M , K > . Furthermore, assume that there exist subsets E ⊂ E and E ⊂ E such that for some M , M , β > and β > . Then Note that Theorem 1.1 (in particular (1.6)), Theorem 1.3 and Corollary 2.1 imply that the assumptions of the above theorem hold for holds for some non-decreasing M and β > that depend only on ν.
We are now ready to prove that the global attractor is bounded in E .

Theorem 2.4. Assume (H1) and let A be the global attractor of the dynamical system (E, S(t)) given by (1.4).
Then for some non-decreasing M that depends only on ν.

Remark 2.2. Note that (2.6) implies the following estimate
We end this section with one more result which will be useful later.
Theorem 2.5. Assume (H1). Then, for any initial data ξ ∈ E , the energy solution u to problem (1.2) is such that ξu ∈ L ∞ (R+; E ) and the following dissipative estimate is valid: The proof is very close to the proof of Lemma 2.1 and for this reason is omitted. We only remark that, since E ⊂ E and the dissipative estimate in E is already known, we see that the quantity u(t) L ∞ (Ω) is bounded. Thus, basically, one applies linear dissipative estimates to the equations for p and q in the proof of Lemma 2.1 with the appropriately changed initial data.

Homogenisation and convergence of global attractors
Let us now consider the dynamical systems Sε(t) and S (t) generated by problems (0.1) and (0.2) respectively. In Theorem 2.4 we established that Sε (respect. S ) has a global attractor A ε (respect. A ). Moreover, Theorem 2.4 informs us that A ε is a, uniformly in ε, bounded subset of E ε and A is a bounded subset of E , where

Additionally for A , as a h is constant, we can readily deduce that A is a bounded subset of E ∩ (H (Ω)) . That is, the inequalities
hold for some non-decreasing function M independent of ε.
The main result of this section is the following theorem which establishes convergence of the global attractors A ε to the global attractor A in the one-sided Hausdor distance.
for any ≤ β < α where α is given in Remark 3.1.
To prove Theorem 3.1 we shall use the following classical homogenisation theorem for elliptic PDEs (see for example [15,Section 1] . To prove the result it is su cient to show that there exists ξ ∈ A such that ξn converges, up to some subsequence, to ξ in (C β (Ω)) as n → ∞. For each n ∈ N, we denote by un ∈ K εn the bounded (for all time) in E solution of (0.1) that satis es It remains to prove that ξ ∈ A , and this is established if we demonstrate that ξ = ξu ( ) for some bounded (for all time) in E solution u to (0.2). The remainder of the proof is to establish the existence of such a u . In what follows convergence is meant up to an appropriately discarded subsequence.
By Remark 3.1 and the strict invariance of A ε (property 2 of De nition (1.1)) there exists M > such that Furthermore, di erentiating (3.5) in t gives This equation, along with (3.4), the boundedness of ∂ t un in W and growth assumption on f imply that Therefore, since the embeddings H (Ω) ⊂ L (Ω) and L (Ω) ⊂ H − (Ω) are compact, by Aubin-Lions lemma we deduce that Due to (3.6) we know that Brought to you by | University of Durham Authenticated Download Date | 9/18/19 12:35 PM Therefore, by an application of the homogenisation theorem (Theorem 3.2), we conclude, that for every t ∈ [z, z + ], un(t) weakly converges in H (Ω) to the solution u (t) of the homogenised problem It follows from (3.6) and the weak convergence un(t) Consequently, from this identity and the above equation, we see that u (weakly) solves Let us argue that the above equation holds for all time. Indeed, by a Cantor diagonalisation argument we see that the convergences (3.6) can be taken to hold for all z ∈ Z. Then, by noting that any ϕ ∈ C ∞ (R; C ∞ (Ω)) can be represented as a nite sum of smooth functions whose individual supports (w.r.t to time) are in some [z, z + ], we deduce that u weakly solves the homogenised equation (0.2). Hence, u is a bounded in E solution to (0.2) for all time.

Rate of convergence to the homogenised global attractor
We shall begin with recalling an important result on error estimates in homogenisation theory of elliptic PDEs. Recall, for xed ε > , the mappings satisfying uniform ellipticity and boundedness assumptions, Aε and A given by (4.1) and g ∈ L (Ω). Let also u ε , u ∈ H (Ω) solve the problems Then, the following estimate u ε − u ≤ Cε g , (4.2) holds for some constant C = C(ν, Ω).

Remark 4.1. Note that inequality (4.2) is equivalent to the following operator estimate on resolvents:
In what follows we wish to compare properties of the semi-groups associated to (0.1) and (0.2) via estimates in terms of ε. In fact, we shall provide stronger estimates in terms of the di erence A − ε − A − L(L (Ω)) . The mentioned ε estimates then immediately follow by Remark 4.1.
Our rst important result is the following continuity estimate. Let us x ξ , set ξ u ε (t) := Sε(t)ξ , ξ u (t) := S (t)ξ , and de ne r ε := u ε − u . Then, r ε solves (4.4) By testing the rst equation in (4.4) with A − ∂ t r ε we deduce that We compute Therefore, we can rewrite (4.5) as We now aim to bound the right-hand-side of (4.6) in terms of A − ε − A − L(L (Ω)) and Λ, then subsequently apply Gronwall's inequality and the following standard estimate to deduce the desired result.
To this end, let us rst estimate the non-linear term. Using the growth restriction on f (see Remark 1.1a) and Hölder's inequality (for exponents (p , p , p ) = ( , , )) we compute Then, by the Sobolev embedding L (Ω) ⊂ H (Ω), the fact that u ε and u are bounded in E (see dissipative estimate (1.3)) and (4.7) we compute for some positive M . By utilising the above inequality in (4.6) we infer that Now, by the dissipative estimate in E ε (Theorem 2.5) we have the following uniform bounds in t and ε: By collecting the above inequalities together we deduce that Consequently, by applying Gronwall's inequality and the initial data ξ r ε | t= = we have Hence, the above two inequalities along with (4.7) and (4.9) demonstrate (4.3) and the proof is complete.
Along with Theorem 4.2, to prove error estimates on the distance between global attractors we need the following exponential attraction property of A : holds for some non-decreasing function M.

(H2)
It is known that, for problem (0.2), the property (H2) is a generic assumption in the sense that it holds for an open dense subset of forces g ∈ L (Ω) (cf. [2]).
We are now ready to formulate and prove our main result of this section. and (E, S (t)) corresponding to the problems (0.1) and (0.2). Then the following estimate , (4.10) holds. Here, K is as in Theorem 4.2, σ as in (H2), and M = M( g ) is a non-decreasing function independent of ε.
Proof. The assertion follows from the already obtained estimate (4.3) and the exponential attraction property (H2). Indeed, let ξε ∈ A ε ⊂ B E ε ( , R ) be arbitrary. Then due to (2.6) there exists a complete bounded trajectory ξ u ε (t) ∈ K ε , such that ξ u ε ( ) = ξε. Let us x an arbitrary T ≥ and consider for some M and K which are independent of ε and ξε ∈ A ε . On the other hand, due to exponential attraction Therefore, using the triangle inequality, we derive We recall that T ≥ is arbitrary and therefore we choose T that minimizes the right hand side of (4.11). For example, taking T = T(ε) such that κe KT = e −σT yields , and since ξε ∈ A ε is arbitrary we obtain the desired inequality (4.10).
Proof. The corollary follows directly from the uniform boundedness of A ε and A in C α (Ω) (Remark 3.1), the estimate on the distance between attractors in E − (cf. (4.10)) and the interpolation inequalities

Approximation of global attractors with error estimates in the energy space E
In addition to the obtained estimates in Section 4 on the distance in E − we would like to obtain estimates in the energy space E. Note that we can not expect, in general, convergence of the global attractors in the strong topology of E, cf. Remark 3.
Consequently, the following inequality holds for some C > independent of ε and w. Indeed, this follows from the above multiplier estimate and the fact N i ∈ L ∞ (Q) (by elliptic regularity). Now, we are ready to present the well-known corrector estimate result in elliptic homogenisation theory which improves the L -estimate given in Theorem 4.1 to H -norm. Then, the following estimate As in Theorem 4.3, we would like to compare the distance between Sε(t)ξ , for ξ ∈ E ε , to some trajectory for S but this time in the energy space E. However, here the trajectory S (t)ξ is not a suitable candidate as it does not have the su cient regularity needed to apply the above corrector estimates. To overcome this di culty we carefully choose our initial data for the homogenised problem (0.2). More precisely, let us recall the spaces E ε , E given in (3.1), (3.2), and introduce the bounded linear operator Πε : E ε → E given by The operator Πε has the following nice properties.
Lemma 5.1. The operator Πε : E ε → E is a bijection that satis es: Proof. The bijective property and equality (5.5) directly follow from the de nitions of E ε , E and the identity Πε(ξ , ξ ) = (A − Aε ξ , A − Aε ξ ). Inequality (5.6) follows from the identity We now compare Sε(t)ξ with S (t)Πε ξ in E for ξ ∈ E ε . The following result is the direct analogue of Theorem 4.2 when one replaces the initial data ξ by Πε ξ in problem (0.2).
Set ξ u ε (t) := Sε(t)ξ , ξ u (t) := S (t)Πε ξ . We begin by noting the following uniform bounds in t and ε: Indeed, these bounds are a consequence of identity Πε B E ε ( , R) = B E ( , R) and the dissipative estimates for u ε and u in E ε and E respectively (Theorem 2.5 for a = a( · ε ) and a = a h respectively). Now, the di erence r ε := u ε − u solves Upon handling the non-linearity as in (4.8), and utilising Lemma 5.1 we conclude that Now, by di erentiating the rst equation in (5.10) in time (and then adding A ∂ t r ε to both sides) we nd that q ε := ∂ t r ε solves Testing the rst equation in the above problem with A − ∂ t q ε gives We aim to prove the inequality for some M and K independent of ε and ξ , which subsequently implies the desired result via an application of Gronwall's inequality and (5.11). As usual, we shall utilise the H − -norm equivalence given by (4.7). So it remains to prove (5.12). By arguing as in Theorem 4.2, we utilise the identity ∂ t q ε = ∂ t u ε − ∂ t u and uniform bounds (5.9) to compute Let us now handle the non-linear term. We compute The arguments to bound I and I will use the uniform bounds on u ε and u given by (5.9). By the growth condition on f and the H − -norm equivalence (4.7), we compute Additionally, by Hölder's inequality (for exponents (p , p , p , p ) = ( , , , )) we compute The above assertion and (5.7) imply Combining the above calculations leads to the inequality (5.12). The proof is complete.
The following estimate is an immediate consequence of Theorem 5.2 and standard elliptic theory.
Proof. Note that u ε ∈ H (Ω) satis es the equation and u ∈ H (Ω) satis es Since ξ ∈ B E ε ( , R) then by ( Then, by Theorem 5.1 we have Therefore, by the triangle inequality, we have The above inequality along with (5.15) imply the desired result and the proof is complete.
Let us now provide estimates on the distance in the energy space. As in Corollary 5.1 this requires adding an appropriate correction to the attractor A . To this end, we introduce the corrector Tε : E → L (Ω) which maps the pair ξ = (ξ , ξ ) to the pair Tε ξ = (Tε ξ , ξ ). (5.16) By (5.2), we readily deduce the following inequality: there exists a constant C > , independent of ε, such that the inequality holds. By inequality (5.8) and Corollary 5.1 we have shown the following result. The following estimate on the global attractors in E holds.

Theorem 5.3. Assume (H1) and (H2). Let A ε and
A the global attractors of problems (0.1) and (0.2) respectively, and let Tε be given by (5.16). Then, the following estimate holds for some M = M( g ) which is independent of ε. Here κ is as in Theorem 4.3.
Proof. The method of proof follows along the same lines as the argument for Theorem 4.3 and so we shall only sketch it here. For ξε ∈ A ε and T > , consider ξ −T,ε ∈ A ε that satis es Sε(T)ξ −T,ε = ξ . Then, by Corollary 5.2 we have Furthermore, by (5.17) we have Now, to control the second term on the above right we use the fact that ΠεA ε and A are bounded subsets of E (see Remark 3.1 and inequality (5.5)) and that we have a dissipative estimate for S (t) on E (see Theorem 2.5). Consequently, we compute and the remainder of the proof utilises the exponential attraction property of A , as in Theorem 4.3.

Remark 5.2.
1. The appearance of √ ε in (5.3) is a well-known consequence of the fact that the correction Tε u does not approximate well the function u ε in a ε-neighbourhood of the boundary. In particular, the reduced power of ε appears in the estimate due to the fact that Tε u does not satisfy the Dirichlet boundary conditions and a 'boundary correction' is needed. In general, the explicit ε-dependence (i.e. leading-order asymptotics) of this boundary correction is not known.

Exponential attractors: existence, homogenisation and convergence rates
Let us recall the de nition of an exponential attractor for a dynamical system.
for some non-decreasing M and constant σ > .

. Existence of exponential attractors and continuity in E −
Let us present our main result for this subsection.
Here α is the same as in Remark 2.1 and the constants M > , σ > , < κ < and D ≥ are independent of ε.
The remainder of the section is dedicated to the proof of Theorem 6.1. First, we recall a variation of an abstract result which establishes the existence of an exponential attractor M ε , for a parameter-dependent family of semi-groups Sε, whose characteristics are independent of ε (see Appendix B, [ r Uε( K , ) where the set Bε ⊂ B E ε ( , R) is closed in E. Furthermore, we assume Sε satis es the following properties: 1. for every ξ and ξ from O(Bε), the di erence Sε ξ − Sε ξ can be represented in the form: for constants c − and L > .
The exponent of attraction σ > is independent of ε ≥ and dim f (M ε , E) ≤ D for some positive D independent of ε (see De nition 6.1). Moreover where the constants C > and κ = κ(c , L, K, δ ) are independent of ε.
The proof of Theorem 6.2 is postponed to Appendix B. We now move on to the proof of Theorem 6.1. As in the usual way, we rst construct exponential attractors for the discrete dynamical systems with maps Sε := Sε(T), S := S (T), for large enough T > . Then by a standard procedure, clari ed below, one arrives at exponential attractors for the continuous dynamical systems (E, Sε(t)), t ≥ .
Proof of Theorem 6.1.
Proof of (ii). We shall provide an explicit construction for the covers. Moreover, it will be important later that we produce a cover such that for some Cr > independent of ε ≥ . For this reason we seek a cover of B E ε ( , r) in the form To ensure ξ iε are in E ε we see that (p i , q iε ) should belong to (H (Ω)) with Aε q iε ∈ L (Ω).  Additionally, due to density arguments, we can suppose Moreover, as the eigenfunctions of Aε form an orthonormal basis for L (Ω) we can nd q iε such that Aε q iε ∈ L (Ω) and N(μ, r).
Therefore, we have the covering Now, for xed ξ ∈ B E ε ( , r) we readily deduce from the ellipticity of a that Furthermore, it is clear that (Aε ξ − g, ξ ) ∈ B L (Ω)×H (Ω) (−g, ), r . Consequently, one can readily check that Additionally, since q iε are obtained by truncating q i with respect to the eigenfunctions of Aε, we compute and so we deduce that Hence, upon settingμ = ( ∨ν − ) µ, we see that the centers satisfy (ii) for δr = ( ∨ ν − )r. Also the additional desired properties (6.3) hold.

Construction of Bε and Sε.
We set Bε := B E ε ( , R ) to be the absorbing ball provided by Theorem 2.5 for E = E ε , and a(·) = a · ε in the case ε > and a(·) ≡ a h for ε = . The radius R is independent of ε and clearly Bε is closed in E.
Since Bε is an absorbing set in E ε and, by (6.3), O(Bε) is a subset of E ε , we can choose T large enough (and independent of ε) such that Sε := Sε(T), ε ≥ , satis es Let us verify properties (1) and (2) ξw| t= = ( , ), w| ∂Ω = , (6.7) Using the fact that our initial data is from E ε we conclude that u i , ∂ t u i are bounded in L ∞ (Ω) uniformly in ε. Then upon testing the rst equation in (6.7) with ∂ t p, rewriting the subsequent right-hand-side in the form we obtain via standard arguments, and the Lipschitz continuity of Sε(t) in E (Corollary 1.1), the uniform esti- Consequently, we use p = ∂ t w and (6.6) to conclude for some positive constants M and K independent of ε and ξ i . Therefore, for T = max{T , T }, property (1) holds.
Proof of (2). This property is given by Corollary 1.2 for a ≡ a h . Hence, the assumptions of Theorem 6.2 hold and therefore Theorem 6.1 holds for the discrete dynamical systems (Bε , Sε(T)) with discrete exponential attractors M ε d . Indeed, Theorem 6.1 (1)-(4) hold due to the choice of Bε and Uε, and (5) follows from (6.2), (6.3), Theorem 4.2, Lemma 5.1 and the fact that the map Πε : Bε → B is a bijection.
Step 2: Discrete to continuous dynamics. From the discrete exponential attractors M ε d we can build exponential attractors M ε for the original dynamical systems (E, Sε(t)) by the following standard construction ( [5]): Indeed, the properties (1)-(4) can be easily veri ed due to dissipative estimate in E ε , Lipschitz continuity with respect to initial data in E (Corollary 1.1) on the bounded set Bε: and Lipschitz continuity with respect to time: for some constant M > (independent of ε). Indeed, the continuity in time follows from the uniform boundedness of Bε in the space E ε . It remains to check the continuity property (5) for the exponential attractors M ε . This readily follows from the fact that (5) holds for the discrete exponential attractors M ε d , Theorem 4.2 and the following computation:

. Continuity of exponential attractors in E.
Theorem 6.1 (5) demonstrates Hölder continuity between the exponential attractors M ε and M in the space E − . In this section we provide continuity results in the energy space E. Unlike in E − , in the stronger topology of E this requires a correction (such as in De nition 5.16) of the exponential attractor M . More precisely, the main result of this section is the following theorem. Theorem 6.3. Assume (H1) and let M ε , M be the exponential attractors constructed in Theorem 6.1. Then, the following estimate is valid: , ε > , (6.9) where the 'correction' operator Tε is given by (5.16), < κ < as in Theorem 6.1 and the constant M > is independent of ε.
To prove this result, we make an important development of Theorem 6.2 to provide estimates between exponential attractors which admit correction. That is we establish the following new result. for some constant Lcor > independent of ε and positive function m(·) with m( + ) = . 5. the maps Sε are uniformly Lipschitz continuous in E with respect to ε > , that is with some constant L > independent of ε > .
Then the following estimate holds for constant C > independent of ε and κ as in Theorem 6.2.
The proof of this result is presented in Appendix C.
We rst establish, based on the abstract result Theorem 6.4, the estimate (6.9) for the discrete exponential attractors M ε d (de ned in the proof of Theorem 6.1). That is we prove the following inequality: Let us check that the assumptions of Theorem 6.4 hold. Indeed, assumption (3) follows from the fact that Bε = B E ε ( , R ) (see the proof of Theorem 6.1) and De nition 5.4 of the projector Πε (where we note that Πε can be trivially extended to the map from E ε onto E , preserving the bijection property). Assumption (4) holds with m(ε) = Cε (for some constant C > , independent of ε) due to the multiplier estimate (5.17) and the fact that O(B ) is a bounded subset of E by construction. Assumption (5) is a consequence of Corollary 1.1. Hence the assumptions of Theorem 6.4 hold and (6.10) holds for the discrete exponential attractors M ε d and M d .
Let us now estimate the terms on the right-hand side of (6.10) in terms of ε. Since Πε : E ε → E is bijective and preserves the norm (Lemma 5.1), and since Therefore, this observation and Corollary 5.2 imply that for some M > independent of ε > . Also from the identity and Remark 5.1 we deduce that for some constant M > independent of ε > . It remains to compare the distance between the covers present in the right-hand side of (6.10). To this end, we notice that if ξ iε := (A − ε (p i + g), q iε ) ∈ Uε(µ, r), then Consequently, due to Remark 5.1 and the properties of q iε (see (6.4)) one can see that dist s E (Uε(µ, r), Tε U (µ, r)) ≤ Cr √ ε, (6.16) for some constant Cr > independent of ε, µ. Upon collecting the above estimates we derive (6.11). It remains to establish (6.9) for the exponential attractors M ε . It is su cient to show that Indeed, since κ < , the above inequality, (6.11), (6.14) and Corollary 5.2 implies (6.9). Let us demonstrate (6.17): Hence the theorem is proved.

The case of di erent boundary conditions
In this section we are going to show that the analogues of the obtained homogenisation error estimates for the global and exponential attractors still hold if we change the Dirichlet boundary conditions to be either Neumann or periodic.
Let Ω ⊂ R be a smooth bounded domain and H := H (Ω) or Ω be a three-dimensional torus T := [ , ) , > , with In both cases we endow H with the norm For the maps Aε be given by (4.1), ε ≥ , we consider the problem boundary conditions. It is well-known that problem (7.1) with either boundary conditions (N) or (P) is well-posed in the energy space E := H × L (Ω) and, therefore, de nes a dynamical system (E, Sε(t)) where for u ε (t) the unique solution of the corresponding problem with initial data ξ . Moreover, is well-known that Aε + : for condition (N) or D(Aε + ) = {u ∈ H | Aε u ∈ L (Ω), ∇u(x + e k ) = ∇u(x), k ∈ { , , }}, ε ≥ , for condition (P). Setting it is straightforward to see from Appendix A and Sections 2-6 that the following theorem holds.
Theorem 7.1. Assume (H1). Then, for every ε ≥ , the dynamical systems (E, Sε(t)) generated by problem (7.1) with boundary conditions (N) or (P) possesses a global attractor A ε , and exponential attractor M ε , of nite fractal dimension such that: Let us now discuss error estimates between the anisotropic and homogenised attractors. It is known that the main homogenisation results, Theorems 4.1 and 5.1, remain valid for the case of Neumann and periodic boundary conditions.

Theorem 7.2 ([16]).
Let Ω ⊂ R be a bounded smooth domain or three-dimensional torus T , ε > , periodic matrix a(·) satisfying uniform ellipticity and boundedness assumptions, Aε and A given by (4.1) and g ∈ L (Ω). Let also u ε ∈ D(Aε + ), u ∈ D(A + ), solve the equations Then, the following estimates where (H ) * stands for the dual space of H . We now draw the reader's attention to the fact that the key theorems (Theorems 4.2 and 5.2) on the distance between trajectories in E − are in terms of resolvents of the operator Aε, ε ≥ . The key point to note is that the proofs of these results essentially rely on the fact Aε is self-adjoint and (uniformly in ε) bounded and positive. Since the operator Aε + , for Neumann (N) or periodic (P) boundary conditions, also possesses these properties one can see that analogues of Theorems 4.2-5.2 readily hold (after appropriately changing the projector Πε). Namely, upon de ning Πε : E ε → E , for E ε given by (7.2), as follows Πε(ξ , ξ ) := (ξ , ξ ), where the term ξ i ∈ D(A + ), i = , , satis es we have the following result. Theorem 7.3. Let E ε be given by (7.2) and Sε(t) be the solution operator to the problem (7.1) with Neumann (N) or periodic (P) boundary conditions. Then, for all ξ ∈ E ε , ξ E ε ≤ R, R > , the inequalities hold for some non-decreasing functions M = M(R, g ) and K = K(R, g ) which are independent of ε > .

A Proof of Theorem 1.3
To prove Theorem 1.3 we perform a splitting of the solution u = v + w to the problem (1.2) into asymptotically contractive and compact parts. This form of splitting was intoduced in [31].
Let us consider where the xed constant L > is speci ed below.
Recall that B denotes a positive invariant absorbing set of the semigroup (E, S(t)) (see (1.6)). Similar to Theorem 1.1 we have the following result.
holds for some non-decreasing function M L that depends only on ν and L.
Lemma A.3 (Modi ed Gronwall's Lemma [31]). Let Λ : R + → R + be an absolutely continuous function satis- where µ > , k ≥ and t s h(τ) dτ ≤ µ(t − s) + m, for all t ≥ s ≥ and some m ≥ . Then We are now ready to show that v exponentially goes to in the energy space E. Proof. Fix κ > to be speci ed below. Multiplying equation (A.1) by ∂ t v + κv in L (Ω) we nd (after some algebraic manipulation) that Now by the lower bound on f (see (H1)) we compute Thus, for L > K , (A.6) implies We shall establish below, for su ciently large L, the equivalence as well as the inequalities From (A.9), and the fact ξv( ) = ξu( ), we prove the desired result. Therefore, to complete the proof it remains to establish (A.9)-(A.11). Let us prove (A.9). We shall prove the upper bound, as the argument for the lower bound is similar. For κ ∈ ( , γ/ ), utilising the dissipative estimate for u (1.3) and the bounds on f (see (H1) and Remark 1.
Then for large enough L, we deduce Λ ≥ ∂ t v + (a∇v, ∇v) and the upper bound in (A.9) holds. To prove (A.10) and (A.11), we use dissipative bounds on u and w (Lemma A.1) plus the growth assumption on f to establish Then the desired inequalities follow by invoking the ellipticity of a and the now established (A.9). The proof is complete.
To complete the proof of Theorem 1.3 it remains to prove that ξw is a bounded trajectory in E , this is the subject of the next result.
holds for some non-decreasing function M L that depends only on ν and L.
The identity (A.12) can be rewritten in the form Arguing in a similar manner as in the proof of (A.9) we have Cν ξq E ≤ Λ (A.14) for some Cν, as long as L = L(γ, ν, f ) is large enough. Using the growth condition of f (see (H1)), the dissipative estimate for u (1.3), energy estimate for w (Lemma A.1) and arguing as in the proof of (A.10), the right-hand side H(t) can be estimated as follows: for any δ > . Choosing < κ < γ , δ small, and collecting (A.13), (A.14), (A.15) we derive Consequently, using Lemma A.2 (with small enough µ) and applying the modi ed Gronwall's lemma we determine that It now readily follows that div(a∇w) ≤ M L ( B E ), t ≥ .

B Proof of Theorem 6.2
The proof of Theorem 6.2 is an adaptation of a construction for exponential attractors presented in [29, Theorem 2.10]. The di erence here is one needs to keep track on the parameter dependence of all the sets used in the construction and incorporate the fact we compare the symmetric distance in a topology di erent to that in which the exponential attractors are constructed. For the reader's convenience we shall provide the details here.

B. Construction of the exponential attractors.
Let us introduce notations for the 'starting' cover Uε( K , R) and the 'model' cover Uε( K , ): where N := cardV (ε) = N( K , R) and N := N( K , ) are, by assumption, independent of ε ≥ . We shall begin with constructing a family of sets V k (ε), k ∈ N, that satisfy⁵ Note that, by the assumptions of Theorem 6.2, the above property holds for k = . We now assume that the set V k (ε) exists, for some xed k, and are going to construct from it the set V k+ (ε). From (B.1) it follows that Let us consider an element Sε ζ ∈ Sε B E (ξ , K k ) for some ξ ∈ V k (ε). Due to the splitting (6.1) we have Therefore, by using the model cover U(ε) of B E ε ( , ), we see that Since Sε ζ = Sε ξ + vε + wε we deduce that As ξ i,ε E ε ≤ δ we conclude that (B.1) holds for Now, it is straightforward to verify the following properties of V k (ε): Based on the sets V k (ε) we construct the sets E k (ε) ⊂ O(Bε): We shall now demonstrate that the sets are exponential attractors for the discrete dynamical systems (Bε , Sε). To this end we use the following result.
The proof of this lemma, basically, repeats the proof of Lemma 2.3 from [29], so we omit the proof. Now, we are ready to verify that the constructed sets M ε satisfy De nition 6.1. The positive invariance and the uniform exponential attraction property (with σ = ln ) holds for some constant M = M(L) independent of ε and k.
Step 1. We rst establish (B.10) for the sets V k (ε), V k ( ). To this end it is convenient to introduce the It is su cient to establish that the following recurrent chain of inequalities Indeed, upon iterating these inequalities one nds Let us prove (B.11). Note that, from the construction of V k (ε) (B.2), we readily have the following inequal- Fixing arbitrary a ∈ A, c ∈ C and using Lipschitz continuity of S in E − we obtain Consequently (B.14) holds. Hence, upon combining (B.13) with (B.14), we deduce (B.11) and step 1 is complete.
Step 2. We claim that the sets E k (ε), E k ( ) satisfy the same inequality as in (B.12), namely Proof. We follow the strategy of Lemma B.2 and x ε ≥ .
We rst derive an estimate on the distance between V k (ε) and TεV k ( ). Let us introduce the notations d k := dist s E V k (ε), TεV k ( ) , k ∈ Z+,d := dist s E U(ε), Tε U( ) ; s := sup We are going to verify the recurrent chain of inequalities d k+ ≤ s +d + Ld k , k ∈ Z+. To derive the estimate (C.1) on the distance dist s E E k (ε), Tε E k ( ) we simply argue as in Step 2 of Lemma B.2.
We are ready to prove the theorem. We x ε ≥ and set As in the proof of Theorem 6.2 we will only consider dist E (M ε , TεM ) as the other side can argued in a similar manner. Let k ∈ N and ξε ∈ E k (ε) be xed. Then according to Lemma C. 1  In order to further improve (5.8) (or rather (D.1)), and achieve the optimal bound with power one, we intend to argue as in the proof of Theorem 4.2. For this reason, we require additional regularity on the initial data ξ .
In particular, we shall show that it is su cient for ξ ∈ E ε to be such that the solution u ε to (0.1) (with initial data ξ ) satis es Aε ∂ t u ε ≤ M, t ≥ .
Then, we shall demonstrate that this additional regularity is 'natural' in the sense that the global attractor A ε possesses such smoothness under the additional mild assumption on the non-linearity f : Let us introduce the mapping Au := − div(a∇u), recall E = ξ ∈ (H (Ω)) | Aξ − g ∈ H (Ω) and Aξ ∈ L (Ω) , and introduce E := ξ ∈ E | A Aξ + f (ξ ) − g ∈ L (Ω) and Aξ ∈ H (Ω) , Our rst result is that a dissipative estimate holds in E .