Robust exponential attractors for Cahn-Hilliard type equations with singular potentials

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2004; 27:545–582 (DOI: 10.1002/mma.464)MOS subject classi�cation: 35B 40; 35B 45; 35 L 30

Robust exponential attractors for Cahn-Hilliard type equationswith singular potentials

Alain Miranville∗;† and Sergey Zelik‡

Universit�e de Poitiers; Laboratoire d’Applications des Math�ematiques-SP2MI; Boulevard Marie et PierreCurie-T�el�eport 2; 86962 Chasseneuil Futuroscope Cedex; France

Communicated by W. Wendland

SUMMARY

Our aim in this article is to study the long time behaviour of a family of singularly perturbed Cahn-Hilliard equations with singular (and, in particular, logarithmic) potentials. In particular, we are able toconstruct a continuous family of exponential attractors (as the perturbation parameter goes to 0). Fur-thermore, using these exponential attractors, we are able to prove the existence of the �nite dimensionalglobal attractor which attracts the bounded sets of initial data for all the possible values of the spatialaverage of the order parameter, hence improving previous results which required strong restrictions onthe size of the spatial domain and to work on spaces on which the average of the order parameter isprescribed. Finally, we are able, in one and two space dimensions, to separate the solutions from thesingular values of the potential, which allows us to reduce the problem to one with a regular potential.Unfortunately, for the unperturbed problem in three space dimensions, we need additional assumptionson the potential, which prevents us from proving such a result for logarithmic potentials. Copyright ?2004 John Wiley & Sons, Ltd.

KEY WORDS: Cahn-Hilliard equation; singular potential; logarithmic potential; exponential attractor;global attractor

1. INTRODUCTION

In this article, we are interested in the study of the long time behaviour of the Cahn-Hilliardequation with singular potentials (and, in particular, with logarithmic potentials).The Cahn-Hilliard equation (see References [1,2]) is very important in material science.

It describes important qualitative behaviour of two-phase systems, namely, the transitions of

∗Correspondence to: Alain Miranville, Universit�e de Poitiers, Laboratoire d’Applications des Math�ematiques-SP2MI,86962 Chasseneuil Futuroscope Cedex, France.

†E-mail: [email protected]‡E-mail: [email protected]

Contract=grant sponsor: INTAS; contract=grant number: 00-899Contract=grant sponsor: CRDF; contract=grant number: 10545

Copyright ? 2004 John Wiley & Sons, Ltd. Received 3 March 2003

546 A. MIRANVILLE AND S. ZELIK

phases in binary alloys. This can be observed when the alloy is cooled down su�ciently. Onecan then observe a partial nucleation, i.e. the apparition of nucleides, or a total nucleation, theso-called spinodal decomposition. In that case, the material quickly becomes inhomogeneousand forms a �ne-grained structure, more or less alternating between the two components ofthe alloy (see References [1,2]). From a mathematical point of view, the spinodal decomposi-tion corresponds to an unstable homogeneous equilibrium. We refer the reader to References[3,4] for a qualitative study of the spinodal decomposition. In a second stage, called coars-ening, the microstructures observed after the spinodal decomposition coarsen. The coarseningoccurs at a slower time scale and is less understood (see e.g. References [5,6]). So, in viewof a better understanding of these phenomena, it seems useful to have a good understandingof the asymptotic behaviour of the equation, and also to have some understanding of thetransient dynamics. In this article, we are more speci�cally interested in the study of �nitedimensional attractors (in the sense of the Hausdor� and the fractal dimensions). In partic-ular, the existence of such sets would indicate that the long time behaviour of the system,and also, in some sense, the phenomena observed, only involve a �nite number of degreesof freedom.The starting point in the Cahn-Hilliard theory is the free energy, also called Ginzburg-

Landau free energy, which goes back to van der Waals (see Reference [7]) and is of theform

= (u;∇u)=�2|∇u|2 + F(u)

where u= u(t; x) is the order parameter (a rescaled density of atoms), F is the potential,a double-well potential whose wells correspond to the phases of the material, and �¿0 isrelated to the surface tension. For simplicity, we will take �=1 in this article. This leads tothe following fourth-order parabolic equation for the order parameter:{

@tu=−�x(�xu− f(u))

@nu|@� = @n�xu|@� =0; u|t=0 = u0(1)

where f(u) :=F ′(u) and � is a bounded smooth domain of R3.A slightly more complicated model, which is based on a new balance law for micro-

forces and which takes into account the working of internal microforces, was introduced inReference [8] (we can note that microforces describe forces which are associated with mi-croscopic con�gurations of atoms, whereas standard forces are associated with macroscopiclength scales, hence a reason to consider separate balance laws for microforces and stan-dard forces). For an isotropic material, this leads to the following generalization ofEquation (1): {

@tu=−�x(�xu− �@tu− f(u))

@nu|@� = @n�xu|@� =0; u|t=0 = u0(2)

where � is a (small) positive parameter and where the term �@tu describes the in�uence of theinternal microforces. This equation can also be viewed as a viscous Cahn-Hilliard equation,see e.g. Reference [9].These equations have been studied intensively; see e.g. the review articles [10] and [11]

and, among many references, [9,12–27]. In particular, in the case of regular potentials, the

Copyright ? 2004 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2004; 27:545–582

ROBUST EXPONENTIAL ATTRACTORS 547

problem is well understood and one has existence and uniqueness of solutions and existenceof the �nite dimensional global attractor.However, logarithmic non-linear interaction functions f(u)=F ′(u) of the form

f(u)=flog(u) :=−�cu+ � ln1 + u1− u

; 0¡�¡�c (3)

are more relevant from the physical point of view, see e.g. References [1,2,10]. Actually,in order to simplify the equation, one often approximates logarithmic potentials by regu-lar potentials such as, in particular, polynomials of degree four. In that case, the additionalcondition

−1¡u(t; x)¡1; for almost every (t; x)∈R+×� (4)

must be imposed to the order parameter u(t; x). It would be important to prove that, in thecase of regular potentials, the order parameter also satis�es (4). At the moment, however,this seems out of reach, due to the lack of a maximum principle for fourth-order parabolicequations.The �rst existence and uniqueness results for the classical Cahn-Hilliard equation with a

logarithmic potential were obtained in Reference [20]; see also References [19,28] for mo-bilities which depend on the order parameter and which degenerate. In particular, in orderto prove the existence of solutions, one approximates the singular potential by regular onesand then passes to the limit. The numerical analysis of such equations can be found inReferences [12,29,30].The long time behaviour of the Cahn-Hilliard equation (1) with a logarithmic potential

was studied in Reference [28]. In particular, the authors obtained the existence of the globalattractor. In that case, however, the study of the �nite dimensionality of the global attractoris much more di�cult than in the case of regular potentials. Indeed, the classical scheme,based on the Lyapunov exponents and on the volume contraction method (see e.g. Reference[27]), requires that the corresponding semigroup is (quasi)di�erentiable on the global attractor.This di�erentiability is a rather delicate problem for the Cahn-Hilliard equation with singular(e.g. logarithmic) potentials and, to the best of our knowledge, it can be solved only whenevery point of the global attractor A is separated from the singular points of the potential,i.e. when

‖A‖L∞(�)61− � (5)

for some constant �∈]0; 1[. Under this assumption, the di�erentiability of the semigroup canbe proven exactly as in the case of regular potentials. Unfortunately, the existence of sucha constant � is also a highly non-trivial problem which has been solved in Reference [28]only for small domains �. Moreover, for technical reasons related to the volume contractionmethod, the authors could only consider subspaces of the phase space on which the aver-age of the order parameter is a given constant, which is not entirely satisfactory. Analogousresults for Equation (2) with a logarithmic potential of the form (3) have been obtained inReference [16].One of the aims of this article is to improve the results of References [16,28]. In particular,

we prove that the global attractors A�; �¿0, associated with problems (2) with a logarithmicpotential have �nite dimension for arbitrary three-dimensional domains � (and not only for



small domains � as it was the case previously). Moreover, we also construct a robust familyof exponential attractors M� for problems (2) which tend to the exponential attractor M0

associated with the limit problem (1) as �→ 0:

distsym; L∞(�)(M�;M0)6C�� (6)

where the constants C and � can be computed explicitly. We note that this result does notrequire that the corresponding semigroups are quasidi�erentiable, and, thus, does not requirethat the analogue of (5) is satis�ed. We also consider a more general class of (singular)non-linearities f∈C1(−1; 1) which satisfy the following assumptions:{

1: limz→±1 f(z)=±∞2: limz→±1 f′(z)=+∞

(7)

We recall that, in contrast to exponential attractors, global attractors are, in general, only uppersemicontinuous with respect to perturbations. So, in general, one does not expect that estimate(6) is satis�ed for global attractors. Indeed, the lower semicontinuity of global attractors ismuch more delicate to prove and usually requires some kind of ‘hyperbolicity assumption’ onthe unperturbed attractor A0. For instance, there exists a rather wide class of global attractors(the so-called regular attractors in the terminology of Babin and Vishik) which are upper andlower semicontinuous and satisfy estimate (6) under natural assumptions on the perturbations(they are also exponential), see e.g. References [31,32]. This theory requires, however, theexistence of a global Lyapunov function and the hyperbolicity of all the equilibria of thesystem under consideration.In the case of the Cahn-Hilliard equation (1), we obviously have a global Lyapunov func-

tion, but the second assumption on the hyperbolicity of the equilibria is rather delicate toprove, even in the case of regular potentials. Indeed, even though this assumption is in somesense ‘generic’, it is very di�cult to verify for concrete values of the physical parameters(say, �; �c and �) and, to the best of our knowledge, this problem has been solved onlyfor one-dimensional domains �, see References [33,34]. Moreover, even in that case, it isunclear how to estimate the constants C and � (in (6)) in terms of the parameters �; �c and �for regular attractors. On the contrary, estimate (6) for exponential attractors requires neitherthe existence of a global Lyapunov function nor any ‘hyperbolicity assumption’ and usuallyholds under natural assumptions on the system under consideration, see References [35–37]for details. Moreover, as it can be seen from the proof of estimate (6) (see Sections 2 and 5),the constants C and � can be estimated explicitly in terms of the physical parameters of theproblem although we do not give the explicit formulae here.Thus, exponential attractors are more robust and, perhaps, more suitable objects than global

attractors for the study of dynamical systems depending on parameters. We refer the reader toReferences [35–38] for more detailed discussions on the robustness of exponential attractorsin di�erent situations including the problem of the approximation of a PDE in an unboundeddomain by the same PDE in bounded domains, which is known to be very singular at thelevel of global attractors, see Reference [36]. Furthermore, the global attractor may be trivial,say, reduced to one point, and may thus fail to capture important transient dynamics. Again,an exponential attractor may be a better object. Indeed, the trajectories will be attracted toany exponential attractor exponentially fast and will then converge to the global attractor ata slower rate. This bears some resemblance to what can be observed in the phase separation



process described by the Cahn-Hilliard equation, i.e. the spinodal decomposition followed bya slower process, the coarsening.This article is organized as follows.In Section 2, we prove an abstract result on the construction of a continuous (with respect to

perturbations) family of exponential attractors. This result generalizes previous constructionsgiven in References [35,37,39]. Also, here, we do not have the continuity up to time shiftsas it was the case in Reference [38]. Such a generalization is necessary in order to be ableto treat the speci�c di�culties induced by singular potentials.In Section 3, we derive several a priori estimates which are of fundamental signi�cance for

what follows. In particular, we prove that every solution u(t) of Equation (2) with �¿0 isseparated from the singular points of the non-linearity. The latter means that, for every �¿0and �¿0, there exists a strictly positive constant ��; � such that every solution u(t) of (2)satis�es

‖u(t)‖L∞(�)61− ��; �; ∀t¿� (8)

This result shows that the additional term �@tu in the right-hand side of Equation (2) reg-ularizes the classical Cahn-Hilliard equation (1) with a singular potential. We note that theconstant ��; � in estimate (8) tends to zero as �→ 0 so that this estimate does not provide anyinformation for the limit equation (1).The existence and uniqueness of a solution u(t) for problem (2), �¿0, is obtained in

Section 4. We prove the existence of solutions for (1) with singular potentials by using the(physically relevant) regularization (2). Hence, our proof is di�erent from those in References[19,20,28], where the problem is regularized by approximating the singular potential by regularones.In Section 5, we apply the abstract scheme given in Section 2 in order to construct a

continuous (as the perturbation parameter goes to 0) family of exponential attractors for ourproblem. As a consequence, we also obtain the existence of the �nite dimensional globalattractor without any assumption on the size of the spatial domain as it was the case inReferences [16,28].In Section 5, we work on spaces on which the average of the order parameter is prescribed.

As a consequence, the exponential attractors (and also the global attractors) constructed dependa priori on this value of the average of the order parameter. However, in Section 5, we areable to construct a continuous family of exponential attractors that do not depend on theaverage of the order parameter. Furthermore, using these exponential attractors, we are ableto construct the �nite dimensional global attractor which attracts the bounded sets of initial datafor all the possible values of the average of the order parameter, which is more satisfactoryfrom the physical point of view. Such a result would be more di�cult to prove for exponentialattractors and will be studied elsewhere (see Remark 6.2).Finally, we obtain in Section 7 uniform (with respect to the perturbation parameter)

L∞-bounds on the solutions. Such estimates allow to separate the solutions of (1) from thesingular points of the potential and thus to reduce the problem to one with a regular potential.Unfortunately, in three space dimensions, we need additional assumptions on the potential andare thus not able to consider logarithmic potentials. However, we are able to prove such aresult for logarithmic potentials in one and two space dimensions.For the reader’s convenience, several properties of the nonlinear interaction function are

collected in Appendix A.



2. PERTURBATIONS OF EXPONENTIAL ATTRACTORS:THE ABSTRACT SETTING

In this section, we give a generalization of an abstract result on the construction of uni-form exponential attractors for a singularity perturbed family of maps, which is given inReferences [35,37]. This result will be applied to our problem in Section 5. In order to doso, we use the concept of the Kolmogorov �-entropy.

De�nition 2.1Let K be a (pre)compact set in a metric space V . Then, due to the Hausdor� criterium, forevery �¿0, the set K can be covered by a �nite number of �-balls in V . Let N�(K; V ) bethe minimal number of such balls. Then, by de�nition, the Kolmogorov �-entropy of K isthe following number:

H �(K; V ) := lnN�(K; V )

(see e.g. Reference [40] for details). We recall that the fractal dimension of the set K can beexpressed in terms of the �-entropy:

dimF(K; V ) := lim sup�→0+

H �(K; V )ln 1

�

We are now in a position to formulate our abstract scheme.Let E(�) and E1(�); �∈[0; 1], be two families of Banach spaces (which are embedded into

a larger topological space V) such that E1(�)bE(�), for every �∈[0; 1]. We also assume thatthese compact embeddings are uniform with respect to � in the following sense:

H �(B(0; 1;E1(�));E(�))6M(�); ∀�¿0 (9)

where M(�) is some monotonic function which is independent of � (here and below, wedenote by B(v; R; V ) the R-ball in V centred at v).We further assume that we are given a family of closed sets B� ⊂E(�) and a family of

maps S� :B� →B� such that

1. The set B0 is a subset of E(�), for every �∈[0; 1], and‖b0‖E(�)6C1‖b0‖E(0) + C2�; ∀b0∈B0 (10)

2. There exists a family of sets C� ⊂B� such that, for every b1� ; b2� ∈C�, the following estimate

holds:

‖S�b1� − S�b2� ‖E1(�)6K‖b1� − b2� ‖E(�) (11)

where K is independent of �.3. There exist non-linear ‘projectors’ � � :B� →B0 such that � �B�=B0 and

‖S (k)� b� − S (k)0 � �b�‖E(�)6C�Lk (12)

for every b�∈B�, where the constants C and L are independent of � (here and in whatfollows, S (k)� denotes the kth iteration of the map S�).



4. The sets C� satisfy the following uniform recurrence property: there exist �¿0 and N ∈N(which are independent of �) such that, for every b�∈B�, at least one of the sets

S�(B� ∩B(b�; �;E(�)); : : : ; S (N−1)� (B� ∩B(b�; �;E(�)) (13)

is a subset of C�.5. The sets B� can be covered by a �nite number of �-balls and

H �(B�;E(�))6M0¡∞ (14)

where M0 is independent of �.6. The maps S� are uniformly Lipschitz on B�:

‖S (k)� b1 − S (k)� b2‖E(�)6CeLk‖b1 − b2‖E(�); ∀b1; b2∈B�; ∀k∈N (15)

where C and L are independent of �.

Thus, in contrast to previous results (see References [35,37]), we now assume the smoothingproperty (11) on the ‘recurrent’ subsets C� of the phase space B� only. This will be importantwhen applying the abstract result given in this section to our problem, since, as we will see inthe next sections below, we cannot prove that in general the solutions of the (classical) Cahn-Hilliard equation are separated from the singular values of the potential and, consequently,we cannot obtain the smoothing property (11) for every b1� and b2� belonging to the wholephase space B�.The main result of this section is the following theorem.

Theorem 2.1Let assumptions (9)–(15) hold. Then there exists a family of exponential attractors M� ⊂B�

for the maps S� such that S�M� ⊂M�, and the following conditions are satis�ed:

1. The rate of exponential attraction is uniform with respect to �:

distE(�)(S (k)� B�;M�)6C3e−�k (16)

where the positive constants C3 and � are independent of � and where dist denotes theHausdor� semidistance.

2. The sets M� are compact in E(�), for every �∈[0; 1], and their Kolmogorov �-entropyis uniformly bounded with respect to �:

H �(M�;E(�))6C4 ln1�+ C ′

4 (17)

where the positive constants C4 and C ′4 are independent of � and �¿0. Therefore, their

fractal dimension is also uniformly bounded with respect to �.3. The symmetric distance between M� and M0 satis�es

distsym;E(�)(M�;M0)6C5�� (18)

Moreover, the constants Ci; i=3; 4; 5, and 0¡�¡1 can be computed explicitly.



ProofWe �rst construct an appropriate exponential attractor M0 for S

(N )0 . To this end, we construct

a family of sets Vi ⊂ S (Ni)0 B0; i=0; 1; 2; : : : ; by the following inductive procedure. We recall

that, due to (14), there exists a covering of the set B0 by a �nite number N0 := eM0 of �-ballsin the metric of E(0). Let V0 := {b10; : : : ; bN0

0 }⊂B0 be the centres of these balls. We assumethat the set Vk is already constructed such that Vk ⊂ S (Nk)

0 B0 and Vk is an Rk := �=2k-net ofS (Nk)0 B0. We then construct the next set Vk+1 preserving these properties. To this end, we �xan arbitrary bk ∈Vk and consider the images of the �=2k-balls centred at bk . Then, accordingto the recurrence property (13) and the Lipschitz continuity (15), there exists N (bk)6N − 1such that

S (N (bk ))0 (B0 ∩B(bk ; Rk ;E(0)))⊂C0 ∩B(S (N (bk ))0 bk ; CRk2L(N−1);E(0)) (19)

Using now the smoothing property (11), we �nd

S (N (bk )+1)0 (B0 ∩B(bk ; Rk ;E(0)))⊂B(S (N (bk )+1)bk ; CKRk2L(N−1);E1(0)) (20)

We now recall that E1(0)bE(0). Consequently, there exists a covering of the right-hand sideof (20) by a �nite number of Rk=(C2L(N−1)+2)-balls in E(0). We �x a covering with a minimalnumber of such balls and denote by Wk(bk) the set of all the centres of this covering (withoutloss of generality, we can assume that Wk(bk)⊂B0). Furthermore, we note that

#Wk(bk) = NRk=(C2L(N−1)+2)(B(S(N (bk )+1)0 bk ; CKRk2L(N−1);E1(0));E(0))

= N1=(C2K22L(N−1)+2)(B(0; 1;E1(0));E(0))

6 exp(M(1=(C2K22L(N−1)+2))) :=P0 (21)

It is essential for us that the number P0 de�ned in (21) be independent of k (and of b∈Vk ;moreover, thanks to (9), this number will be independent of � as well if we replace the spacesE(0) and E1(0) by E(�) and E1(�)).Thus, due to (19)–(21), we have the following embedding:

S (N (bk )+1)0 (B0 ∩B(bk ; Rk ;E(0)))⊂⋃

b∈Wk (bk )B(b; Rk=(C2L(N−1)+2);E(0)) (22)

Using (22), the Lipschitz continuity (15) and the fact that N (bk) + 16N , we obtain

S (N )0 (B0 ∩B(bk ; Rk ;E(0)))⊂⋃

b∈Wk (bk )B(S (N−N (bk )−1)b; Rk=4);E(0)) (23)

and, therefore, the Rk=4-balls centred at the points of⋃

bk∈Vk S(N−N (bk )−1)W (bk) cover the set

S((k+1)N )0 B0. Moreover, increasing the radius of the balls by a factor of two, we can assumethat the centres of the covering belong to S((k+1)N )B0. We denote by Vk+1 the set consistingof all these centres.



Thus, we have constructed by induction a family of sets Vk ⊂ S (Nk)0 B0; k∈N, which satisfy

the following properties: {1: #Vk6N0Pk

0

2: distE(0)(S(Nk)0 B0; Vk)6�2−k

(24)

We now de�ne a new family of sets Ek; k∈N, by the following inductive formula:E1 :=V1; Ek+1 := S (N )0 Ek ∪Vk+1

Then, obviously 1: Ek ⊂ S (Nk)

0 B0; S (N )0 Ek ⊂Ek+1

2: #Ek6N0Pk+10

3: distE(0)(S(Nk)0 B0; Ek)6�2−k

(25)

We �nally set

M0′ :=

⋃k∈N

Ek; M0 := [ M′0]E(0) (26)

where [·]V denotes the closure in V . It is not di�cult to verify that the set M0 is an exponentialattractor for the map S0 := S (N )0 on B0. Indeed, (25)1 and (26) imply that S0 M0⊂ M0 and theexponential attraction (16) is an immediate corollary of (25)3. Furthermore, it follows from(25)2, together with (25)1, that, for every k∈N, we have

N�2−k ( M0;E(0))6k∑

l=1#El6kN0Pk+1

0 (27)

which immediately implies (17) (see Reference [39] for details).We now construct the exponential attractors M� for the maps S� := S (N )� in the case �¿0,

using the construction of the attractor M0 as in Reference [37]. For the reader’s convenience,we brie�y recall this construction.We �x inverse images of the sets Ek under the maps S

(k)0 (this is possible due to (25)1).

To be more precise, we assume that the family of sets Ek ⊂B0 is such that{1: S

(k)0 Ek =Ek

2: #Ek =#Ek6N0Pk+10

(28)

We �x an arbitrary �∈(0; 1] and arbitrary liftings of the sets Ek ⊂B0 to B� with respect tothe projectors � �, i.e. the Ek(�)⊂B� are such that{

1: � �Ek(�)= Ek

2: #Ek(�)=#Ek6N0Pk+10

(29)

Such liftings exist since � �B�=B0. We �nally set Ek(�) :=S(k)

� Ek(�). Then, as proven inReference [37], estimates (10), (12) and (25)3 imply that

distE(�)(S(k)� B�; Ek(�))62C�LNk + C1�2−k + C2� (30)



where the constants C; C1; C2 and L are de�ned in (10) and (12). Let now k(�) and 0¡�¡1be the solutions of

�LNk =2−k = �� (31)

i.e. k(�)= ln1=�=(N ln L+ ln 2) and �= ln 2=(N ln L+ ln 2). Then it follows from (12), (30)and (31) that {

1: distsym;E(�)( Ek(�); Ek)6C ′��

2: distE(�)(S(k)� B�; Ek(�))6C ′′�2−k

(32)

for every 16k6k(�), where the constants C ′ and C ′′ are independent of k and �.Thus, we can take Ek(�) := Ek(�), if k6k(�). In order to construct the sets Ek(�) for

k¿k(�), we forget the sets Ek(�) and construct them by the inductive procedure describedabove (for the case �=0), based on (11) and (13), but starting from Ek(�)(�). Finally, wehave a family of sets Ek(�) which satisfy the following conditions:{

1: Ek(�)⊂S(k)� B�; S�Ek(�)⊂Ek+1(�); #Ek(�)6N0Pk+1

0

2: distE(�)(S(k)� B�; Ek(�))6C ′�2−k

(33)

Moreover, for k6k(�), we have

distsym;E(�) (Ek(�); Ek)6C ′′�� (34)

We then de�ne the exponential attractor M� as follows:

M′� :=

⋃k∈N

Ek(�); M� := [M�′]E(�) (35)

It is proven in References [35,37] that the family of sets M� thus de�ned is indeed a familyof robust exponential attractors for the maps S� := S (N )� , which satis�es all the assumptions ofTheorem 2.1.The robust family of exponential attractors M� for the initial maps S�; �∈[0; 1], can now

be de�ned (in terms of the attractors M� for S� := S (N )� constructed above) by the followingstandard expression:

M� :=N−1⋃i=0

S (i)�M� (36)

and Theorem 2.1 is proven.

Remark 2.1It is essential for the next sections below to note that M� satis�es

M� ⊂ S�B� (37)



3. UNIFORM A PRIORI ESTIMATES

In this section, we derive several a priori estimates on the solutions of Equation (2), whichplay a fundamental role for what follows. In order to obtain these results, we rewrite (followingReferences [18,35]) the initial problem (2) in the following equivalent form (see Remark 4.1):{

�@tu+ (−�x)−1N @tu = �xu− f(u) + 〈f(u)〉; �¿0

@nu|@� = 0; u|t=0 = u0(38)

where 〈v〉 := (1=|�|) ∫� v(x) dx denotes the spatial average of the function v and (−�x)−1N isthe inverse Laplace operator in � associated with Neumann boundary conditions, which iswell de�ned on the space L20(�) of the functions belonging to L2(�) with zero average:

(−�x)−1N : L20(�)→L20(�); L20(�) := {v∈L2(�); 〈v〉=0} (39)

We also recall that Equation (38) (or, equivalently, Equation (2)) possesses a conservationlaw

〈u(t)〉≡ 〈u0〉; ∀t¿0 (40)

Therefore, we impose the additional condition

〈u0〉=m0; |m0|61− �; for some �xed �∈]0; 1[ (41)

to the initial value u0. We also de�ne, for every m0∈[−1 + �; 1 − �], the phase space Dm0�

for this problem as follows:

Dm0� := {v∈H 2(�); @nv|@� =0; ‖v‖L∞(�)61; 〈v〉=m0

f(v)∈L2(�); �1=2�∈L2(�) and �∈H−1(�) where

� := (�+ (−�x)−1N )−1[�xv− f(v) + 〈f(v)〉]} (42)

and the ‘norm’ in the space Dm0� is de�ned as follows:

‖v‖2Dm0�:= ‖v‖2H 2(�) + ‖f(v)‖2L2(�) + �‖�‖2L2(�) + ‖�‖2

H−1N (�)

(43)

where H−1N (�) := [H 1(�)]′.

In this section, we only give a formal derivation of the a priori estimates for u(t), assumingthat u(t) is a su�ciently regular function which satis�es the additional assumption

‖u‖L∞(R+×�)¡1 (44)

This assumption will be relaxed in the next section (see De�nition 4.1).The main result of this section is the following theorem which gives a dissipative estimate

in the space Dm0� .



Theorem 3.1Let the non-linearity f satisfy assumptions (7) and let u(t) be a solution of Equation (38)which satis�es (44). Then the following estimate holds:

‖u(t)‖2Dm0�+

∫ t+1

t‖@tu(s)‖2H 1(�) ds6Q�(‖u(0)‖Dm0

�)(1− t) + C�; ∀t¿0 (45)

where (z) is the Heaviside function and where the constant C� and the monotonic functionQ� may depend on �, but are independent of �∈[0; 1] and u(t).

ProofWe �rst prove the non-dissipative analogue of estimate (45).

Lemma 3.1Let the above assumptions hold. Then, for every T ¿0, there exists a monotonic functionQ�;T depending on � and T such that

‖u(t)‖2Dm0�+

∫ t+1

t‖@tu(s)‖2H 1(�) ds6Q�;T (‖u(0)‖Dm0

�) (46)

holds uniformly with respect to � and t6T .

ProofWe di�erentiate Equation (38) with respect to t and set �(t) := @tu(t). Then we have

�@t�+ (−�x)−1N @t�=�x�− f′(u)�+ 〈f′(u)�〉; @n�|@� =0 (47)

Multiplying this equation by �(t), integrating over � and noting that 〈�(t)〉≡ 0 (due to theconservation law (40)) and that f′(v)¿− K (due to assumption (7)2), we obtain

@t[�‖�(t)‖2L2(�) + ‖�(t)‖2H−1

N (�)] + ‖∇x�(t)‖2L2(�)62K‖�(t)‖2L2(�) (48)

Applying the interpolation inequality ‖�‖2L2(�)6C‖∇x�‖L2(�)‖�‖H−1N (�) and Gronwall’s in-

equality to (48), we have

�‖�(t)‖2L2(�) + ‖�(t)‖2H−1

N (�)+

∫ t+1

t‖�(s)‖2H 1(�) ds

6C ′eK′t(�‖�(0)‖2L2(�) + ‖�(0)‖2

H−1N (�)

) (49)

where the constants C ′ and K ′ are independent of � and u. We note that the right-hand sideof (49) can be uniformly (with respect to �) estimated by ‖u(0)‖Dm0

�. Thus, estimate (49)

implies the �-part of estimate (46). In order to obtain the u-part, we rewrite Equation (38)as follows:

�xu(t)− f(u(t)) + 〈f(u(t))〉= h(t) := ��(t) + (−�x)−1N �(t); @nu(t)|@� =0 (50)

and interpret this equation as an elliptic problem, for every �xed t¿0. We multiply (50)by �xu(t) and integrate over �. Then, since 〈�xu(t)〉=0 and f′(v)¿−K , we �nd, using



estimate (49),

‖�xu(t)‖2L2(�)6K‖∇xu(t)‖2L2(�) + C ′′eK′t‖u0‖2Dm0

�(51)

Using now the interpolation inequality ‖∇xv‖2L2(�) 6C ′′′‖�xv‖L2(�)‖v‖L2(�) and the fact that‖u(t)‖L∞(�)61, we deduce from (51) the inequality

‖u(t)‖2H 2(�)6C1eK′t‖u0‖2Dm0

�+ C ′

1 (52)

It then follows from (49), (50) and (52) that

‖f(u(t))− 〈f(u(t))〉‖2L2(�)6C2eK′t‖u0‖2Dm0

�+ C ′

2 (53)

and, consequently (due to (40) and Proposition A.2),

|〈f(u(t))〉|6Q�;T (‖u0‖Dm0�); ∀t6T (54)

where the monotonic function Q�;T is independent of �. Estimates (49) and (52)–(54) imply(46), hence Lemma 3.1.

The next lemma gives the uniform (with respect to �) smoothing property for the solutionsof Equation (38).

Lemma 3.2Let the above assumptions hold. Then, for every �¿0, there exists a constant C�;� such that

‖u(t)‖2Dm0�+

∫ t+1

t‖@tu(s)‖2H 1(�) ds6C�;� (55)

holds uniformly with respect to �∈[0; 1]; t¿� and u0∈Dm0� .

ProofWe �rst note that, rewriting Equation (38) in the form (50) arguing as in the end of theproof of Lemma 3.1, we can easily obtain estimate (55), provided that the following estimateholds:

�‖@tu(t)‖2L2(�) + ‖@tu(t)‖2H−1N (�)

6C�;�; ∀t¿� (56)

where C�;� is independent of � and �. Moreover, it is su�cient to verify (56) for t=� only(the general case can be reduced to this one by an appropriate time shift). To this end, weset w(t) := u(t)−m0, where m0 := 〈u(t)〉≡ 〈u0〉, multiply Equation (38) by w(t) and integrateover [0; �]×�. Then, employing 〈w(t)〉=0, we have∫ �

0‖∇xu(s)‖2L2(�) ds+

∫ �

0

∫�f(m0 + w(s)):w(s) dx ds

=−12[�‖w(s)‖2L2(�) + ‖w(s)‖2

H−1N (�)

]�0 (57)



Since ‖w(t)‖L∞(�)62 and |m0|61− �, we �nd, using estimates (A3) and (A4)∫ �

0‖∇xw(s)‖2L2(�) ds+

∫ �

0

∫�F(u(s)) dx ds+

∫ �

0

∫�|f(u(s))| dx ds6 C ′

�(�+ 1) (58)

where F(v) :=∫ v0 f(z) dz is the potential. Multiplying Equation (38) by t@tu(t), integrating

over [0; �]×� and using estimate (58), we obtain∫ �

0t(�‖@tu(t)‖2L2(�) + ‖u(t)‖2

H−1N (�)

) dt +12�‖∇xu(�)‖2L2(�) + �

∫�F(u(�)) dx

=12

∫ �

0‖∇xu(t)‖2L2(�) dt +

∫ �

0

∫�F(u(t)) dx dt6C ′′

�; � (59)

Di�erentiating now Equation (38) with respect to t, setting (t) := @tu(t), multiplying theequation that we obtain (i.e (47)) by t2(t) and integrating over [0; �]×�, we have

�2(�‖@tu(�)‖2L2(�) + ‖@tu(�)‖2H−1N (�)

) + 2∫ �

0t2‖@tu(t)‖2H 1(�)2 dt

62K∫ �

0t2‖@tu(t)‖2L2(�)dt +

∫ �

0t(�‖@tu(t)‖2L2(�) + ‖@tu(t)‖2H−1

N (�)) dt (60)

where we have used the inequality f′(v)¿−K . Using �nally (59) and the interpolation in-equality ‖v‖2L2(�)6C‖v‖H 1(�)‖v‖H−1

N (�) in order to estimate the right-hand side of (60), weobtain estimate (56), which �nishes the proof of Lemma 3.2.

It �nally remains to note that estimate (45) is an immediate consequence of (46) and (55)to �nish the proof of Theorem 3.1.

Remark 3.1We note that the proof of Theorem 3.1 does not make use of the space dimension. Conse-quently, the a priori estimate (45) remains valid in any space dimension.The next corollary gives an estimate of f(u) in the space L2([t; t+1]; L∞(�)), which is of

fundamental signi�cance for what follows.

Corollary 3.1Let the assumptions of Theorem 3.1 hold. Then the solution u(t) of problem (38) satis�esthe following estimate: ∫ t+1

t‖f(u(s))‖2L∞(�) ds6C�;�; ∀t¿� (61)

where the constant C�;� depends on � and �, but is independent of �, t and u(t).

ProofWe �rst note that it is su�cient to prove (61) for t=� only (the general case can be easilyreduced to this one by an appropriate time shift). To this end, we rewrite problem (38) as asecond-order parabolic equation:

�@tu−�xu+ f(u)= h := 〈f(u)〉 − (−�x)−1N @tu (62)



Then, according to estimate (55) and to the embedding H 3(�)⊂L∞(�), we have

‖h‖L2([t; t+1]; L∞(�))6C ′�; �; ∀t¿�=2 (63)

with an appropriate constant C ′�; �. We set h±(t) :=±‖h(t)‖L∞(�) and consider the following

two auxiliary ODEs:

�y′± + f(y±)= h±; ∀t¿�=2; y±(�=2) :=±‖u(�=2)‖L∞(�) (64)

The solutions y±(t) are well de�ned, due to assumption (44). Moreover, due to the comparisonprinciple for second-order parabolic equations, we have

y−(t)6u(t; x)6y+(t); ∀t¿�=2; ∀x∈� (65)

On the other hand, it follows from Proposition A.4 and estimate (63) that∫ �+1

�|f(y±(t))|2 dt6C ′′

�; � (66)

Estimate (61) is then an immediate consequence of (65) and (66). This �nishes the proof ofCorollary 3.1.

To conclude this section, we also derive L∞-bounds on the solutions of (38), with �¿0.We emphasize that, in contrast to all the previous estimates, these bounds are not uniformwith respect to � → 0+.

Corollary 3.2Let the assumptions of Theorem 3.1 hold and let � be strictly positive. Then u(t) satis�es

‖u(t)‖L∞(�)61− ��; �; �; ∀t¿� (67)

where the constant ��; �; �¿0 depends on �; � and �, but is independent of u(t).Moreover, if, in addition, we have

‖u(0)‖L∞(�)61− �0; for some �0¿0 (68)

then

‖u(t)‖L∞(�)61− �′�; �; �0 (‖u(0)‖Dm0�); ∀t¿0 (69)

where the constant �′�; �; �0¿0 depends on �; �; �0 and ‖u(0)‖Dm0�.

ProofIt follows from (55) that

�‖@tu(t)‖2L2(�)6C�;�; ∀t¿�=2

with an appropriate constant C�;�, and, consequently (due to the embedding H 2(�)⊂C(�)and to the fact that �¿0), we can improve (63) as follows:

‖h(t)‖L∞(�)6C ′�; �; �; ∀t¿� (70)



which, however, is not uniform with respect to �. Then, arguing as in the proof of Corollary3.1, but using estimate (A16) for the solutions y±(t) instead of estimate (A14), we �ndestimate (67).We now assume that (68) holds. Then, using (45) instead of (55), we can prove that

‖h(t)‖L∞(�)6C ′′�; �(‖u(0)‖Dm0

�); ∀t¿0 (71)

In order to obtain estimate (69), it is su�cient to consider the auxiliary solutions y±(t) of(64) with the initial conditions y±(0)=± (1− �0) de�ned on the interval [0;∞), and to useestimate (A13) in order to obtain upper and lower bounds on these solutions. This �nishesthe proof of Corollary 3.2.

Remark 3.2Using more accurate estimates on the term W (t) := (−�x)−1N @tu(t), we can slightly improvethe result of Corollary 3.1 as follows:∫ t+1

t‖f(u(s))‖8L∞(�) ds6C�;�; ∀t¿� (72)

where the constant C�;� is independent of �. Indeed, it follows from estimate (55) and theelliptic regularity of the Laplace operator that

‖W‖L∞([t; t+1]; H 1(�)) + ‖W‖L2([t; t+1]; H 3(�))6C ′′� ; ∀t¿� (73)

where C ′� is independent of �. Using an appropriate interpolation inequality, it follows from

this estimate that

‖W‖L8([t; t+1]; L∞(�))6C ′′� ; ∀t¿�

Using now estimate (A17) instead of (A14) and arguing as in the proof of Corollary 3.1, wehave estimate (72).

4. EXISTENCE AND UNIQUENESS OF SOLUTIONS.THE CORRESPONDING SEMIGROUPS

In this section, we prove the existence and uniqueness of solutions for problem (38) in thecorresponding phase spaces and obtain several auxiliary results which will be used in the nextsection below for the study of exponential attractors. We �rst give the de�nition of a solutionof problem (38).

De�nition 4.1A function u(t) is a solution of Equation (38) with u0∈Dm0

� if

u∈L∞([0; T ];Dm0� )∩C([0; T ]; H−1

N (�)); ∀T ¿0 (74)

and Equation (38) is satis�ed in the sense of distributions.



Remark 4.1We note that, for regular solutions u(t) (u(t)∈H 4(�) and ‖u‖L∞(R+×�)¡1), Equations (2)and (38) are equivalent. If u(t) only belongs to Dm0

� (as in De�nition 4.1), Equation (38) isthus viewed as a de�nition of solutions for Equation (2). It is not di�cult to verify that thisde�nition of a solution is equivalent to the standard de�nition of the variational solution forEquation (2) (see e.g. Reference [27]).

The following theorem gives the uniqueness of solutions in the above class.

Theorem 4.1Let the function f satisfy (7) and let u1(t) and u2(t) be two solutions of (38) (in the senseof De�nition 4.1). Then the following estimate is valid:

�‖u1(t)− u2(t)‖2L2(�) + ‖u1(t)− u2(t)‖2H−1N (�)

+∫ t+1

t‖u1(s)− u2(s)‖2H 1(�) ds

6CeKt(�‖u1(0)− u2(0)‖2L2(�) + ‖u1(0)− u2(0)‖2H−1N (�)

) + C�eKt |m1 −m2| (75)

for all t¿0, where mi := 〈ui(0)〉; i=1; 2, and the constants C;K and C� depend only on fand � and are independent of �∈[0; 1] and ui(0); i=1; 2.

ProofSince f′(v)¿−K , for every v∈(−1; 1), then

(f(v1)− f(v2)) (v1 − v2)¿− K |v1 − v2|2 ∀v1; v2∈(−1; 1) (76)

We now set v(t) := u1(t)− u2(t). This function satis�es the equation

�@tv+ (−�x)−1N @tv=�xv− (f(u1)− f(u2)) + 〈f(u1)− f(u2)〉; @nv|@� =0 (77)

Multiplying this equation by v(t), integrating over � and using (76) and the conservation law(40), we have

@t[�‖v(t)‖2L2(�) + ‖v(t)‖2H−1

N (�)] + ‖∇xv(t)‖2L2(�)

6 2K‖v‖2L2(�) + C|m1 −m2| · |〈f(u1(t))− f(u2(t))〉| (78)

Applying Gronwall’s inequality, together with an appropriate interpolation inequality, to (78),and using (58) in order to estimate the last term in the right-hand side of (78), we haveestimate (75) (analogously to (48) and (49)) and Theorem 4.1 is proven.We are now in a position to verify the existence of a solution for Equation (38) in the

class (74). We start with the case �¿0.

Corollary 4.1Let the function f satisfy (7) and let � be strictly positive. Then, for every u0∈Dm0

� , thereexists a unique solution u(t) of Equation (38) which satis�es estimates (45), (55), (61)and (67).



ProofWe �rst assume that the initial condition u0∈Dm0

� is separated from the singular points −1and 1 of the non-linearity f, i.e.

‖u0‖L∞(�)61− �; �¿0 (79)

Then, due to estimate (69), every solution u(t) of Equation (38) is a priori also separated fromthe singular points of the non-linearity f. Consequently, the existence of such a solution can beveri�ed exactly as in the case of regular non-linearities f (see e.g. References [13,15,23,25];see also References [12,17,19,20]).We now assume that u0∈Dm0

� , but that assumption (79) is not satis�ed. Then we approxi-mate the initial condition u0 = u0(x) by a sequence un

0(x) as follows:

un0(x) := nu0(x) (80)

where 0¡n¡1 is an arbitrary sequence such that limn→∞ n=1. Then, obviously, un0 ∈Dnm0

�and un

0 satis�es (79), with �=1 − n¿0. Moreover, it is not di�cult to verify, noting that�¿0 and using Lebesgue’s theorem, that

‖un0‖Dnm0

�→ ‖u0‖Dm0

�as n → ∞ (81)

Indeed, since �¿0, the operator (�+(−�x)−1N )−1 is bounded in L2(�) and, consequently, in or-

der to prove (81), it is su�cient to verify only that ‖un0‖H 2(�) → ‖u0‖H 2(�) and ‖f(un

0)‖L2(�) →‖f(u0)‖L2(�) as n → ∞. The �rst convergence is obvious, thanks to (44) and the conver-gence n → 1. The second one is also obvious, due to the inequalities |un

0(x)|6 |u0(x)| and‖f(u0)‖L2(�)¡∞ and Lebesgue’s theorem.Let un(t) be the corresponding solution of Equation (38) (whose existence is proven above).

Then it follows from (81) that these solutions satisfy estimates (45), (55), (61) and (67)uniformly with respect to n. Passing to the limit n → ∞ in these estimates and in Equations(38) for un and using (67) and (81), we obtain the desired solution u(t). The passage to thelimit n→∞ in Equation (38) arises no di�culty since, due to (67), the solutions un(t) areuniformly (with respect to n) separated from the singular points u=±1 of the non-linearityf if t¿0. This �nishes the proof of Corollary 4.1.

We now prove the existence of a solution of equation (38) in the limit case �=0.

Corollary 4.2Let the function f satisfy (7). Then the limit equation (38) with �=0 possesses a uniquesolution u(t) in the class (74) which satis�es estimates (45), (55) and (61) (with �=0).

ProofLet u0 belong to Dm0

0 . Then, obviously, u0∈Dm0� , for every �¿0, and

‖u0‖Dm0�6C‖u0‖Dm0

�(82)

where C is independent of �→ 0. We consider a sequence �n¿0; �n → 0, and the correspond-ing sequence u�n(t) of solutions of Equation (38) with �= �n, whose existence is proven inCorollary 4.1. Then, due to (82), estimates (45), (55) and (61) are satis�ed uniformly withrespect to �n. The desired solution u(t) can be obtained by passing to the limit �n → 0. Indeed,



since the u�n are uniformly bounded in the space L∞([0; T ]; H 2(�))∩H 1([0; T ]×�), for every�xed T (due to estimate (45)), we can assume, without loss of generality, that this sequenceconverges ∗-weakly to some function u in this space. Moreover, arguing in a standard manner(see e.g. References [19,20,28]), we can prove that f(u�n(t)) converges ∗-weakly to the limitfunction f(u(t)) in L∞([0; T ]; L2(�)). Passing now to the limit �n → 0 in Equations (38) foru�n(t), we verify that the limit function u(t) solves Equation (38) with �=0. Passing �nallyto the limit �n → 0 in estimates (45), (55) and (61), it follows that the limit function u(t)also satis�es these estimates and belongs, in particular, to the space (74). This �nishes theproof of Corollary 4.2.

Remark 4.2We note that, in contrast to the case �¿0, we cannot prove that the solution u(t) of Equation(38) is separated from the singular points −1 and 1 of the non-linearity f in the limitcase �=0 in general (even when (79) is satis�ed; see Section 7 for a discussion on someparticular cases), which is the main di�culty in the study of global and exponential attractors.Nevertheless, it follows from estimate (61) that

‖u(t)‖L∞(�)¡1; for almost every t∈R+ (83)

As shown in the next section, this will be su�cient to develop the theory of exponentialattractors for this class of equations. We note, however, that the term �@tu; �¿0, in (38)regularizes the classical Cahn-Hilliard equation (�=0) with respect to the separation from thesingular points (see also Section 7).It follows from Corollaries 4.1 and 4.2 that Equation (38) de�nes solving semigroups

St(�); �∈[0; 1], in the phase spaces Dm0� :

St(�) : Dm0� → Dm0

� ; St(�)u0 := u�(t) (84)

where u�(t) is the solution of Equation (38) with u�(0)= u0. Moreover, according to Theorem4.1, these semigroups are uniformly Lipschitz (with respect to the initial data). We considerthe spaces E(�) de�ned by the following norms:

‖v‖2E(�) := �‖v‖2L2(�) + ‖v‖2H−1

N (�)(85)

(i.e. E(�)=L2(�) if �¿0 and E(0)=H−1N (�)). By continuity, the semigroups St(�) can be

extended in a unique way to semigroups acting on the larger phase space Lm0 :

St(�) : Lm0 → Lm0

Lm0 := [Dm0� ]E(�) = {u0∈L∞(�); ‖u0‖L∞(�)61; 〈u0〉=m0}

(86)

where, for u0 =∈Dm0� , we set

St(�)u0 :=E(�)− limn→∞ St(�)un

0 (87)

and where un0 ∈Dm0

� is an arbitrary sequence satisfying the condition ‖un0 − u0‖E(�)→ 0 as

n→∞. Such a sequence exists, since Dm0� contains regular functions w0(x) (e.g. such that

w0∈C3(�); ‖w0‖L∞(�)¡1 and 〈w0〉=m0) and these regular functions are dense in Lm0 for the



topology of E(�). Moreover, u�(t) := St(�)u0 belongs to C([0; T ];E(�)), for every u0∈Lm0 , and,due to (55), we have

St(�) : Lm0 →Dm0� ; ∀t¿0 (88)

Thus, by passing to the limit n→∞, it is not di�cult to verify the formula u(t) := St(�)u0,which allows us to de�ne a solution of Equation (38) (in the sense of distributions) for everyinitial datum u0∈Lm0 .To conclude this section, we derive several useful properties of the semigroups constructed

above. We start with the uniform Holder continuity in the C-norm.

Corollary 4.3Let the function f satisfy (7) and let u1(t) := St(�)u10 and u2(t) := St(�)u20 be two solutions of(38), with initial data in Lm0 . Then, for every �¿0, the following estimate is valid:

‖u1(t)− u2(t)‖L∞(�)6C�eKt‖u1(0)− u2(0)‖�E(�); ∀t¿� (89)

where the constants C� ; K and � are independent of �; u10 and u20 .

ProofAccording to (55), we have

‖u1(t)− u2(t)‖H 2(�)6C�;�; ∀t¿� (90)

Estimate (89) is then an immediate consequence of Theorem 4.1, estimate (90) and thefollowing interpolation inequality:

‖v‖L∞(�)6C‖v‖1−�H 2(�)‖v‖�H−1

N (�)(91)

for an appropriate exponent 0¡�¡1, and Corollary 4.3 is proven.

The next corollary gives the uniform Lipschitz continuity of the solutions with respect to tin the H−1

N (�)-norm.

Corollary 4.4Let the non-linearity f satisfy (7) and let u�(t) := St(�)u0; u0∈Lm0 , be a solution of (38).Then, for every �¿0, the following estimate holds:

‖u�(t + s)− u�(t)‖E(�)6C�s; ∀t¿�; ∀s∈[0; 1] (92)

where the constant C� is independent of �; t; u0 and s. Moreover

‖u�(t + s)− u�(t)‖L∞(�)6C ′�s

�; ∀t¿�; ∀s∈[0; 1] (93)

where � is the same as in Corollary 4.3 and C ′� is independent of �; t; u0 and s.

ProofAccording to (55), we have

‖@tu�(t)‖E(�)6C1�; ∀t¿�



where C1� is independent of t; � and u0, which immediately implies estimate (92). Estimate(93) now follows from (92), the interpolation inequality (91) and the fact that the H 2(�)-normof the solutions is uniformly bounded on [�;+∞). This �nishes the proof of Corollary 4.4.

We �nally give an estimate on the di�erence of solutions of (38) with �¿0 and �=0,respectively, in terms of the parameter �.

Theorem 4.2Let f satisfy (7) and the initial datum u0∈Lm0 be such that

‖u0‖2H 1(�) + 2∫�F(u0) dx¡∞ (94)

Then the following estimate is valid for the solutions u�(t) := St(�)u0 and u0(t) := St(0)u0:

‖u�(t)− u0(t)‖2H−1N (�)

6CeKt(‖u0‖2H 1(�) + 2

∫�F(u0) dx

)�2 (95)

where the constants C and K are independent of � and u0 and F is the potential.

ProofThanks to de�nition (87), we can assume, without loss of generality, that u0∈Dm0

0 . Multiplyingthen Equation (38) for u�(t) by @tu�(t) and integrating over [0; T ]×�, we obtain, after standardtransformations

‖∇xu�(T )‖2L2(�) + 2∫�F(u�(T )) dx + 2

∫ T

0(�‖@tu�(t)‖2L2(�) + ‖@tu�(t)‖2H−1

N (�)) dt

= ‖∇xu�(0)‖2L2(�) + 2∫�F(u�(0)) dx (96)

We now set v�(t) := u�(t)− u0(t). This function satis�es the equation

(−�x)−1N @tv� =�xv� − (f(u�)− f(u0)) + 〈f(u�)− f(u0)〉+ �@tu�(t)

v�|t=0 = 0(97)

Multiplying (97) by v�(t), integrating over � and using inequality (76) and the fact that〈v�〉=0, we obtain

12 @t‖v�(t)‖2H−1

N (�)+ 1

2 ‖∇xv�(t)‖2L2(�)6K‖v�(t)‖2L2(�) + �2‖@tu�(t)‖2H−1N (�)

(98)

Applying Gronwall’s inequality to (98) and using (96) and a proper interpolation inequality,we �nally derive (analogously to (48) and (49)) estimate (95). Theorem 4.2 is proven.



5. EXPONENTIAL ATTRACTOR FOR THE CAHN-HILLIARD EQUATION

In this section, we apply the abstract scheme of Section 2 to construct a robust family ofexponential attractors for the Cahn-Hilliard equation (38) with singular potentials as �→ 0.The main result of the section is the following theorem.

Theorem 5.1Let the non-linearity f satisfy assumptions (7). Then, for every m0 satisfying |m0|61 −�; 0¡�¡1, there exists a robust family of exponential attractorsM�=M�(m0)⊂Dm0

� ; �∈[0; 1],for the semigroups (86) associated with equations (38), which satis�es the following proper-ties:

1. Semi-invariance: St(�)M� ⊂M�, for every t¿0.2. Uniform exponential attraction:

distE(�)(St(�)Lm0 ;M�)6Ce−�t (99)

where the positive constants C and � are independent of �.3. Finite dimensionality:

H �(M�;E(�))6M ln1�+M ′ (100)

where the constants M and M ′ are independent of � and �¿0.4. Continuity as �→ 0:

distsym;E(�)(M�;M0)6C�� (101)

where C¿0 and 0¡�¡1 are independent of �.

ProofWe apply Theorem 2.1 in order to construct the desired family M�. To this end, we set

B�=B0≡B(R) :={u0∈Lm0 ; ‖∇xu0‖2L2(�) + 2

∫�F(u0) dx6R

}(102)

where R¿0 is large enough so that, due to (55)

S1(�)Lm0 ⊂B(R); ∀�∈[0; 1] (103)

Thus, it is su�cient to construct the exponential attractors M� in the phase space B(R) insteadof Lm0 . Moreover, it follows from estimate (96) that

St(�)B(R)⊂B(R); for every �∈[0; 1] and t¿0 (104)

We now set S� := S1=N (�), where N ∈N is su�ciently large and will be speci�ed below,and verify the conditions of Theorem 2.1 for this family of maps. According to (104), themaps S� are well de�ned on B� :=B(R). Moreover, since the H 1-norm of u0 is uniformlybounded on B(R), estimate (10) (with � replaced by �1=2) is an immediate consequence ofde�nition (85) of the norm in E(�). The Lipschitz continuity (15) is veri�ed in Theorem 4.1.Furthermore, if we set �� := Id, then estimate (12) is satis�ed (with � replaced by �1=2), due



to Theorem 4.2 and estimate (10). Property (14) is also obviously satis�ed for every �¿0,since the embedding H 1(�)⊂L2(�) is compact.Thus, it remains to construct the set C� ⊂B� which possess the recurrence property (13)

and �nd the family of spaces E1(�) which satisfy (9) and for which we have estimate (11).Let u0∈B(R) be an arbitrary point and let u�(t) := St(�)u0 be the corresponding trajectory.

Then, according to estimate (61), we have∫ 3=4

1=4‖f(u�(t))‖2L∞(�) dt6C (105)

where C is independent of u0 and �. Consequently, there exists a time T =T (u0; �)∈[ 14 ; 34 ]such that

‖f(u�(T ))‖L∞(�)6(2C)1=2

and, consequently,

‖u�(T )‖L∞(�)61− 4�f (106)

where the strictly positive constant �f depends only on C and f. Moreover, it follows fromthe Holder continuities (89) and (93) that there exist su�ciently small constants s¿0 and�¿0 which are independent of �; T and u0 such that

‖St(�)u′0‖L∞(�)61− 2�f; ∀t∈[T; T + s]; ∀u′0∈B(u0; �;E(�))∩B� (107)

Setting now

C� := S1=4(�)B(R)∩{v∈Lm0 ; ‖v‖L∞(�)61− 2�f} (108)

and �xing N ∈N large enough, we can prove that there exists M=M (u0; �)∈N; M6N − 1,such that

SM=N (�)(B� ∩B(u0; �;E(�)))⊂C� (109)

Since u0∈B(R) is arbitrary, (109) implies the recurrence property (13) for the maps S� :=S1=N (�). We �nally de�ne the family of spaces E1(�) by the following norms:

‖u0‖2E1(�) := �‖v‖2H 1(�) + ‖v‖2L2(�) (110)

(i.e. E1(�)=H 1(�) if �¿0 and E1(0)=L2(�)). Then the entropy condition (9) is also,obviously, satis�ed and it only remains to verify the uniform smoothing property (11). Letu0∈C� be an arbitrary point and let u�(t) := St(�)u0 be the corresponding solution. Then, sinceC� ⊂ S1=4(�)B(R), we have, thanks to the Holder continuity (93) and assuming that N is largeenough

‖u�(t)‖L∞(�)61− �f; ∀t∈[0; 1=N ] (111)

Thus, in order to verify assumption (11), it su�ces to prove the following lemma.

Lemma 5.1Let the above assumptions hold and let u1� (t) and u2� (t) be two solutions of (38) which satisfythe inequality

‖ui�(t)‖L∞(�)61− �; ∀t∈[0; 1=N ] (112)



for i=1; 2 and for some constant �∈]0; 1[. Then the following estimate holds:t‖u1� (t)− u2� (t)‖2E1(�)6C�eKt‖u1� (0)− u2� (0)‖2E(�); ∀t∈[0; 1=N ] (113)

where the constants C� and K are independent of �.

ProofThe function v�(t) := u1� (t)− u2� (t) is solution of the equation

�@tv� + (−�x)−1N @tv�=�xv� − l�v� + 〈l�v�〉 (114)

where l�(t) :=∫ 10 f′(su1(t) + (1− s)u2(t)) ds satis�es, in view of (112)

‖l�(t)‖L∞(�)6C�; ∀t∈[0; 1=N ] (115)

for some constant C� which is independent of �. Multiplying Equation (114) by t�xv�(t),integrating over � and using (115), we �nd, after simple calculations

12@t[t‖v�(t)‖2E1(�)]6C ′

�t‖v�(t)‖2L2(�) + 12‖v�(t)‖2E1(�) (116)

where t61=N and C ′� is independent of �. Applying Gronwall’s inequality to (116) and using

(75) in order to estimate the right-hand side of (116), we then obtain estimate (113) andLemma 5.1 is proven.Thus, all the assumptions of Theorem 5.1 are satis�ed for the discrete maps S� := S1=N (�)

and, consequently, there exists a uniform family of discrete exponential attractors Md� ⊂ S1=NB�

⊂Dm0� for the maps S�, which satis�es all the assertions of Theorem 5.1. The desired family

of exponential attractors for the continuous semigroups St(�) can be now constructed by thefollowing standard expression:

M� :=⋃

t∈[0;1=N ]St(�)Md

� (117)

Since the semigroups St(�) are uniformly (with respect to t and u0) Lipschitz on Dm0� , (117)

de�nes indeed the desired family of exponential attractors (see Reference [38] for details) andTheorem 5.1 is proven.

Remark 5.1Since all the exponential attractors M� are uniformly bounded in H 2(�), it follows from theinterpolation inequality (91) that estimates (99)–(101) remain valid (with di�erent constants)if we replace the spaces E(�) by the space L∞(�). Thus, the family M�; �∈[0; 1], is a robustfamily of exponential attractors for problems (38) in the space L∞(�) as well.

Remark 5.2We recall that an exponential attractor always contains the global attractor. Consequently,Theorem 5.1 implies the existence of the �nite dimensional global attractors A�=A�(m0) forthe semigroups (86) associated with Equations (38) and the following uniform estimate ontheir fractal dimension holds:

dimF(A�; L∞(�))6C¡∞ (118)

where the constant C is independent of �∈[0; 1].



6. DEPENDENCE OF THE ATTRACTORS ON THE AVERAGE m0

In this section, we study the dependence of the global attractors A�=A�(m0) and of theexponential attractors M�=M�(m0) of the semigroups (86) associated with problems (38) onthe average m0 of the initial datum (de�ned in (41)). We �rst note that all the estimatesobtained in Sections 3 and 4 depend on the parameter �¿0 introduced in (41). Furthermore,all these estimates diverge as �→ 0. Therefore, estimates (99)–(101) for the exponentialattractors M�(m0) also a priori diverge as |m0|→ 1. Nevertheless, it is possible to obtain auniform (with respect to m0) family of exponential attractors M�(m0) based on the followingsimple proposition which shows that Equation (38) is uniformly exponentially stable if |m0|is large enough.

Proposition 6.1Let the non-linearity f satisfy assumptions (7) and let, in addition, the average m0 := 〈u0〉 ofthe initial datum u0 satisfy the condition |m0|¿M0, where M0¡1 is de�ned by formula (A2).Then there holds the following estimate on the solution u(t) := St(�)u0 of problem (38):

‖u(t)−m0‖2E(�)6Ce−�t‖u0 −m0‖2E(�) (119)

where the positive constants C and � are independent of u0; � and m0.

ProofMultiplying Equation (38) by w(t) := u(t)−m0, integrating over � and employing 〈w(t)〉=0,we have

12@t‖w(t)‖2E(�) + ‖∇xw(t)‖2L2(�) +

∫�(f(u(t))− f(m0)) · (u(t)−m0) dx = 0 (120)

Since |m0|¿M0, estimate (A1) implies that the last term in (120) is non-negative and, con-sequently

12 @t‖w(t)‖2E(�) + ‖∇xw(t)‖2L2(�)60 (121)

Moreover, since 〈w〉=0, then ‖w‖E(�)6 ‖∇xw‖L2(�), for some positive constant dependingonly on �. Applying Gronwall’s inequality to (121), we �nally obtain estimate (119) andProposition 6.1 is proven.

We note that, obviously, u(t) ≡ m0 solves (38). Thus, Proposition 6.1 implies that every so-lution u(t) of this equation converges to this spatially homogeneous equilibrium exponentiallyif |m0|¿M0. Therefore

A�(m0)= {m0} if |m0|¿M0 (122)

Moreover, this attractor is exponential and is independent of �. Consequently, we can rede�nethe family of exponential attractors M�(m0) constructed in Theorem 5.1 as follows:

M�(m0) :=

{M�(m0) if |m0|6M0

{m0} if |m0|¿M0

(123)



and the family of exponential attractors that we obtain is uniform with respect to m0 (i.e. allthe constants in estimates (5.1)–(5.3) are independent of m0∈(−1; 1)). In order to simplifythe notations, we write below M�(m0) again instead of M�(m0).Using the exponential attractors constructed above, we are now in a position to prove

the existence of the universal (global) attractor A� for Equation (38), which attracts all thesolutions of this equation with all possible averages m0∈[−1; 1]. To this end, we �rst extendthe action of the semigroups (86) to the points u0≡± 1 by the following natural formula:

St(�)(± 1) :=±1 (124)

Employing this extension, the semigroups (86) will act on the whole unit ball of L∞(�):

St(�) : L→ L; L :=B(0; 1; L∞(�))=⋃

m0∈[−1;1]Lm0 (125)

The main result of this section is the following theorem which establishes the existence ofthe �nite dimensional global attractor A� for St(�).

Theorem 6.1Let the non-linearity f satisfy assumptions (7) and let the set L be endowed with the topologyof E(�). Then the semigroup (125) possesses the global attractor A� which has the followingstructure:

A�=⋃

m0∈[−1;1]A�(m0) (126)

Moreover, its fractal dimension is �nite and uniformly bounded with respect to �:

dimF(A�; L∞(�))6C (127)

where C is independent of �∈[0; 1].ProofWe �rst prove the existence of the global attractor A�, for the semigroup St(�) de�ned by(125). To this end, we note that, due to Proposition 6.1, it is su�cient to verify the existenceof the global attractor only for the following restriction of the semigroup (125):

St(�) : LM0 → LM0 ; LM0 :=⋃

|m0|6M0

Lm0 (128)

where M0¡1 is de�ned in (A2). It remains to note that the semigroup (128) is Holdercontinuous for the E(�)-metric (due to estimate (75)) and possesses a compact absorbingset (due to estimate (55)). Consequently, according to the standard existence theorem (seee.g. References [17,27,31,41–43]), the semigroup possesses a compact global attractor. Thus,the existence of A� is veri�ed. Description (126) now follows from the general fact that theglobal attractor is generated by all the complete bounded trajectories of the correspondingsemigroup. So, it only remains to verify the �nite dimensionality (127). To this end, we needthe following lemma.



Lemma 6.1Let the above assumptions hold. Then, for every m1; m2∈[−M0; M0] with m1 �=m2, the fol-lowing estimate holds:

distL∞(�)(St(�)Lm1 ;M�(m2))6C|m1 −m2|�; ∀t¿1 + L ln2

|m1 −m2| (129)

where the positive constants �; C and L are independent of t; � and m1; m2.

ProofLet u0∈Lm1 be an arbitrary point. Then, obviously, v0 := u0 +m2 −m1∈Lm2 and, according toestimate (75), we have

‖u(t)− v(t)‖E(�)6CeKt |m1 −m2|1=2 (130)

where u(t) := St(�)u0; v(t) := St(�)v0 and the constants K and C are independent of �. On theother hand, since M�(m2) is an exponential attractor in Lm2 , then, due to (99), we have

distE(�)(v(t);M�(m2))6Ce−�t ; �¿0 (131)

and, consequently,

distE(�)(St(�)Lm1 ; M�(m2))6C(eKt |m1 −m2|1=2 + e−�t) (132)

Setting now t=T := (1=2(K + �)) ln(2=|m1 −m2|) + 1, we obtaindistE(�)(ST (�)Lm1 ;M�(m2))6C ′|m1 −m2|�1 (133)

where �1 := �=2(K + �) and the constant C ′ is independent of � and m1; m2. Moreover, due toestimate (55) and to the interpolation inequality (91), it follows from (133) that

distL∞(�)(ST (�)Lm1 ;M�(m2))6C ′′|m1 −m2|��1 (134)

where � is de�ned in (91). Finally, since St(�)Lm1 ⊂ Lm1 for t¿0, (134) implies estimate (129),for every t¿T , which �nishes the proof of Lemma 6.1.

We are now in a position to verify the �nite dimensionality of the attractor A� and �nishthe proof of Theorem 6.1. To this end, we note that, due to Proposition 6.1 and formula(122), it is su�cient to verify the �nite dimensionality of the following set:

A�(M0) :=⋃

|m0|6M0

A�(m0) (135)

Moreover, it follows from estimate (129) and the embedding A�(m1)⊂ St(�)Lm1 for every t¿0,that

distL∞(�)(A�(m1);M�(m2))6C|m1 −m2|� (136)

for every m1; m2∈[−M0; M0]. Now, let us �x an arbitrary �¿0 and let V� ⊂ [−M0; M0] be a(�=2C)1=�-net of the interval [−M0; M0]. For every m∈V�, we also �x a �=2-net W�;m ⊂M�(m)of the exponential attractor M�(m) for the metric of L∞(�) (which exists, thanks to Theorem



5.1 and Remark 5.1). Then, according to (136), the system of �-balls centred at the points of⋃m∈V� W�;m covers the set (135). Therefore, due to estimate (100) and Remark 5.1, we have

H �(A�(M0); L∞(�))6H (�=2C)1=�([−M0; M0];R) +M ln12�+M ′ (137)

which implies the uniform (with respect to �∈[0; 1]) �nite dimensionality of the set (135)and �nishes the proof of Theorem 6.1.

Remark 6.1We recall that Theorem 4.1 gives the Lipschitz continuity of the semigroup St(�) with respectto initial data belonging to Lm0 with the same average only, whereas for initial data withdi�erent averages, we only have Holder continuity with Holder exponent 12 . That is the reasonwhy we cannot apply the results of Section 1 in order to construct exponential attractors forthe semigroup (128) and verify in this way that the global attractor is �nite dimensional.Therefore, we have to use di�erent arguments in the proof of Theorem 6.1 in order to verifythis property.

Remark 6.2Analogously to formula (126), we can try to construct a robust (with respect to �) family ofexponential attractors for the semigroups (125) in the following ‘naive’ way:

M� :=⋃

m0∈ [−1;1]M�(m0) (138)

Then the uniform exponential attraction property provided by estimates (99) and (101), i.e.continuity as �→ 0, will obviously be satis�ed for this family of ‘exponential attractors’. Un-fortunately, de�nition (138) does not allow to control the fractal dimension of these attractorsand, therefore, does not allow to construct exponential attractors for the semigroups (125).Nevertheless, there exists a slightly more sophisticated way to construct a robust (with respectto �) family of exponential attractors M� for the semigroups (125), based on the exponentialattractors M�(m0) constructed above, namely, by employing

M′� :=

∞⋃n=1

Sn(�)

{2Kn⋃

l=−2Kn

M�

(12Kn

)}; M� := [M′

� ]L∞(�) (139)

Indeed, by standard arguments, it can be veri�ed that de�nition (139) gives a family ofexponential attractors for the semigroups (125), which satis�es (99)–(101), uniformly withrespect to � if the constant K is large enough. We will give the details of the proof of thisresult in a forthcoming article.

7. UNIFORM L∞-BOUNDS ON THE EXPONENTIAL ATTRACTORS

In this concluding section, we discuss the problem of obtaining uniform (with respect to �)L∞-bounds on the solutions of problem (38) as in (69), which allow to separate the solutionsof Equation (38) from the singular points ±1 of the non-linearity f. Estimates of this typeare crucial for the study of equations of the type (1) with singular potentials, since they allow



to reduce the problem to one with regular potentials, whose theory is now developed (seee.g. References [12,17,19,22,26,27] and the references therein). Unfortunately, in the three-dimensional case (even for the logarithmic non-linearities (3)), we are not able to obtain suchestimates for general non-linearities f satisfying (7). Nevertheless, in this section, we obtainsuch estimates when f has a su�ciently strong singularity at u=± 1 and we also discuss thecase of lower dimensions dim�¡3. We begin with the following result.

Theorem 7.1Let the non-linearity f satisfy (7) together with the following additional assumption:

|f′(u)|6C(|f(u)|2 + 1) (140)

Then, for every �¿0, there exists a positive constant �= ��;� which is independent of �∈[0; 1]such that

‖u(t)‖L∞(�)61− �; ∀t¿� (141)

for every solution u(t) of (38) with |〈u0〉|61− �.

ProofDue to estimate (55), we can assume, without loss of generality, that u0∈Dm0

� and that itsnorm in this space is uniformly bounded. Moreover, due to estimate (72), we can also assumethat ∫ t+1

t‖f(u(s))‖8L∞(�) ds6C�;�; ∀t¿0 (142)

Then assumption (140) implies that

‖f′(u)‖L4([t; t+1]; L∞(�))6C ′�; �; ∀t¿0 (143)

We now rewrite Equation (47) for �(t) := @tu(t) as follows:

�@t�+ (−�x)−1N @t�−�x�=Hu := 〈f′(u)�〉 − f′(u)�

�|t=0 =�0; @n�|@� =0 (144)

We then estimate the L2-norm of the function Hu(t). To this end, we note that, due to estimate(45), we have

‖�‖L∞([t; t+1]; H−1N (�)) + ‖�‖L2([t; t+1]; H 1(�))6C ′′

�; �; ∀t¿0 (145)

and, consequently,

‖�‖2L4([t; t+1]; L2(�))6C‖�‖L∞([t; t+1]; H−1N (�))‖�‖L2([t; t+1]; H 1(�))6C ′′′

�; � ; ∀t¿0 (146)

due to a proper interpolation inequality. Estimates (143) and (146), together with Holder’sinequality, imply that

‖Hu‖L2([t; t+1]×�)6K�;�; ∀t¿0 (147)



where the constant K�;� is independent of � and t. Multiplying Equation (144) by t�x�(t),integrating over � and using estimates (145) and (147), we obtain, after some calculations

t‖@tu(t)‖2L2(�)6K ′�; �; ∀t¿0 (148)

where K ′�; � is also independent of � and t. Arguing �nally as in the proof of Corollary 3.2, but

using (148) in order to estimate the function h(t) (de�ned in (62)), we obtain the analogueof estimate (70), but with a constant C ′

�; � which is independent of �. Consequently, estimates(65) and (A16) imply (141), which �nishes the proof of Theorem 7.1.

Corollary 7.1Let the assumptions of Theorem 7.1 be satis�ed. Then the exponential attractors M�=M�(m0),�∈[0; 1], |m0|61− �, constructed in Theorem 5.1 satisfy

‖M�‖L∞(�)61− �� (149)

where the positive constant �� is independent of �∈[0; 1].Inequality (149) is an immediate consequence of (141) and of the fact that M�(m0)⊂

S1=N (�)Lm0 .

Remark 7.1Assumption (141) is satis�ed, for instance, for the following class of non-linearities:

f(u)=�(u)

(1− u2)�; where �∈C1([−1; 1]); �(± 1) �=0 and �¿1 (150)

Consequently, we have estimate (141) in that case. Moreover, we can slightly relax assumption(150) (say, up to �¿ 3

7 instead of �¿1). Nevertheless, we do not know how to adapt thisscheme for obtaining estimate (141) for the logarithmic non-linearities (3) which are the mostimportant ones from the physical point of view (see Reference [2]).To conclude, we brie�y consider the case of lower dimensions dim�¡3. We start with

the simplest case, dim�=1.

Proposition 7.1Let dim�=1 and let the non-linearity f satisfy assumptions (7). Then every solution u(t)of Equation (38) satis�es estimate (141).

ProofSince in one space dimension we have the Sobolev embedding H 1(�)⊂C(�), the uniform(with respect to �) analogue of estimate (70) follows from estimate (55) which gives uniformbounds on ‖@tu(t)‖H−1

N (�). Arguing now as in the proof of Corollary 3.2, we derive estimate(141) and �nish the proof of Proposition 7.1.

Thus, in one space dimension we have estimates (141) and (149) for all the admissiblenon-linearities f, in particular, for the logarithmic non-linearities (3).



We now consider the case dim�=2. In that case, we do not have the embedding H 1(�)⊂C(�) and, consequently, we are not able to obtain estimate (141) for all the non-linearitiessatisfying (7). Nevertheless, using the embedding of H 1(�) into an appropriate Orlicz space,we obtain this result for a wide class of non-linearities, which includes the logarithmic non-linearities (3).

Theorem 7.2We assume that dim�=2 and that the non-linearity f satis�es assumptions (7) and thefollowing additional condition:

|f′(u)|6eC1|f(u)|+C2 (151)

with some positive constants C1 and C2. Then every solution u(t) of Equation (38) satis�esestimate (141).

The proof of this result is based on the following lemma.

Lemma 7.1We assume that dim�=2 and that the non-linearity f satis�es assumptions (7). Then, forevery L¿0 and every �¿0, the following estimate holds:∫

[t; t+1]×�eL|f(u(s))| dx ds6CL;�; � (152)

where t¿� and the constant C is independent of � and of the initial datum u0 satisfying|〈u0〉|61− �.

ProofLet u(t) be an arbitrary solution of (38). Then, due to estimates (55), (61) and (83), thereexists a strictly positive constant �=�u6� such that

‖u(�)‖Dm0�6C�;� and ‖u(�)‖L∞(�)61− ��;� (153)

where the positive constants ��;� and C�;� are independent of � and u(t). We now rewriteEquation (38) in the form (62). Then, due to estimate (45), the function h(t) in (62) satis�es

‖h(t)‖H 1(�)6C ′�; �; ∀t¿� (154)

where C ′�; � is independent of �. We can also assume, without loss of generality, that

f′(v)¿0; ∀v∈(−1; 1) (155)

since otherwise it is su�cient to add the term Ku to both parts of Equation (62), where Kis su�ciently large, and consider the new function f(u) :=f(u) + Ku which now satis�esassumption (155).



Let L¿0 be an arbitrary positive number. We multiply (62) by f(u(t))eL|f(u(t))|, integrateover [�; �+ 1]×� and, after elementary transformations, have

�∫�FL(u(�+ 1)) dx +

∫[�; �+1]×�

|∇xu(t)|2f′(u(t))[1 + L|f(u(t))|]eL|f(u(t))| dx dt

+∫[�; �+1]×�

|f(u(t))|2eL|f(u(t))| dx dt

= �∫�FL(u(�)) dx +

∫[�; �+1]×�

h(t)f(u(t))eL|f(u(t))| dx dt (156)

where FL(v) :=∫ v0 zeL|z| dz. Since f′(v)¿0, (153) and (156) imply the estimate∫

[�; �+1]×�|f(u(t))|2eL|f(u(t))| dx dt

6CL; k; � +∫[�; �+1]×�

|h(t)| · |f(u(t))|eL|f(u(t))| dx dt (157)

where the constant CL;�; � is independent of �. In order to estimate the last term in the right-hand side of (157), we use the following version of Young’s inequality:

a · b6�(a) + �(b); ∀a; b¿0 (158)

where

�(s) := es − s− 1; �(s) := (s+ 1) ln(1 + s)− s (159)

see e.g. References [44–46]. Applying this inequality to a :=N |h(t)| and

b :=N−1|f(u(t))|eL|f(u(t))|

where N =N (L) is su�ciently large, we obtain

|h(t)| · |f(u(t))|eL|f(u(t))|6 12 |f(u(t))|2eL|f(u(t))| + eN |h(t)| + C (160)

where C is independent of � and u(t). Thus, inserting (160) into the right-hand side of (157),we have ∫

[�; �+1]×�|f(u(t))|2eL|f(u(t))| dx dt 62

∫[�; �+1]×�

eN |h(t)| dx dt + C ′�; �; L (161)

We now recall that, due to the Orlicz embedding theorem (see e.g. References [44–46]), wehave the estimate ∫

�eN |v(x)| dx6e�N (‖v‖

2H1(�)

+1) (162)



for every v∈H 1(�), where �N only depends on N and �. Estimates (154), (161) and (162)�nally imply that ∫

[�; �+1]×�|f(u(t))|2eL|f(u(t))| dx dt6C ′′

�; �; L (163)

where the constant C ′′′�; �; L is independent of �∈[0; 1]. Estimate (163) implies (152) for t=�,

which, in turn, implies estimate (152) for every t¿� by employing an appropriate time shift.This �nishes the proof of Lemma 7.1.We are now in a position to prove Theorem 7.2. Indeed, estimates (151) and (152) imply

that, for every p¿1, there exists a constant Cp;�; � which is independent of � such that

‖f′(u)‖Lp([t; t+1]×�)6Cp;�; �; ∀t¿� (164)

Arguing as in the proof of Theorem 7.1, but using now estimate (164) instead of (143),we can easily verify the analogues of estimates (147) and (148) and �nish the proof ofTheorem 7.2.

Corollary 7.2In the case dim�=2, every solution u(t) of Equation (38) with a logarithmic non-linearity (3)satis�es estimate (141) and, consequently, the exponential attractors M�(m0) for this equationconstructed in Theorem 5.1 satisfy estimate (149).

Indeed, for dim�=2, any function of the form (3) satis�es condition (151) of Theorem 7.2.

Remark 7.1Since the non-linearities (3) belong to C∞(−1; 1) and the domain � with dim �=2 is smooth,estimate (141) implies the smoothing property

St(�) : Lm0 →C∞(�) (165)

and, consequently, M�(m0) is uniformly bounded (with respect to �) in C∞(�).

APPENDIX A: SOME AUXILIARY ESTIMATES

In this appendix, we formulate and prove several properties on the singular interaction functionf(u), which are essential in our study of the Cahn-Hilliard equation (2). We begin with thefollowing proposition.

Proposition A.1Let the function f∈C1(−1; 1) satisfy assumptions (7). Then:1. For every m∈(−1; 1), there exists a constant Cm, Cm6C (where C is independent of

m) such that

[f(m+ v)− f(m)]:v¿− Cm|v|2; for every v∈(−1−m; 1−m) (A1)



In particular, if |m|¿M0, where

M0 :=max{|w|: ∃z∈(−1; 1); f(z)=f(w) and f′(z)=0}¡1 (A2)

then Cm=0.2. Let F(u) :=

∫ u0 f(v) dv be the potential. Then, for every m∈(−1; 1) and w∈(−1 − m;

1−m), the following estimate holds:

F(m+ w)6f(m+ w):w + F(m) + C (A3)

where the constant C is independent of m and w.3. For every m∈(−1; 1), there exist positive constants Cm and C ′

m such that

|f(v+m)|6Cmf(v+m)·v+ C ′m (A4)

for every v∈(−1−m; 1−m).

ProofDue to assumption (7)2, we have f′(v)¿−C, for every v∈(−1; 1), which immediately impliesestimate (A1). Let us now verify that Cm=0, if |m|¿M0. We �rst note that M0¡1, due toassumption (7)2. Let us assume that m¿M0 (the case m6−M0 can be treated analogously).Then the function f is monotonic (f′¿0) on the interval [m; 1) and, consequently,

f(m+ v)− f(m)¿0; ∀v¿0; m+ v¡1 (A5)

On the other hand, it follows from the de�nition of M0 that f(m)¿f(z) for every z6m, ifm¿M0, and, consequently,

f(m+ v)− f(m)60; ∀v60; v+m¿−1 (A6)

Estimates (A5) and (A6) imply that (A1) holds with Cm=0.Let us now verify estimate (A3). To this end, we recall that, due to conditions (7), the

function f can be split as follows:

f=f0 + � (A7)

where f0 is monotonic (f′0¿0) and � is bounded on (−1; 1). We can thus assume, without

loss of generality, that f′(u)¿0, ∀u∈(−1; 1). In that case, (A3) is an immediate consequenceof the following formula:

F(m+ w)− F(m)− f(m+ w):w=∫ 1

0sf′(m+ sw)|w|2 ds¿0

Let us now verify estimate (A4). We only consider the case m¿0 (the case m¡0 can betreated analogously). Moreover, due to the splitting (A7), we can also assume that f ismonotonic and that f(0) = 0. Then we have two possibilities:{

1: m+ v∈[−1 +m=2; (1 +m)=2]

2: m+ v =∈ [−1 +m=2; (1 +m)=2]



In the �rst case, the value m+ v is separated from the singular points of the non-linearity fand, consequently,

|f(v+m)|6C ′m

where C ′m is independent of v. In the second case, we have |v|¿(1−m)=2 and v · (m+v)¿0.

Noting that v ·f(v+m)¿0, we �nd the estimate

|f(v+m)|6 21−m

f(v+m) · v

which �nishes the proof of estimate (A4) and Proposition A.1 is proven.

The next proposition allows us to estimate the average of f(v(x)) in terms of L1-boundsof f(v(x))− 〈f(v)〉 and of the average of v.Proposition A.2Let the function f satisfy (7), � be a bounded domain in Rn and the functions v∈L∞(�);‖v‖L∞(�)61; f(v)∈L1(�), and ∈L1(�) satisfy

f(v(x))− 〈f(v)〉= (x); |〈v〉|61− �; x∈� (A8)

with some �¿0. Then there exist positive constants C� and C ′� (depending only on �; f and

�) such that

|〈f(v)〉|6C�‖‖L1(�) + C ′� (A9)

ProofWe set m := 〈v〉 and w := v − m. Multiplying Equation (A8) by w, integrating over � andemploying ‖w‖L∞(�)62, we have∫

�f(m+ w(x)) ·w(x) dx62‖‖L1(�) (A10)

Estimate (A9) is then an immediate consequence of (A4) and (A10) and Proposition A.2 isproven.

To conclude, we obtain several estimates on the solutions of the following ODE, which arenecessary to apply the comparison principle to the Cahn-Hilliard equation (see Section 3):

�y′ + f(y)= h; y(0)=y0; |y0|¡1 (A11)

We begin with the following result.

Proposition A.3Let the function f satisfy (7) and let us assume that �¿0 and that

|y0|61− �0 and h∈L∞([0; T ]) (A12)

with some positive constants �0 and T . Then there exists a constant �= �(�0; ‖h‖L∞([0; T ]))¿0which is independent of � and T , such that

|y(t)|61− �; ∀t∈[0; T ] (A13)



This assertion is a consequence of (7)1 and of the comparison principle for the solutionsof �rst-order ODEs.We also formulate the following version of the ‘smoothing’ property.

Proposition A.4Let the non-linearity f satisfy (7) and let, in addition, h belong to L2([0; T ]). Then everysolution y(t) of Equation (A11) satis�es∫ T

0t|f(y(t))|2 dt6CT (1 + ‖h‖2L2([0;T ])) (A14)

where the constant CT is independent of �¿0 and y(0).

ProofMultiplying Equation (A11) by tf(y(t)) and integrating over [0; T ], we have

�TF(y(T )) +12

∫ T

0t|f(y(t))|2 dt6�

∫ T

0F(y(t)) dt +

12

∫ T

0t|h(t)|2 dt (A15)

We note that, due to assumptions (7), we have F(u)¿− C. So, it only remains to estimatethe �rst integral in the right-hand side of (A15). To this end, we multiply Equation (A11)by y(t), integrate over [0; T ] and note that |y(t)|61 to obtain the estimate∫ T

0f(y(t)):y(t) dt=

�2(y(0)2 − y(T )2) +

∫ T

0h(t):y(t) dt6C ′

T (1 + ‖h‖2L2([0; T ]))

where C ′T is independent of �. Combining this estimate with (A3) and (A15), we derive (A14)

and Proposition A.4 is proven.

Combining Propositions A.3 and A.4, we have the following result.

Corollary A.1Let the non-linearity f satisfy (7) and let h belong to L∞([0; T ]). Then, for every �¿0,there exists a constant �= �(�; ‖h‖L∞([0; T ]))¿0 which is independent of �; T and y(0), suchthat

|y(t)|¡1− �; ∀t∈[�; T ] (A16)

Indeed, it follows from estimate (A14) that there exist a time T�∈[�=2; �] and a constantK =K(�; ‖h‖L2([0; T ])) such that

|f(y(T�))|6K

and, consequently, there exists �0 = �0(f;K)¿0 such that |y(T�)|61−�0. Estimate (A16) isnow a consequence of Proposition A.3.

Remark A.1It is not di�cult to verify that, for every y(0) satisfying |y(0)|¡1, there exists a uniquesolution y(t) of problem (A11), which satis�es |y(t)|¡1 for every t∈R.



Remark A.2If, under the assumptions of Proposition A.4, we have, in addition, h∈Lp([0; T ]) for somep¿2, then, multiplying Equation (A11) by f(y(t))|f(y(t))|p−2 and arguing analogously, wecan prove that, for every �¿0, the following estimate holds:∫ T

�|f(y(t))|p dt6C�(1 + ‖h‖pLp([0; T ])) (A17)

where the constant C� is independent of � and y(t).

ACKNOWLEDGEMENTS

The authors wish to thank Professor W. Wendland for his careful reading of this article and for hismany remarks which helped improve the presentation of the article. This work was partially supportedby INTAS project no. 00-899 and CRDF grant no. 10545.

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Robust exponential attractors for Cahn-Hilliard type equations with singular potentials

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