Inertial Manifolds for the 3D Cahn-Hilliard Equations with Periodic Boundary Conditions

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.


Introduction
It is believed that the dynamics generated by dissipative PDEs in bounded domains is typically finite-dimensional. The latter means that despite the infinite-dimensionality of the initial phase space, the limit dynamics, say, on the so-called global attractor can be effectively described by finitely many parameters which satisfy a system of ODEs -the so-called inertial form of the dissipative PDEs considered, see [1,4,16,18,22] and references therein. This reduction clearly works when the underlying PDE possesses an inertial manifold (IM) that is a finite-dimensional invariant C 1 -smooth manifold with exponential tracking property. Then the desired inertial form can be constructed just by restricting the considered PDE to the invariant manifold, see [9,15,19]. However, the existence of an inertial manifold requires rather strong spectral gap assumptions which are usually satisfied only for the parabolic equations in the space dimension one and, despite a big permanent interest and many results obtained in this direction, the finitedimensional reduction for the case where the IM does not exist remains unclear and there are even some evidence that the dissipative dynamics may be infinite-dimensional in this case, see [6,14,23] and reference therein.
It is also known that the above mentioned spectral gap assumptions are sharp and cannot be relaxed/removed at least on the level of the abstract functional models associated with the considered PDE, see [6,15,19]. However, an IM may exist for some concrete classes of PDEs even when the spectral gap condition is violated. The most famous example is a scalar reactiondiffusion equation on a 3D torus x ∈ [−π, π] 3 . Here the spectral gap condition reads where λ 1 ≤ λ 2 ≤ · · · are the eigenvalues of the minus Laplacian on a torus enumerated in the non-decreasing order and L is a Lipschitz constant of the nonlinearity f . The eigenvalues of the Laplacian are all natural numbers which can be presented as sums of 3 squares and by the Gauss theorem, there are no gaps of length more than 3 in the spectrum, so the spectral gap condition clearly fails if the Lipschitz constant L is large enough. Nevertheless, the corresponding IM can be constructed (for all values of the Lipschitz constant L) using the so-called spatial averaging principle introduced in [13]. One more example is the 1D reaction-diffusion-advection problem x ∈ [0, π], u x=0 = u u=π = 0, where the spectral gap condition is also not satisfied initially, but is satisfied after the proper change of the dependent variable u, see [24] and also [20,21] where the Lipschitz continuous inertial form is constructed.
Although the spatial averaging principle has been used to get the IM for reaction-diffusion equations in some non-toroidal domains, see [12], to the best of our knowledge, it has been never applied before to the equations different from the scalar reaction-diffusion ones. The aim of the paper is to cover this gap by extending the method to the so-called Cahh-Hilliard equation on a 3D torus. To be more precise, we consider the following 4th order parabolic problem: where Ω is a bounded 3D domain and f (u) is a given non-linear interaction function, see [3,7,17] and the references therein concerning the physical background of this equation. We also assume that this function satisfies some standard dissipativity assumptions, so the associated semigroup possesses a global attractor A which is bounded in H 2 (Ω) ⊂ C(Ω), see e.g., [5,16,22] for more details. By this reason, without loss of generality, we may assume from the very beginning that the function f : R → R is globally bounded and is globally Lipschitz continuous with the Lipschitz constant L. It worth mentioning that this equation possesses a mass conservation law so we assume from now on that u(t) = u(0) = 0. Thus, the natural phase space of the problem is Note that the spectral gap condition for the IM existence for equation (1.4) reads This condition is clearly satisfied for 1D domains only, in the 2D case it is still an open problem whether or not the spectral gaps of arbitrary width exist for any/generic domains Ω although it will be so for some special domains like 2D sphere or 2D torus. In these cases the construction of the IM is straightforward, see [2,22] for more details. However, it is extremely unlikely that the spectral gap condition is satisfied for more or less general 3D domains (in a fact, we know the only example of a 3D sphere where it is true). In particular, it obviously fails for the case of a 3D torus Ω = T 3 = [−π, π] 3 (endowed by periodic boundary conditions), therefore, the problem of finding the IM for the 3D Cahn-Hilliard equation with periodic boundary conditions becomes non-trivial and to the best of our knowledge, has been not considered before.
The next theorem gives the main result of the paper.
Theorem 1.1. For infinitely many values of N ∈ N there exists an N -dimensional IM M N for the Cahn-Hilliard problem (1.4) with periodic boundary conditions which is a graph of a Lipschitz continuous function over the N -dimensional space spanned by the first N eigenvectors of the Laplacian. Moreover, this function is C 1+ε -smooth for some small ε = ε(N ) > 0 and the manifold possesses the so-called exponential tracking property, see Section 3 for the details.
The paper is organized as follows.
In Section 2, we consider the functional model related with the problem considered and prepare some technical tools which will be used later.
In Section 3, for the reader convenience, we remind the invariant cone and squeezing property as well as give the proof of the IM existence theorem for our class of equations under the assumption that the cone and squeezing property are satisfied (following mainly [23]).
In Section 4, we reformulate the cone and squeezing property in a more convenient form of a single differential inequality and derive some kind of normal hyperbolicity (dominated splitting) estimates which are necessary to verify the smoothness of an IM.
In Section 5, we verify that the constructed manifold is C 1+ε -smooth if the nonlinearity is smooth enough. This improves the result of [13] even on the level of reaction-diffusion equations where only C 1 -smoothness has been verified.
The abstract form of spatial averaging principle has been stated in Section 6 and the existence of the IM is verified under the assumption that this principle holds.
Finally, in Section 7, we verify this principle for the case of the Cahn-Hilliard equation on a 3D torus and finish the proof of the main Theorem 1.1.

Preliminaries
We consider the following equation: where A : D(A) → H is a linear self-adjoint positive operator with compact inverse, D(A) is the domain of the operator A, and non-linearity F : H → H is a globally Lipschitz with Lipschitz constant L and globally bounded, i.e., It is well known that problem (2.1) is globally well-posed and generates a non-linear semigroup S(t) in H. From Hilbert-Schmidt theorem we conclude that the operator A possesses the complete orthonormal system of eigenvectors {e n } ∞ n=1 in H which corresponds to eigenvalues λ n numerated in the non-decreasing way: Ae n = λ n e n , 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ ... and due to the compactness of A −1 , we have λ n → ∞ as n → ∞. Thus, we may represent u in the form: u n e n , u n = (u, e n ).
Then, as usual, the normed spaces H s := D(A s/2 ), s ∈ R + , is defined as follows For s < 0 such defined space H s is not complete. Thus for negative s we define H s as completion of H with respect to corresponding norm · H s . Let us introduce the orthoprojector to the first N Fourier modes: u n e n and denote by Q N = Id−P N , H + := P N H and H − := Q N H. Obviously, the following estimates are valid: Throughout the work we will use notations u + := P N u and u − := Q N u for given element u ∈ H.
The next proposition collects the standard dissipativity and smoothing properties of the solution semigroup associated with equation (2.1), see [10,23,22] for more details.
Proposition 2.1. Let the non-linearity F and operator A satisfy the above assumptions. Then, problem (2.1) is uniquelly solvable for any u 0 ∈ H −1 and, therefore, the solution semigroup S(t) : H −1 → H −1 is well-defined. Moreover, the following properties hold for any solution u(t) of problem (2.1): 1. Dissipativity in H s for s ∈ [−1, 2]: where C, γ and R * are some positive constants which are independent of the solution u and t; 2. Smoothing property: where C and R 0 are independent of u and t 3. Dissipativity of the Q N component: for all N ∈ N and κ ∈ (0, 3]. Here C, γ and C κ are independent of N , u and t. We are now ready to give the key definition of the paper, namely, to define the inertial manifold (IM) associated with the Cahn-Hilliard equation. 3. The exponential tracking property holds, i.e., there exist positive constants C and α such that for every u 0 ∈ H −1 there is v 0 ∈ M such that As usual, to verify the existence of the IM, we will use invariant cones method. Namely, introduce the following quadratic form in H −1 : to be the associated cone. Definition 2.3. Let the above assumptions hold. We say that equation (2.1) possesses the cone property (invariance of the cone K + ) if (2.14) where ξ 1 , ξ 2 ∈ H −1 and S(t) is a solution semigroup associated with (2.1). Analogously, we say that (2.1) possesses the squeezing property if there exists positive γ and C such that

Invariant cones, squeezing property and inertial manifolds
The aim of this section is to remind the reader how to construct an IM based on the cone and squeezing property, see [8,9,13,19] for more details. So, the main result of the section is the following theorem. Proof. Step 1. Let us consider the following boundary value problem: We claim that it has a unique solution for any T > 0 and any u + 0 ∈ H + . Indeed, introduce the map G T : H + → H + by the following rule: where S(t) is a solution operator of problem (2.1). Obviously this map is continuous. We want to prove that this map is invertible. Indeed, let u 1 (t), u 2 (t) be two solutions of the problem (3.1) (with different initial data u 1 (−T ) and u 2 (−T ) belonging to H + ). Then, their difference v(t) = u 1 (t) − u 2 (t) lies at the cone K + at the moment t = −T . Thus, from the cone property we conclude that The next lemma is the main technical tool for verifying the one-to-one property.
Lemma 3.2. Let the above assumptions hold. Then, the following estimate hold for the solutions u 1 (t) and u 2 (t): for some constants C and α which are independent of u i .
Proof of the lemma. Since v(t) ∈ K + , we have the estimate Multiplying now the equation for the difference v by A −1 v + and using that the nonlinearity is globally Lipschitz, we have Integrating this inequality over s ∈ (t, 0) and using the interpolation between H 1 and H −1 together with estimate (3.5), we end up with To estimate the last term in the right-hand side, we multiply the equation for v by A −1 v − and integrate over s ∈ (t, 0). With the help of (3.5) again, this gives Inserting the last estimate into the RHS of (3.6), we finally arrive at and the Gronwall inequality finishes the proof of the lemma.
We are now ready to finish the first step of the proof of the theorem. Indeed, since u 1 (t) and u 2 (t) were chosen arbitrary, then we conclude that the map G T : ) and the first step is completed.
Step 2. Let u T,u + 0 be the solution of the boundary value problem (3.1). We claim that for all t ≤ 0, there exists a limit Indeed, since solution of the problem (3.1) starts from u − (−T ) = 0, according to Proposition 2.1, we have: for all T ≥ 0 and u + 0 ∈ H and κ ∈ (0, 3]. In particular, the choice κ = 3 gives the control of the H −1 -norm. Let us introduce the following notations u i (t) := u T i ,u + 0 (t) and v(t) = u 1 (t) − u 2 (t). Then we know that at the moment t = 0 we have v + (0) = 0 and consequently v(0) ∈ K + . By the cone property (2.14) Thus, using squeezing property (2.15) we get: and consequently thanks to (3.10), Thus, u T i ,u + 0 is a Cauchy sequence in C loc ((−∞, 0), H −1 ). Consequently, there exists limit (3.9) and u u + 0 is a backward solution of the problem (2.1).
Step 3. Let us define a set N ⊂ C loc (R, H) as the set of all solutions of the problem (2.1) obtained as a limit (3.9). Then, by the construction, N is invariant with respect to the solution semigroup S(t), i.e.
Consider function Φ : H + → H − acting by the rule: Indeed, according to Steps 1 and 2, the trajectory u ∈ N exists for any u 0 ∈ H + . Moreover, as not difficult to see by approximating the solutions u 1 , u 2 ∈ N by the solutions of the boundary value problem (3.1), that for any u 1 , u 2 ∈ N . Therefore, Φ(u 0 ) is well defined and Lipschitz continuous. Thus, it remains to note that manifold Step 4. In order to prove that M is a desired inertial manifold, it remains to show that exponential tracking property holds. Let u(t) , t ≥ 0, be a forward solution of the problem (2.1) and u T (t) ∈ N , T > 0, be the solution of (2.1) which belongs to the manifold M such that Then, due to the cone property (2.14) On the other hand, using squeezing property (2.15) we obtain: Since the manifold M is finite-dimensional and u T (0) is uniformly bounded, we may assume without loss of generality that u Tn (0) →ũ(0) for some T n → ∞ andũ(0) ∈ M. Then, according to (2.12) and Lemma 3.2, u Tn →ũ ∈ N in C loc (R, H −1 ). Passing to the limit n → ∞ at (3.20) we see that Thus, the exponential tracking is verified and the theorem is proved.
Remark 3.3. Note that, according to Lemma 3.2 and the property (3.16), any two solutions u 1 , u 2 ∈ N satisfy the backward Lipschitz continuity where the positive constants C and α are independent of u 1 , u 2 ∈ N and T ≥ 0. One more simple but important observation is that the above given proof uses the cone and squeezing property not for all u 1 , u 2 ∈ H −1 but only for those which satisfy estimate (3.10). By this reason, we actually need to verify the cone and squeezing property only for u 1 , u 2 ∈ H 2−κ for some κ ∈ (0, 3] such that where C κ is some constant depending only on κ. Indeed, this inequality is automatically satisfied for all trajectories involving into the construction of the set N and the associated inertial manifold M (due to estimate (2.10) and the fact that Q N u(−T ) = 0 in the boundary value problem (3.1)). The exponential tracking a priori holds for all trajectories starting from u(0) ∈ H −1 , however, due to the smoothing property (see Proposition 2.1), it is sufficient to verify it only for u(0) satisfying (3.23). This observation plays a crucial role in the construction of a special cut-off for the spatial averaging method, see below.

Invariant cones and normal hyperbolicity
In this section, we reformulate the cone and squeezing property in a more convenient (at least for our purposes) form of a single differential inequality on the trajectories of (2.1) and its equation of variations and state a number of technical results related with the invariant cones and exponential dichotomy/normal hyperbolicity for the trajectories of the corresponding equation of variations. These results will be used in the next section for establishing the smoothness of the IM. We further assume that the nonlinearity F (u) is at least Gateaux differentiable and there exists a derivative F ′ (u) ∈ L(H, H) for any u ∈ H. Then, obviously and we also assume that the following integral version of the mean value theorem holds: Then, the difference v(t) = u 1 (t)−u 2 (t) of any two solutions u 1 (t) and u 2 (t) of the Cahn-Hilliard problem (2.1) solves the following linear equation: however, it will be more convenient for us to study more general linear equations Definition 4.1. We say that equation (4.4) satisfies the strong cone condition (in a differential form) if there exist a positive number µ and a function α : R → R such that and for any solution v(t), t ∈ [S, T ], S < T , of problem (4.4), the following inequality holds: Here and below V is the quadratic form defined by (2.13).
As easy to see from inequality (4.6), the above assumptions guarantee that the cone K + is invariant with respect to the evolution generated by equation (4.4). Moreover, as will be shown below, the squeezing property is also incorporated in our version of the strong cone condition (due to the strict positivity of the exponent α(t)), but we first need to remind some elementary properties of the introduced strong cone condition. We start with reformulating it in a pointwise form applicable to the non-homogeneous form of equation (4.4).
Lemma 4.2. The strong cone condition in the differential form is equivalent to the following condition: Proof. Indeed, let condition (4.7) be satisfied. Then, multiplying equation (4.4) by the expression ) and using equation (4.4), we have Estimating the right-hand side of this inequality by (4.7) with w = v(t), we end up with the desired cone inequality (4.6). Vise versa, let the cone condition (4.6) hold and let w ∈ H 1 and t 0 ∈ R be arbitrary. Let us consider the solution v(t) of equation (4.4) satisfying v(t 0 ) = w.
Using then the cone condition (4.6) with t = t 0 and formula (4.8) for the derivative of the quadratic form V , we end up with the desired inequality (4.7) with t = t 0 . Thus, the lemma is proved.
with h ∈ L ∞ (R, H −1 ) the following analogue of (4.6) holds: Indeed, analogously to (4.8), but using the non-homogeneous equation (4.9), we have and estimating the terms in the right-hand side of this inequality with the help of (4.7) with w = v(t), we end up with the desired inequality (4.10).
At the next step, we show that the strong cone condition is robust with respect to perturbations of the cone and this will give us the main technical tool for proving the smoothness of the IM. Namely, for any ε ∈ R, we define Let equation (4.4) satisfy the strong cone condition in the differential form. Then, there exists ε 0 > 0 such that, for any 0 < ε < ε 0 , the following inequalities hold for any solution v(t) of equation (4.4).
Proof. Let us check the first inequality of (4.13). Multiplying equation (4.4) by 2εA −1 v and using the Lipschitz continuity, we have Taking a sum of this inequality with (4.6), using (4.5) and fixing ε > 0 to be so small that we end up with H . Combining this inequality with the obvious estimate and assuming that ε ≤ 1 2 , we prove the first formula of (4.13). Let us prove the second inequality of (4.13). Multiplying (4.4) by −4εA −1 v + (t) and using Multiplying now inequality (4.6) by (1 − ε) > 0, taking a sum with the last inequality and fixing ε > 0 in such way that 4ε(λ N + L) ≤ 1 4 (1 − ε)µ, we end up with Using the inequality (4.16) again and assuming that ε < 1 4 , we end up with the desired second estimate of (4.13) and finish the proof of the lemma.
The next corollary shows that the strong cone condition implies some kind of normal hyperbolicity in the sense that the trajectories outside of the cone squeeze stronger than the trajectories inside of the cone may expand.
The next corollary shows that the strong cone property in the differential form implies both cone and squeezing properties for the solutions of equation (4.4). (4.23) v(0) ∈ K + ⇒ v(t) ∈ K + , for all t ≥ 0.
2. Squeezing property: there exists positive γ and C such that where the constants γ and C are independent of v and T .
Proof. Indeed, the first assertion is an immediate corollary of inequality (4.6). To verify the squeezing property, it is sufficient to use estimate (4.20) on the interval [0, T ] instead of [−T, 0]. This together with inequality (4.5) give and the squeezing property is verified. Thus, the corollary is also proved.
We are now ready to return to the non-linear equation (2.1) and state for it the analogous strong cone condition in the differential form.
the following analogue of (4.6) holds H and the function α satisfies The relation between the strong cone conditions for the non-linear equation (2.1) and for the equation (4.3) for the differences of its solutions is clarified in the following lemma.  Proof. Indeed, according to Lemma 4.2, the strong cone condition for equation (2.1) is equivalent to ∀w ∈ H 1 and every u ∈ H. Replacing u = su 1 (t) + (1 − s)u 2 (t) in this inequality and integrating over s ∈ [0, 1], we end up with H , ∀w ∈ H 1 which according to Lemma 4.2 again is equivalent to the strong cone condition for equation (4.3) and the lemma is proved.
We summarize the obtained results in the following theorem which can be considered as the main result of the section.  To conclude this section, we show that the classical spectral gap condition implies the strong cone condition and, therefore, guarantees the existence of the IM. Proposition 4.11. Let the nonlinearity F (u) be globally bounded and the following spectral gap condition be satisfied for some N ∈ N: where L is a Lipschitz constant of the non-linearity F . Then for every two solutions u 1 (t) and u 2 (t) of the equation (2.1), the associated equation  Proof. Let v(t) be a solution of (4.3). Then, multiplying equation for v(t) first by A −1 v − (t), second by −A −1 v + (t) and taking sum of them we obtain: By the definition of α and µ, we have Analogously, Inserting these estimates to (4.33) and using that l u 1 (t),u 2 (t) L(H,H) ≤ L, we have Thus, the strong cone condition for (4.3) is verified and the proposition is proved.

Smoothness of the inertial manifolds
The aim of this section is to obtain the extra smoothness of the function Φ : H + → H − which determine the inertial manifold M under the assumption that non-linearity F is C 1+δ (H, H) for some δ ∈ (0, 1), i.e., To be more precise, the main result of this section is the following theorem.
Theorem 5.1. Let the assumptions of Theorem 4.9 hold and let also the assumption (5.1) on F be valid for some δ > 0. Then the map Φ is Frechet differentiable and C 1+ε -smooth for some ε > 0, i.e, H . Proof. To verify (5.2), we first need to study the Frechet derivative Φ ′ (u 1 + ). Following the definition of Φ, it is natural to expect that this derivative is defined as follows: Here and below w + ∈ H + and u i (t), t ≤ 0, i = 1, 2, are the solutions of (2.1) belonging to the inertial manifold M and satisfying P N u i (0) = u i + .
Step 1. Well-posedness of Φ ′ (u 1 (t)). The existence of a solution for problem (5.4) can be verified exactly as in Theorem 3.1 and to define the operator Φ ′ , we only need to check the existence of the limit (5.3). Indeed, since the trajectory w T ∈ K + , according to Corollary 4.5 and estimate (4.19), we get Hereᾱ(t) = 0 t α(u 1 (s)) ds and µ is the same as in the strong cone inequality. Consider now another approximation w T 1 (t), T 1 ≥ T , and their difference w T,T 1 (t) := w T 1 (t) − w T (t). This trajectory does not belong to the cone K + at t = 0 and, therefore, it is not in K + for all t ∈ [−T, 0]. Using now (5.5) together with estimate (4.20) of Corollary 4.5, we end up with Thus, w T (0) is a Cauchy sequence and the limit (5.3) exists and, therefore, the operator Φ ′ (u 1 (t)) is well-defined. Moreover, according to (5.5), we have the following estimate for the limit function w(t): Let u 1 (t) and u 2 (t) be two trajectories on the inertial manifold defined by the limit (3.9) which correspond to the initial data u 1 + ∈ H + and u + 2 ∈ H + respectively and w + := u + 1 − u + 2 . Since v(t) ∈ K + then, due to estimate (4.19) and the assumption that the exponent α(t) is globally bounded, we have Our aim at this step is to improve (5.8) and to obtain the estimate which is analogous to (5.7).
To this end, we note that the function v solves the equation where l u 1 (t),u 2 (t) := 1 0 F ′ (u 1 (t) + sv(t)) ds. Since F satisfies (5.1) and v(t) ∈ K + , we have Thus, treating equation (5.9) as a non-homogeneous problem in the form of (4.9) with the righthand side h(t) := [l u 1 (t),u 2 (t) − F ′ (u 1 (t))]v(t) and according to (4.10) and (4.13) (see also (4.18)), we get that, for a sufficiently small ε > 0, the following estimate holds: Thus, since v ∈ K + , analogously to (4.19), we end up with the following estimate: which differs from (5.7) only by the presence of the lower bound for t.
Remark 5.2. Applying the parabolic smoothing property to the equation for θ(t), it is not difficult to verify the stronger version of estimate (5.2), namely H , where κ > 0 is arbitrary and C κ depends only on κ. Moreover, as follows from the proof, estimate (5.1) is actually used for u 1 and u 2 satisfying (3.23) only and can be replaced by for some κ ∈ (0, 2]. As we will see below, assumption (5.22) is much easier to verify in applications than the initial assumption (5.1) which is more natural for the abstract theory.
Note also that the result of Theorem 5.1 is in a sense optimal since the typical regularity of the inertial manifolds is exactly C 1+ε for some small ε > 0. The further regularity (C 2 or more) requires essentially stronger spectral gap assumptions which are usually satisfied only in the case of small Lipschitz constant L, see [11,23] for more details.

Spatial averaging: an abstract scheme
In this section, we adapt the method of spatial averaging developed in [13] to the class of abstract Cahn-Hilliard equations (2.1). To this end, we first need to introduce some projectors.
Let N ∈ N and k > 0 be such that λ N > k. Then, As has been observed in [13] (at least on the level of reaction-diffusion equations, see also [23]), the spectral gap condition is actually used only for the control the norm of the "intermediate" Moreover, if this intermediate part is close to the scalar operator then the spectral gap condition may be relaxed. The following theorem adapts this result to the case of the Cahn-Hilliard equations.
Theorem 6.1. Let the function F be globally Lipschitz with the Lipschitz constant L, globally bounded and differentiable and let the number N be such that where a(u) ∈ R is a scalar depending on u and δ < L. Assume also that as before θ = λ N +1 − λ N and k is chosen in such a way that Then, equation (2.1) possesses the strong cone property in the differential form and, consequently, there exists a Lipschitz N-dimensional inertial manifold for this equation.
Proof. Due to Theorem 4.9, we know that in order to prove the existence of inertial manifold it is sufficient to check the validity of the strong cone inequality To this end, we first need the following estimate for the norm of P k,N w in H −1 (here and below α = λ N λ N +1 ): and the similar estimate for the norm of Q k,N v in H Let w(t) be a solution of the equation of variations (4.25). Then, arguing analogously to the derivation of (4.33) but using estimates (6.5) and (6.6) together with (4.34) and (4.35), we get: Estimating the last term, we have: . It would be convenient to define two more spectral projectors: (6.9)P k,N w := n: λ N −k≤λn≤λ N (w, e n )e n andQ k,N w := n: λ N+1 ≤λn≤λ N +k (w, e n )e n . Remark 6.2. The typical situation to apply the above proved theorem is when, for sufficiently small δ > 0 and any k there exists an infinite sequence of N ∈ N such that λ N +1 − λ N ≥ ρ > 0 (ρ is independent of N and k) such that the spatial averaging assumption (6.2) hold for every such N . Then, for very large N , the main condition (6.3) reads and we see that it is indeed satisfied if δ is small enough (say, δ < ρ 4 ) and k = k(ρ, L) is large enough (say, k = 4L + 12L 2 ρ ). This gives the existence of the desired inertial manifold for these large N s.
As in the case of reaction-diffusion equations, see [13,23], estimate (6.2) is too restrictive since the constant δ is uniform with respect to u ∈ H and in applications it usually depends on the higher norms of u. Namely, similar to [13,23], we give the following definition. Definition 6.3. We say that the non-linearity F : H → H satisfies the spatial averaging condition if it is globally bounded, Lipschitz continuous, differentiable in the sense that the mean value theorem (4.2) holds and there exist a positive exponent κ and a positive constant ρ such that for every δ > 0, R > 0 and k > 0 there exists infinitely many values N ∈ N satisfying (6.17) λ N +1 − λ N ≥ ρ and (6.18) sup for some scalar multiplier a(u) = a N,k,δ (u) ∈ R which is assumed to be bounded Borel measurable as a function from H to R.
In this case, although we do not know how to construct the IM for the initial problem (2.1), it is possible to modify this equation outside of the absorbing ball in such a way that the new equation will possess the IM. Then, the obtained IM will still be invariant with respect to the solution semigroup S(t) of the initial equation (2.1) at least in the neighborhood of the global attractor A and therefore will contain all of its non-trivial dynamics. By this reason, the IM for the modified equation is often referred as the IM for the initial problem (2.1), see e.g., [9,22] for more details.
Thus, (6.21) gives us the restriction Then, using Lemma (6.5) and (4.35), we get: Fix an arbitrary point t ≥ 0 and assume first that where R 1 is the same as in the definition of the cut-off function ϕ. Thus, we see that the structure of (6.29) is exactly the same as the structure of (6.7). In addition, due to (2.10), we may assume without loss of generality that see Remarks 3.3 and 4.10. Therefore, assumption (6.30) implies that where R is independent of the choice of N . Hence, using spatial averaging assumption (6.18) (with this value of the parameter R, sufficiently small δ and sufficiently large k in order to satisfy (6.16)) and repeating word by word the proof of Theorem (6.1), we conclude that there exist a sequence of N s such that where α(u) := λ N λ N +1 − λ N a(u) and µ > 0 is independent of u, N and w. Thus, we have verified the strong cone condition (4.26) in the case when (6.30) is satisfied. Let us now consider the opposite case The situation here is much simpler. Indeed, instead of estimate (6.25), we may use better identity (T ′ (u(t))w, w) = 1 2 (Aw + , w + ). Then, using the Lipschitz continuity of F and the fact that both (αA −1 − A)Q N and (A − αA −1 )P N are negatively definite (see estimates (4.34) and (4.35)), we transform identity (6.28) as follows: Thus, if L < 1 4 λ N +1 − µ (µ is the same as in (6.33)), we end up with estimate (6.33) with α(u) = 3 4 λ N λ N +1 . Therefore the desired strong cone estimate (4.26) is verified for the case when (6.34) is satisfied as well and the theorem is proved.
so the spatial averaging assumption for our case coincides with the analogous assumption for reaction-diffusion equations. By this reason, the proof of the proposition follows word by word to the one given in [13,23] for the case of reaction-diffusion equations. For the convenience of the reader, we sketch this proof below. It is based on the following non-trivial result from the number theory.
We are now ready to verify the spatial averaging principle for the non-linearity f . To this end, we first note that according to the Weyl asymptotics λ N ∼ CN 2/3 , so without loss of generality, we may replace the projector R k,N by the projector to the Fourier modes belonging to C k N which (in slight abuse of notations), we also denote it by R k,N , so we use below the definition Denote w(x) := f ′ (u(x)). Then, the multiplication w(x)v(x) is a convolution in Fourier modes (7.11) [wv] m = l∈Z 3 w m−l v l and, due to condition (7.10), where w >r (x) := |l|>r w l e il.x . Therefore, Furthermore, due to the interpolation, for κ < 1/2, w >r L ∞ ≤ C w >r . Finally, using that H 2−κ is an algebra for κ < 1 2 and that f ′ ∈ C 2 , we have (7.14) for some monotone increasing function Q. Thus, the right-hand side of (7.14) can be made arbitrarily small by increasing r, so estimate (6.18) indeed holds with a(u) = w = 1 (2π) 3 Since the eigenvalues of A are integers, assumption (6.17) also holds with ρ = 1 and the nonlinearity F (u) satisfies indeed the spatial averaging assumption. Thus, the proposition is proved.
The next simple proposition shows that the map F is smooth if f is smooth. Proposition 7.3. Let the nonlinear function f ∈ C 2 and satisfy (7.5). Then, the non-linear operator F (u) defined by (7.6) satisfies estimate (5.22) for any δ ∈ [0, 1] and any κ ∈ (0, 1 2 ).
Proof. We first note that it is sufficient to verify (5.22) for δ = 0 and δ = 1 only. For simplicity, we verify estimate (5.22) for the first term f (u(x)) in the definition (7.6) of the nonlinearity F (u) only. The estimate for the remaining term f (u) can be obtained analogously. Let first δ = 0 and u 1 , u 2 ∈ H. Then, since f is globally Lipschitz continuous, and estimate (5.22) is verified for that case. Let now δ = 1 and u 1 , u 2 ∈ H 2−κ . Then, according to the integral mean value theorem and, due to assumption (7.5) and the embedding H 2−κ ⊂ C, Therefore, estimate (5.22) holds for this case as well and the proposition is proved.
Thus, all abstract assumptions from the previous sections are verified and we have proved the following theorem which is the main result of the paper.
Theorem 7.4. Let the non-linear function f ∈ C 3 (R, R) and satisfy assumptions (7.5). Then, there exists an infinite sequence of N s such that the classical Cahn-Hilliard problem (1.4) on a 3D torus T 3 = [−π, π] 3 possesses an N dimensional inertial manifold containing the global attractor. Moreover, these inertial manifolds are C 1+ε -smooth for some ε = ε N > 0.
Indeed, the existence of IMs follows from Theorem 6.4 and Proposition 7.1 and its smoothness follows from Theorem 6.6 and Proposition 7.3.