A new formulation of the Cahn–Hilliard equation

Nonlinear Analysis: Real World Applications 7 (2006) 285–307www.elsevier.com/locate/na

A new formulation of the Cahn–Hilliard equation

Alain Miranvillea,∗, Alain Piétrusb

aUniversité de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086, SP2MI, BoulevardMarie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France

bUniversité des Antilles-Guyane, Laboratoire A.O.C., Campus de Fouillole, 97159 Pointe à Pitre Cedex,Guadeloupe, France

Received 30 November 2001; accepted 8 March 2005

Abstract

We obtain in this article a new formulation of the generalizations of the Cahn–Hilliard equationbased on constitutive equations proposed by Gurtin. This formulation, which can be seen as a compat-ibility equation, is obtained by making a suitable change of function. The interest is that it allows, oncethis equation is solved, to obtain the order parameter and the chemical potential by simple explicitexpressions. Indeed, we can note that, in the original setting, once the order parameter is known (bysolving the (generalized) Cahn–Hilliard equation), then, the expressions giving the chemical potentialin terms of the order parameter are, for nonisotropic materials, intricate and not convenient (in viewof numerical simulations for instance); in the classical Cahn–Hilliard theory, the chemical potential isgiven, constitutively, as a function of the order parameter. We then study the existence and uniquenessof solutions.� 2005 Elsevier Ltd. All rights reserved.

MSC: 35A05; 35B40; 35B45

Keywords: Cahn–Hilliard equation; Constitutive equations; Compatibility equations; Existence and uniquenessof solutions

0. Introduction

The Cahn–Hilliard equation is very important in materials science (see [3,4,13] andthe references therein). It is a conservation law, in the sense that the average of the order

∗ Corresponding author. Tel.: +33 5 49 49 68 91; fax: +33 5 49 49 69 01.E-mail addresses: [email protected] (A. Miranville), [email protected] (A. Piétrus).

1468-1218/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2005.03.003

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286 A. Miranville, A. Piétrus / Nonlinear Analysis: Real World Applications 7 (2006) 285–307

parameter, which corresponds to a density of atoms, is conserved. Many articles havebeen devoted to the mathematical and numerical studies of this equation (see for instance[1,7–9,12,15,20,22] and the references therein).

In [13], Gurtin makes interesting and important generalizations of the originalCahn–Hilliard equation. He notes that, although it is physically sound and important, theoriginal setting should not be regarded as basic, but rather as a precursor of more com-plete theories. Indeed, he notes that the classical derivation limits the manner in whichrate terms enter the equation and requires a priori specifications of the constitutive equa-tions (in particular, the chemical potential is given, constitutively, as a function of theorder parameter). Also, it is not clear how this derivation should be generalized inthe presence of processes such as deformation and heat transfer. Now, the originality ofthe development presented by Gurtin, when compared with other macroscopic theories oforder parameters, is that it separates balance laws from constitutive equations and introducesa new balance law for microforces. This will be explained in more details in Section 1.1below.

We studied in [2,5,6,16–19] some models of (generalized) Cahn–Hilliard equations basedon the constitutive equations derived by Gurtin.

In the classical theory, once the order parameter is known (by solving the Cahn–Hilliardequation), then, we can obtain immediately the chemical potential. Indeed, as already men-tioned, it is given explicitly in terms of the order parameter. We can also derive explicitexpressions of the chemical potential in terms of the order parameter for the generalizedmodels proposed by Gurtin. However, for nonisotropic materials, these expressions arerather intricate and not convenient (say, for numerical purposes), see Remark 1.1.

When deriving the model, we actually obtain a system of two equations for the orderparameter and the chemical potential (in some situations, one can actually only study thissystem, see [18,19] for discussions on this subject). The Cahn–Hilliard equation, whichis obtained by eliminating the chemical potential in the system, can then be seen as acompatibility equation that the order parameter must satisfy. In Section 1.2, we obtain,by making a suitable change of function, another compatibility equation, which actuallybears some resemblance to the Cahn–Hilliard equation (of course, their mathematical studydiffers, since we no longer have a conservation law). We also note that, once this newequation is solved, then, the chemical potential, but also the order parameter, can be obtainedby very simple (and explicit) expressions. We then study, in Section 2, the existence anduniqueness of solutions for this equation.

Throughout this article, the same letter c (and sometimes c′, c′′ and c′′′) will denote(positive) constants that may change from line to line.

1. Setting of the problem

1.1. Physical derivation of the constitutive equations

We follow in this article the theory developed in [13] by Gurtin. For simplicity, werestrict ourselves to two space dimensions (the changes are however easily made in threespace dimensions) and we assume that all the physical quantities are periodic in space;

A. Miranville, A. Piétrus / Nonlinear Analysis: Real World Applications 7 (2006) 285–307 287

the spatial domain is thus represented by the rectangle � = (0, L1) × (0, L2). Finally, weassume that all the external actions are negligible.

The primitive physical quantities of the problem are the order parameter � (which cor-responds to a density of atoms), the chemical potential �, the mass flux h, the microstressfield � and the internal microforce field �. They are related by the mass balance

��

�t= −div h, (1.1)

the (micro)force balance

div � + � = 0, (1.2)

where

� = �∇�, � > 0, (1.3)

and the dissipation inequality

�dis(Z)��

�t+ h(Z) · ∇��0, (1.4)

which has to be valid for every Z = (�, ∇�, ��/�t, �, ∇�), where

�dis(Z) = �(Z) + f ′(�) − �. (1.5)

Here, f is a double-well potential whose wells correspond to the phases of the material.Furthermore, we have implicitly taken a free energy of the form�=�(�, ∇�)=(�/2)|∇�|2+f (�); such a choice is classical in the Cahn–Hilliard theory and is consistent with the secondlaw of thermodynamics (see below).

A first difference/improvement, when compared with the classical theory, is the introduc-tion of the microforce field � (and thus of the microstress field � and the microforce balance(1.2)). The reason for introducing this quantity is the belief that fundamental physical lawsinvolving energy should account for the working associated with each operative kinemat-ical process (that associated with the order parameter in the Cahn–Hilliard theory). Thus,it seems plausible that there should be microforces whose working accompanies changesin the order parameter. Fried and Gurtin describe this working through terms of the form��/�t (see [10,11]); as a consequence, ��/�t will enter the list of independent constitutivevariables (which are, in the classical theory, � and ∇�). Also, Gurtin assumes that � and ∇�enter the list of independent constitutive variables. Indeed, contrary to the classical theory,� will no longer be given, constitutively, as a function of �; such a relation being no longervalid far from equilibrium. Therefore, the quantities h, � and � (and also the free energy �)depend a priori on Z.

In order to obtain the dissipation inequality (1.4) (and also relation (1.3)), Gurtin thenconsiders the restrictions imposed by the second law of thermodynamics (to be more precise,he considers a purely mechanical version of the second law, see [13] for more details). Theserestrictions also imply that the free energy � and the microstress field � are independent of�, ∇� and ��/�t .


There now remains, in order to complete the theory, to derive the constitutive equationsrelating �dis and h to the constitutive variables. To do so, Gurtin shows that (1.4) yields theexistence of constitutive moduli �(Z) (a scalar), a(Z) and b(Z) (two vectors; the vector bvanishes for isotropic materials) and B(Z) (a matrix; it is called the mobility tensor) suchthat

�dis(Z) = −��

�t− b · ∇�, (1.6)

h(Z) = −a��

�t− B∇�, (1.7)

and (1.4) is satisfied.We assume from now on that the constitutive moduli are constant. We thus deduce from

(1.1)–(1.3) and (1.5)–(1.7) the following system of equations:

��

�t− a · ∇ ��

�t= div(B∇�), (1.8)

� − b · ∇� = −�� + f ′(�) + ��

�t, (1.9)

which we complete with the boundary conditions

� and � are �-periodic, (1.10)

and the initial condition

�|t=0 = �0. (1.11)

Furthermore, it follows from (1.4) that the relation

�x2 + (a + b) · yx + By · y�0, ∀x ∈ R, ∀y ∈ R2, (1.12)

must be satisfied.Assuming now that the following stronger assumption holds:

�x2 + (a + b) · yx + By · y�c(x2 + |y|2), ∀x ∈ R, ∀y ∈ R2, c > 0, (1.13)

then, we can prove, say, for a polynomial potential f (s) =∑2p+2i=0 ais

i , a2p+2 > 0, p�1,and for an initial data �0 ∈ H 1

per(�), the existence and uniqueness of a solution (�, �)

such that � ∈ C([0, T ]; L2(�)) ∩ L∞(0, T ; H 1per(�)), ��/�t ∈ L2(0, T ; L2(�)) and � ∈

L2(0, T ; H 1per(�)), ∀T > 0. To do so, we proceed as in [19] (where we actually considered

Neumann-type boundary conditions and could thus only consider a vector a of the forma = (

a10

)and a mobility tensor of the form B = I, > 0; the changes are easily made for

the more general problem when considering periodic boundary conditions).A second formulation, which actually gives the Cahn–Hilliard equation, consists in elim-

inating the chemical potential in (1.8) and (1.9). To do so, we take the div B∇ of (1.9) andfind, injecting the expression of div(B∇�) given by (1.8) into the equation obtained

��

�t− d · ∇ ��

�t− div

(B∇ ��

�t

)+ � div(B∇��) − div(B∇f ′(�)) = 0, (1.14)


� is �-periodic, (1.15)

�|t=0 = �0, (1.16)

where d = a + b and B = �B − 12 (atb + bta). We can easily prove that a consequence of

(1.12) is that B is positive (we note that, assuming that B is symmetric, it is symmetric).This equation was studied in [2,16]. In particular, we proved, for a polynomial potential asabove and for �0 ∈ H 1

per(�), the existence and uniqueness of a solution � such that � ∈C([0, T ]; H 1

per(�)) and ��/�t ∈ L2(0, T ; H 1per(�)′), ∀T > 0. Furthermore, if the stronger

condition (1.13) holds, in which case B is positive definite, then, for �0 ∈ H 2per(�), we have

� ∈ C([0, T ]; H 2per(�)) and ��/�t ∈ L2(0, T ; H 1

per(�)), ∀T > 0.

Remark 1.1. Once � is obtained, we then find � by solving the equation

� − b · ∇� = −�� + f ′(�) + ��

�t,

� is �-periodic,

(see Section 1.2 below for the explicit expression of the solution of this equation; theexpression that we shall obtain is rather complicated). We note that we cannot obtain �completely by solving (1.8). Indeed, � would then be obtained up to an unknown constant:its average; it follows from (1.9) that it is equal to the spatial average of f ′(�). We thus have

� = −(−div B∇)−1(

��

�t− a · ∇ ��

�t

)+ 1

Vol(�)

∫�

f ′(�) dx.

Again, this expression is not fully satisfactory.

Remark 1.2. For an isotropic material, we have a = b = 0 and B = I, > 0 and I beingthe identity matrix. We thus recover the viscous Cahn–Hilliard equation proposed in [21].If we further assume that � = 0, then we recover the classical Cahn–Hillard equation.

1.2. A new formulation of the problem

Our aim in this subsection is to obtain an equation from which the chemical potential,but also the order parameter, can be easily deduced.

We assume here that all the functions are regular enough to justify the calculations thatwe shall make.

We set, for � regular enough

La(�) = � − a · ∇�. (1.17)

Let us solve (on R2; we extend, by periodicity, all the functions to R2) the equation

La(�) = g, (1.18)


where g is regular enough and �-periodic. We find that � is of the form

�(x1, x2)=ex1/a1h

(x2−a2

a1x1

)− 1

a1ex1/a1

∫ x1

0e−s/a1g

(s, x2−a2

a1x1+a2

a1s

)ds,

(1.19)

where h is to be determined (we assume here that a1 �= 0; the case a1 = 0 is easier to treat).We compute h by noting that � must be �-periodic. Indeed, let us assume, for instance, thata1 > 0 (the case a1 < 0 is similar). Then, there exists a sequence (kn) ∈ ZN such that

knL2 − na2

a1L1 ∈ [−L2, 0] (1.20)

(we take for instance the integer part of na2L1/a1L2). Thus, by taking subsequences ifnecessary, we have two sequences (k1

n) and (k2n) in ZN such that

k2nL2 − k1

n

a2

a1L1 → � ∈ [−L2, 0] as n → +∞,

k1n → +∞ as n → +∞. (1.21)

Now, we have

�(x1 + k1nL1, x2 + k2

nL2) = �(x1, x2), ∀n ∈ N, ∀x1, x2 ∈ R,

which is equivalent to

h

(x2 − a2

a1x1 + k2

nL2 − k1n

a2

a1L1

)− 1

a1

∫ x1+k1nL1

0e−s/a1g

(s, x2 − a2

a1x1

+k2nL2 − k1

n

a2

a1L1 + a2

a1s

)ds = e−k1

nL1/a1

[h

(x2 − a2

a1x1

)

− 1

a1

∫ x1

0e−s/a1g

(s, x2 − a2

a1x1 + a2

a1s

)ds

], ∀n ∈ N, ∀x1, x2 ∈ R. (1.22)

Passing to the limit as n → +∞ and using Lebesgue’s theorem (which is justified if, say,g is continuous (we note that it is then bounded); also, the function h is then continuous),we obtain

h(x) = 1

a1

+∞∫0

e−s/a1g

(s, x + a2

a1s

)ds, ∀x ∈ R, (1.23)

so that

L−1a (g)(x1, x2) = �(x1, x2) = 1

a1e−x1/a1

∫ +∞

x1

e−s/a1g

(s, x2 − a2

a1x1 + a2

a1s

)ds,

(1.24)

∀x1, x2 ∈ R. We easily see that, if g is, say, twice continuously differentiable with respectto the second variable, then, � is indeed solution of the problem.


We now set

(x1, x2, t) =∫ t

0L−1

a (�)(x1, x2, s) ds. (1.25)

Then

� = La

(�

�t

), (1.26)

and we deduce from (1.8) that

� = div(B∇) + �0. (1.27)

Injecting (1.26) and (1.27) into (1.9), we have

La

(�

�t

)− b · ∇La

(�

�t

)= − ��(div(B∇) + �0) + � div

(B∇ �

�t

)+ f ′(div(B∇) + �0).

We have thus obtained the following problem:

�

�t− d · ∇ �

�t− div

(B∇ �

�t

)+ ��(div(B∇) + �0)

− f ′(div(B∇) + �0) = 0, (1.28)

is �-periodic, (1.29)

|t=0 = 0. (1.30)

Our aim now is to get rid of the function �0 in (1.28); indeed, this will allow us to treatthe nonlinear term more easily. To do so, we consider the solution � of

div(B∇�) = �0 − m, (1.31)∫�

� dx = 0, (1.32)


where

m = 1

Vol(�)

∫�

�0 dx. (1.34)

We then set � = + �. We thus have the following equation for �:

��

�t−d · ∇ ��

�t−div

(B∇ ��

�t

)+� div(B∇��) − f ′(div(B∇�) + m)=0, (1.35)


�|t=0 = � = (−div B∇)−1(m − �0). (1.37)


Furthermore, we have

� = La

(��

�t

)= ��

�t− a · ∇ ��

�t(1.38)

and

� = div(B∇�) + m. (1.39)

We thus see that � (and also �) can be easily (and explicitly) deduced from �. In particular,this expression of � is simpler (and more convenient) than those given in Remark 1.1, i.e.

� = L−1b

(−�� + �

��

�t+ f ′(�)

),

where the explicit expression of L−1b is given with (1.24), and

mu = −(−div B∇)−1La

(��

�t

)+ 1

Vol(�)

∫�

f ′(�) dx.

Remark 1.3. We see that Eq. (1.35) bears some resemblance to the Cahn–Hilliard equation(1.14). It is however not a conservation law (the conservation property is important in themathematical treatment of the Cahn–Hilliard equation, see for instance [2,16,20,22]); wehave nevertheless the conservation of � in (1.39).

Remark 1.4. Taking a = b = 0, � = 0 and B = I, > 0, in (1.35), we obtain the corre-sponding equation for the classical Cahn–Hilliard theory:

��

�t+ ��2� − f ′(�� + m) = 0, (1.40)


�|t=0 = 1

(−�)−1(m − �0), (1.42)

and we have

� = ��

�t, (1.43)

� = �� + m. (1.44)

2. Mathematical study of the problem

We set g(s) = f (s + m), s ∈ R, and we consider the following problem:

��

�t− d · ∇ ��

�t− div

(B∇ ��

�t

)+ � div(B∇��) − g′(div(B∇�)) = 0, (2.1)



�|t=0 = �0. (2.3)

We make the following assumptions on the potential g:

g is of class C1, (2.4)

c0s2p+2 − c1 �g(s)�c2s

2p+2 + c3, ∀s ∈ R, c0, c2 > 0, c1, c3 �0,

c′0s

2p+2 − c′1 �g′(s)s�c′

2s2p+2 + c′

3, ∀s ∈ R, c′0, c′

2 > 0, c′1, c′

3 �0,

p ∈ [1, +∞[, (2.5)

|g′(s)|�c4|s|2p+1 + c5, ∀s ∈ R, c4, c5 �0, where p is the same as in (2.5). (2.6)

For instance, polynomials of the form considered in Section 1.1 satisfy (2.4)–(2.6). We alsonote that, if the initial potential f satisfies (2.4)–(2.6), then, the potential g, as defined above,satisfies, for fixed m, (2.4)–(2.6).

Finally, we assume that B is a symmetric and positive definite matrix.We then associate with (2.1)–(2.3) the variational formulation (see Proposition 2.1 below)Find �: [0, T ] → {v ∈ H 2

per(�), div(B∇v) ∈ L2p+2(�)} such that

d

dt[(�, q) − (d.∇�, q) + (B∇�, ∇q)] + �(∇B1/2∇�, ∇B1/2∇q)

− (g′(div(B∇�)), q) = 0, ∀q ∈ H 2per(�), (2.7)

�(0) = �0, (2.8)

where (., .) denotes the usual L2-scalar product (|.| denotes the associated norm).

2.1. Preliminary results

We first prove the

Lemma 2.1. Let A be a symmetric and positive definite matrix. Then, the mapping q →|div(A∇q)| + |q| defines a norm on H 2

per (�) that is equivalent to the usual H 2-norm.

Proof. We first consider the case where A is diagonal,

A =(

1 00 2

), 1 > 0, 2 > 0.

In that case, we proceed exactly as for the Laplace operator, by considering Fourier seriesexpansions (see e.g. [22]). We now consider the general case. Since A is symmetric andpositive definite, there exist E1 diagonal,

E1 =(

1 00 2

), 1 > 0, 2 > 0,

and E2 orthonormal such that E2AtE2 = E1 and we consider the change of coordinates

x′=E2x. We deduce from the structure of E2 that the domain �′=E2(�) is still a rectangle.We endow L2(�′) with the scalar product


(u, v)′ =∫�′

uv� dx′, (2.9)

where � is the jacobian associated with the above change of coordinates (it is a con-stant here); | · |′ denotes the associated norm. We have, for q ∈ H 2

per(�), div(A∇q(x)) =div′(E1∇′q ′(x′)) (with q ′ ∈ H 2

per(�′); ′ means that we consider the variable x′) and we are

in the first case considered. We thus deduce that

|div(A∇q)| + |q|�c‖q ′‖H 2(�′) �c′‖q‖H 2(�), (2.10)

∀q ∈ H 2per(�), which finishes the proof of the lemma (the converse inequality is straight-

forward).

We then have the

Proposition 2.1. We assume that the matrix B is symmetric and positive definite.

(i) We have, for every � and q regular enough and �-periodic

(div(B∇��), q) = (∇B1/2∇�, ∇B1/2∇q). (2.11)

(ii) We have, for q regular enough and �-periodic

(b · ∇q, q) = 0, (2.12)

(b · ∇q, div(B∇q)) = 0. (2.13)

(iii) We assume that B is symmetric. Then, we have, for every � and q regular enough and�-periodic

(div(B∇�), div(B∇q)) = (B∇B1/2∇�, ∇B1/2∇q), (2.14)

(div(B∇�), div(B∇�q)) = −(B∇B1/2∇∇�, ∇B1/2∇∇q). (2.15)

(iv) The mapping q → |∇B1/2∇q| + ‖q‖H 1(�) defines a norm on H 2per(�) that is equiv-

alent to the usual H 2-norm.(v) The mapping q → |∇div(B∇q)| + ‖q‖H 2(�) defines a norm on H 3

per(�) that is

equivalent to the usual H 3-norm.(vi) We assume that the matrix B is symmetric and positive definite. The mapping q →

|B1/2∇B1/2∇∇q| + ‖q‖H 2(�) defines a norm on H 3per(�) that is equivalent to the

usual H 3-norm.(vii) The mapping q → |div(B∇�q)| + ‖q‖H 3(�) defines a norm on H 4

per(�) that is

equivalent to the usual H 4-norm.

Proof. (i) We have

(div(B∇��), q) = − (B1/2∇��, B1/2∇q) = −(�(B1/2∇�), B1/2∇q)

= (∇B1/2∇�, ∇B1/2∇q),

∀� ∈ H 4per(�), ∀q ∈ H 2

per(�).


(ii) The proof of (2.12) and (2.13) is straightforward, noting that (b ·∇q, r)=−(q, b ·∇r),∀q, r ∈ H 1

per(�).(iii) We have

(div(B∇�), div(B∇q)) = − (B1/2∇div(B∇�), B1/2∇q)

= − (div(B∇B1/2∇�), B1/2∇q),

∀� ∈ H 3per(�), ∀q ∈ H 2

per(�); hence (2.14). Similarly, we have

(div(B∇�), div(B∇�q))=(div(B∇�), �div(B∇q))=−(∇div(B∇�), ∇div(B∇q))

= −(div(B∇∇�), div(B∇∇q)),

∀� ∈ H 3per(�), ∀q ∈ H 4

per(�), and we conclude by using (2.14).

(iv) It follows from Lemma 2.1 that q → |div(B1/2∇q)|+‖q‖H 1(�) is a norm on H 2per(�)

that is equivalent to the usual H 2-norm. We then have

|∇B1/2∇q|2 =2∑

i,j=1

∣∣∣∣ �

�xi

(B1/2∇q)j

∣∣∣∣2

�2∑

i=1

∣∣∣∣ �

�xi

(B1/2∇q)i

∣∣∣∣2

� 1

2|div(B1/2∇q)|2,

where (u)i denotes the ith component of the vector u. This yields that

|∇B1/2∇q| + ‖q‖H 1(�) �c(|div(B1/2∇q)| + ‖q‖H 1(�))�c′‖q‖H 2(�),

∀q ∈ H 2per(�).

The converse inequality is straightforward.(v) We have, thanks to Lemma 2.1,

|∇div(B∇q)| + ‖q‖H 2(�) = |div(B∇∇q)| + ‖q‖H 2(�)

�c(|div(B∇∇q)| + ‖∇q‖H 1(�)2 + ‖q‖H 2(�))

�c(‖∇q‖H 2(�)2 + ‖q‖H 2(�))

�c(|�∇q| + ‖q‖H 2(�))

�c(|∇�q| + ‖q‖H 2(�))

�c‖q‖H 3(�),

∀q ∈ H 3per(�). The converse inequality is straightforward.


(vi) We have, noting that B is positive definite and thanks to (iv)

|B1/2∇B1/2∇∇q| + ‖q‖H 2(�) �c(|∇B1/2∇∇q| + ‖∇q‖H 1(�)2 + ‖q‖H 2(�))

�c(‖∇q‖H 2(�)2 + ‖q‖H 2(�))

�c‖q‖H 3(�),


(vii) We have

|div(B∇�q)| + ‖q‖H 3(�) �c(|div(B∇�q)| + |�q| + ‖q‖H 3(�))

�c(‖�q‖H 2(�) + ‖q‖H 3(�))

�c(|�2q| + ‖q‖H 3(�))

�c‖q‖H 4(�),


Remark 2.1. We set H kper(�)={q ∈ Hk

per(�),∫� q dx =0}, k ∈ N (for k=0, Hk

per(�)=L2(�)). Let A denote the operator −divB∇ (we assume here that B is symmetric andpositive definite) with domain H 2

per(�). Then, A is an unbounded, strictly positive and

selfadjoint operator with compact inverse on L2(�). Furthermore, proceeding as in theproof of Lemma 2.1, we prove that D(Ak/2) = H k

per(�). Indeed, in the case where B isdiagonal, we again proceed as for the Laplace operator. In the general case, we have,considering the same change of coordinates as in the proof of Lemma 2.1 and keeping thesame notations, (Ak/2u(x))′ = (A′)k/2u′(x′), ∀u ∈ D(Ak/2), where A′ = −div′B ′∇′ withdomain H 2

per(�′),B ′ diagonal, and we conclude with the first case (we note that u ∈ D(Ak/2)

if and only if u′ ∈ D((A′)k/2) and u ∈ H kper(�) if and only if u′ ∈ H k

per(�′)).

2.2. A priori estimates

We multiply (2.1) by � and obtain, integrating over � and thanks to Proposition 2.1,(i)

1

2

d

dt(|�|2+(B∇�, ∇�))+�|∇B1/2∇�|2=(g′(div(B∇�)), �)+

(d · ∇ ��

�t, �

).

(2.16)

We have, thanks to (2.6) and to Hölder’s inequality

|(g′(div(B∇�)), �)|�c

(∫�

|div(B∇�)|2p+2 dx

)(2p+1)/(p+1)

‖�‖2H 2(�)

+ c′|�|2 + c′′, (2.17)

and we write∣∣∣∣(

d · ∇ ��

�t, �

)∣∣∣∣ ��1

∣∣∣∣∇ ��

�t

∣∣∣∣2

+ c|�|2, ∀�1 > 0. (2.18)


We thus obtain, thanks to Proposition 2.1(iv), the inequation

d

dt(|�|2 + (B∇�, ∇�)) + c‖�‖2

H 2(�)

�c′(∫

�|div(B∇�)|2p+2 dx

)(2p+1)/(p+1)

‖�‖2H 2(�)

+ c′′‖�‖2H 1(�)

+ c′′′

+ �1|∇ ��

�t|2, ∀�1 > 0. (2.19)

We now multiply (2.1) by ��/�t and obtain, integrating over � and thanks to Proposition2.1(ii)

�

2

d

dt|∇B1/2∇�|2 +

∣∣∣∣��

�t

∣∣∣∣2

+(

B∇ ��

�t, ∇ ��

�t

)=(

g′(div(B∇�)),��

�t

). (2.20)

We have, thanks to (2.6) and noting that q → |B1/2∇q| + c|q| defines a norm on H 1per(�)

that is equivalent to the usual H 1-norm, ∀c > 0∣∣∣∣(

g′(div(B∇�)),��

�t

)∣∣∣∣ �c

∫�(|div(B∇�)|2p+1 + 1)

∣∣∣∣��

�t

∣∣∣∣ dx

�c

∥∥∥∥��

�t

∥∥∥∥L2p+2(�)

(∫�(|div(B∇�)|2p+2+1)dx

)(2p+1)/(2p+2)

��

∣∣∣∣��

�t

∣∣∣∣2

+ �2

∣∣∣∣B1/2∇ ��

�t

∣∣∣∣2

+c′(∫

�|div(B∇�)|2p+2 dx

)(2p+1)/(p+1)

+c′′, (2.21)

∀�, �2 > 0; which yields, taking � small enough and noting that B is positive

d

dt|∇B1/2∇�|2+c

∣∣∣∣��

�t

∣∣∣∣2

��2

∣∣∣∣B1/2∇ ��

�t

∣∣∣∣2

+c′(∫

�|div(B∇�)|2p+2 dx

)(2p+1)/(p+1)

+ c′′, ∀�2 > 0. (2.22)

We then multiply (2.1) by div(B∇��/�t) and have, thanks to Proposition 2.1(ii) and (iii)

d

dt(�|∇div(B∇�)|2 + 2

∫�

g(div(B∇�)) dx) + 2

∣∣∣∣B1/2∇ ��

�t

∣∣∣∣2

+ 2

(B∇B1/2∇ ��

�t, ∇B1/2∇ ��

�t

)= 0. (2.23)

Multiplying (2.1) by div(B∇�) and integrating over �, we have, similarly

1

2

d

dt(|B1/2∇�|2 + (B∇B1/2∇�, ∇B1/2∇�)) +

(d · ∇ ��

�t, div(B∇�)

)

+ �|∇div(B∇�)|2 +∫�

g′(div(B∇�))div(B∇�) dx = 0,


which yields, noting that, thanks to (2.5)

∫�

g′(div(B∇�))div(B∇�) dx�c

∫�

g(div(B∇�)) dx + c′|div(B∇�)|2 − c′′,

d

dt(|B1/2∇�|2 + (B∇B1/2∇�, ∇B1/2∇�)) + c(|div(B∇�)|2 + |∇div(B∇�)|2)

+ c′∫�

g(div(B∇�)) dx�c′′∣∣∣∣B1/2∇ ��

�t

∣∣∣∣2

+ c′′′. (2.24)

Next, we assume that g satisfies, in addition to (2.4)–(2.6), the following assumptions:

g is of class C2, (2.25)

g′′(s)� − c6, ∀s ∈ R, c6 �0 (2.26)

and we multiply (2.1) by div(B∇��) and integrate over �. We obtain, thanks to Proposition2.1(i) and (iii) and to (2.26)

1

2

d

dt(|∇B1/2∇�|2 + (B∇B1/2∇∇�, ∇B1/2∇∇�)) + �|div(B∇��)|2

�c6|∇div(B∇�)|2 +(

d · ∇ ��

�t, div(B∇��)

),

which yields, thanks to Proposition 2.1(vii)

d

dt(|∇B1/2∇�|2 + (B∇B1/2∇∇�, ∇B1/2∇∇�)) + c‖�‖2

H 4(�)

�c′‖�‖2H 3(�)

+ c′′∣∣∣∣∇ ��

�t

∣∣∣∣2

. (2.27)

Finally, if g satisfies

|g′′(s)|�c7s2p + c8, ∀s ∈ R, c7, c8 �0, (2.28)

we find, multiplying (2.1) by div(B∇��/�t) and integrating over �

�

2

d

dt|div(B∇��)|2 +

∣∣∣∣∇B1/2∇ ��

�t

∣∣∣∣2

+(

B∇B1/2∇∇ ��

�t, ∇B1/2∇∇ ��

�t

)

= −∫�

g′′(div(B∇�))∇ div(B∇�) · B∇��

�tdx.


We have, thanks to (2.28) and to Hölder’s inequality

∣∣∣∣∫�

g′′(div(B∇�))∇ div(B∇�) · B∇��

�tdx

∣∣∣∣�c

∫�(|div(B∇�)|2p + 1)|∇ div(B∇�)| ·

∣∣∣∣B∇��

�t

∣∣∣∣ dx

�c(‖div(B∇�)‖2p

L6p(�)+ 1)‖∇div(B∇�)‖L6(�)

∥∥∥∥��

�t

∥∥∥∥H 3(�)

�c(‖�‖2p

H 3(�)+ 1)‖�‖H 4(�)

∥∥∥∥��

�t

∥∥∥∥H 3(�)

,

which yields, assuming that B is positive definite and thanks to Proposition 2.1(iv) and (vi)

d

dt|div(B∇��)|2 + c

∥∥∥∥��

�t

∥∥∥∥2

H 3(�)

�c′(‖�‖4p

H 3(�)+ 1)‖�‖2

H 4(�)+ c′′

∣∣∣∣��

�t

∣∣∣∣2

. (2.29)

When B is only positive semi-definite, we write

−∫�

g′′(div(B∇�))∇div(B∇�) · B∇��

�tdx

=∫�

g′′′(div(B∇�))��

�tB∇ div(B∇�) · ∇div(B∇�) dx

+∫�

g′′(div(B∇�))��

�tdiv(B∇div(B∇�)) dx.

We have∣∣∣∣∫�

g′′′(div(B∇�))��

�tB∇ div(B∇�) · ∇div(B∇�) dx

∣∣∣∣�c

∫�(|div(B∇�)|2p−1 + 1)

∣∣∣∣��

�t

∣∣∣∣ |∇div(B∇�)|2 dx

�c(‖div(B∇�)‖2p−1L∞(�)

+ 1)

∣∣∣∣��

�t

∣∣∣∣ ‖∇div(B∇�)‖2L4(�)

,

which yields, noting that, for n= 2, H 2p/(2p−1)(�) ⊂ L∞(�) and H 1/2(�) ⊂ L4(�) withcontinuous injection and using the interpolation inequalities

‖q‖H 2p/(2p−1)(�) �c‖q‖(2p−2)/(2p−1)

H 1(�)‖q‖1/(2p−1)

H 2(�), ∀q ∈ H 2

per(�)

and

‖q‖H 1/2(�) �c|q|1/2‖q‖1/2H 1(�)

, ∀q ∈ H 1per(�),

300 A. Miranville, A. Piétrus / Nonlinear Analysis: Real World Applications 7 (2006) 285–307∣∣∣∣∫�

g′′′(div(B∇�))��

�tB∇div(B∇�) · ∇div(B∇�) dx

∣∣∣∣�c(‖div(B∇�)‖2p−2

H 1(�)‖div(B∇�)‖H 2(�) + 1)

∣∣∣∣��

�t

∣∣∣∣· |∇div(B∇�)|‖∇div(B∇�)‖H 1(�). (2.30)

Furthermore, we have∣∣∣∣∫�

g′′(div(B∇�))��

�tdiv(B∇div(B∇�)) dx

∣∣∣∣�c

∫�(|div(B∇�)|2p + 1)|��

�t‖div(B∇div(B∇�))| dx

�c(‖div(B∇�)‖2p

L∞(�)+ 1)|��

�t‖div(B∇div(B∇�))|,

which yields, since H(2p+1)/2p(�) ⊂ L∞(�) with continuous injection and using theinterpolation inequality

‖q‖H(2p+1)/2p(�) �c‖q‖(2p−1)/2p

H 1(�)‖q‖1/2p

H 2(�), ∀q ∈ H 2

per(�),

∣∣∣∣∫�

g′′(div(B∇�))��

�tdiv(B∇div(B∇�)) dx

∣∣∣∣�c(‖div(B∇�)‖2p−1

H 1(�)‖div(B∇�)‖H 2(�)+1)|��

�t‖div(B∇div(B∇�))|. (2.31)

Finally, using (2.30) and (2.31), we obtain, instead of (2.29)

d

dt|div(B∇��)|2 + c

∥∥∥∥��

�t

∥∥∥∥2

H 2(�)

�c′(‖�‖4p−4H 3(�)

+ ‖�‖4p−2H 3(�)

+ 1)(‖�‖2H 3(�)

+ 1)

× (‖�‖2H 4(�)

+1)‖�‖2H 4(�)

+c′′∣∣∣∣��

�t

∣∣∣∣2

. (2.32)

2.3. The case B positive definite

We assume in this subsection that the matrix B is positive definite (we recall that it issymmetric). We have the following result:

Theorem 2.1. (i) We assume that (2.4)–(2.6) hold and that �0 ∈ H 3per(�). Then, (2.7) and

(2.8) possesses at least one solution � such that � ∈ C([0, T ]; H 3−�(�)) ∩ L∞(0, T ;H 3

per(�)), ∇� ∈ L∞(R+; H 2per(�)2), div(B∇�) ∈ L∞(R+; L2p+2(�)) and ��/�t ∈

L2(0, T ; H 2per(�)), ∀� > 0, ∀T > 0.

(ii) If we further assume that g ∈ C2(R) and that (2.28) holds, then, the solution obtainedin (i) is unique.


(iii) If we further assume that g is of class C2 and that (2.26) (resp. (2.26) and (2.28))holds, then, we have � ∈ L2(0, T ; H 4

per(�)) (resp. � ∈ L∞(0, T ; H 4per(�)) and ��/�t ∈

L2(0, T ; H 3per(�)), if �0 ∈ H 4

per(�)), ∀T > 0.

Proof. (i) The proof of existence of solutions is based on (2.19) and (2.22)–(2.24). Todo so, we consider a Galerkin approximation based on the eigenfunctions of the operatorA = −div B∇ associated with periodic boundary conditions (we recall that this operatoris an unbounded, strictly positive and self-adjoint operator with compact inverse on L2(�)

and we note that L2(�)= L2(�)⊕R). We call 0 = 0 < 1 � · · · � m � · · · the associatedeigenvalues and w0, . . . , wm, . . . corresponding eigenfunctions. We take w0 = 1/

√Vol(�)

and we assume that the wj are orthonormal in L2(�) and orthogonal in H 1per(�) (for

the scalar product (., .) + (B∇·, ∇·); we note that, for q ∈ H 1per(�), we have |A1/2q| =

|B1/2∇q|). Furthermore, we set Vm = Span(w0, . . . , wm).

We then obtain the existence of a local (in time) solution �m=∑mi=0 um

i wi for the approx-imate problem on, say, [0, T �[, T � > 0, and we easily prove that (2.19) and (2.22)–(2.24)hold for � replaced by �m.

We first note that, thanks to (2.23), we have, for t < T �∫�

g(div(B∇�m)) dx� �

2|∇div(B∇�m

0 )|2 +∫�

g(div(B∇�m0 )) dx��(�0), (2.33)

where �(�0) is independent of m (and of t). Indeed, we take for �m0 the projection of �0

onto the m+1 first eigenvectors of the operator A and we note that, thanks to (2.5), we have| ∫� g(div(B∇�m

0 )) dx|�c∫� |div(B∇�m

0 )|2p+2 dx + c′ �c′′‖�m0 ‖2p+2

H 3(�)+ c′ �c′′′(|A3/2

�m0 |2p+2+|(1/Vol(�))

∫� �m

0 dx|2p+2)+c′ (see Remark 2.1), where �m0 =�m

0 −(1/Vol(�))∫� �m

0 dx and (1/√

Vol(�))∫� �m

0 dx = (�0, w0), hence (2.33).Summing then (2.19), (2.22) and (2.23), we obtain, noting that B is positive definite,

using Proposition 2.1(iv) and (v), and choosing �1 and �2 small enough, an inequation ofthe form

dEm0

dt+ c

∥∥∥∥��m

�t

∥∥∥∥2

H 2(�)

�c′(�0)‖�m‖2H 2(�)

+ c′′, (2.34)

where

Em0 = |�m|2 + |B1/2∇�m|2 + |∇B1/2∇�m|2 + �|∇div(B∇�m)|2

+ 2∫�

g(div(B∇�m)) dx, (2.35)

satisfies, thanks to (2.5)

Em0 �c‖�m‖2

H 3(�)+ c′

∫�

|div(B∇�m)|2p+2 dx − c′′. (2.36)

This yields that the solution is global (i.e. T � = T ) and that �m is bounded in L∞(0, T ;H 3

per(�)), that div(B∇�m) is bounded in L∞(0, T ; L2p+2(�)) and that ��m/�t is bounded


in L2(0, T ; H 2per(�)), independently of m. We can then pass to the limit; the passage to the

limit in the nonlinear term is based on classical compactness results (see [14]), which yield,for instance, that

�m → � in C([0, T ]; H 3−�(�)) strong, ∀� > 0, and a.e.

In order to obtain uniform estimates, we sum (2.23) and �3(2.24), where �3 > 0 is smallenough. We then have, noting that, for instance, |∇q|�c|div(B∇∇q)| = c|∇div(B∇q)|,∀q ∈ H 3

per(�)

dEm1

dt+ cEm

1 + c′(∣∣∣∣B1/2∇ ��m

�t

∣∣∣∣2

+∣∣∣∣B1/2∇B1/2∇ ��m

�t

∣∣∣∣2)

�c′′, (2.37)

where

Em1 = �|∇div(B∇�m)|2 + 2

∫�

g(div(B∇�m)) dx + �3(|B1/2∇�m|2

+ |B1/2∇B1/2∇�m|2). (2.38)

(ii) Let �1 and �2 be two solutions of (2.7) and (2.8). We set � = �1 − �2. We have

��

�t− d.∇ ��

�t− div

(B∇ ��

�t

)+ � div(B∇��)

− g′(div(B∇�1)) + g′(div(B∇�2)) = 0, (2.39)

� is �-periodic, �(0) = 0. (2.40)

We multiply (2.39) by ��/�t and have, integrating over � and thanks to (2.28)

�

2

d

dt|∇B1/2∇�|2 +

∣∣∣∣��

�t

∣∣∣∣2

+∣∣∣∣B1/2∇ ��

�t

∣∣∣∣2

�c

∫�(|div(B∇�1)|2p + |div(B∇�2)|2p + 1)|div(B∇�)|

∣∣∣∣��

�t

∣∣∣∣ dx. (2.41)

We have∫�

|div(B∇�1)|2p|div(B∇�)|∣∣∣∣��

�t

∣∣∣∣ dx

�c‖div(B∇�1)‖2p

L6p(�)|div(B∇�)|

∥∥∥∥��

�t

∥∥∥∥L6(�)

�c‖�1‖4p

H 3(�)|div(B∇�)|2 + �

∥∥∥∥��

�t

∥∥∥∥2

H 1(�)

,

∀� > 0. We thus obtain, proceeding similarly for the other terms and taking � small enough

d

dt|∇B1/2∇�|2 + c

∥∥∥∥��

�t

∥∥∥∥2

H 1(�)

�h(t)|div(B∇�)|2, (2.42)

where h(t) = c(‖�1‖4p

H 3(�)+ ‖�2‖4p

H 3(�)+ 1) belongs to L∞(0, T ).


We now multiply (2.39) by � and have, proceeding similarly

d

dt(|�|2 + |B1/2∇�|2) + c‖�‖2

H 2(�)� l(t)|div(B∇�)|‖�‖H 1(�)

+ c′‖�‖2H 1(�)

+ �4

∥∥∥∥��

�t

∥∥∥∥2

H 1(�)

, ∀�4 > 0, (2.43)

where l(t) belongs to L∞(0, T ).Summing (2.42) and (2.43), we obtain, taking �4 small enough

dE2

dt�M(t)‖�‖2

H 2(�), (2.44)

where M(t) belongs to L∞(0, T ) and where

E2 = |�|2 + |B1/2∇�|2 + |∇B1/2∇�|2, (2.45)

is, thanks to Proposition 2.1(iv), equivalent to the usual H 2-norm on H 2per(�). This yields,

using Gronwall’s lemma, the uniqueness.(iii) We sum here (2.34) and �5(2.27) (resp. (2.34) and �5(2.29)), where �5 is small

enough to absorb the term c′′|∇��/�t |2 in (2.27) (resp. the term c′′|��/�t |2 in (2.29)), with� replaced by �m. Actually, if Ei, i ∈ N, denote the eigenspaces of the operator A, wetake Vm = E0 ⊕ · · · ⊕ Em. We note that the Ei have finite dimension, that the sequence ofspaces Vm introduced here is a subsequence of that considered in (i) and that, if q ∈ Vm,then, �q ∈ Vm. This justifies the a priori estimates (2.27) and (2.29) (as well as the other apriori estimates). We then conclude thanks to Proposition 2.1(vi) and (vii).

Remark 2.2. We would obtain similar results in three space dimensions, except that, dueto the restrictions in the Sobolev embedding theorems, we would obtain the existence forp ∈ [1, 2] only. Here, for the last estimate in (iii) (i.e. the equivalent of (2.29)), we write,for p ∈]1, 2] (the case p=1 can be treated similarly by using Agmon’s inequality, see [22])∣∣∣∣

∫�

g′′(div(B∇�))∇div(B∇�).B∇��

�tdx

∣∣∣∣�c(‖div(B∇�)‖2p

L6p/(p−1)(�)+ 1)

∣∣∣∣B∇��

�t

∣∣∣∣ ‖∇div(B∇�)‖L6/(5−2p)(�).

Noting that H(2p+1)/2p(�) ⊂ L6p/(p−1)(�) with continuous injection and using the inter-polation inequality

‖q‖H(2p+1)/2p(�) �c‖q‖(2p−1)/2p

H 1(�)‖q‖1/2p

H 2(�), ∀q ∈ H 2

per(�),

we then obtain∣∣∣∣∫�

g′′(div(B∇�))∇div(B∇�).B∇��

�tdx

∣∣∣∣c(‖div(B∇�)‖2p−1

H 1(�)‖div(B∇�)‖H 2(�) + 1)|B∇�

��

�t|‖∇div(B∇�)‖H 1(�),


and we have, instead of (2.29)

d

dt|div(B∇��)|2+c

∥∥∥∥��

�t

∥∥∥∥2

H 3(�)

�c′(‖�‖4p−2H 3(�)

‖�‖2H 4(�)

+1)‖�‖2H 4(�)

+c′′∣∣∣∣��

�t

∣∣∣∣2

.

Furthermore, we have the uniqueness for p = 1 only with regularity (i) and for p ∈[1, 2] with regularity (iii) (in that case, the term

∫�(|div(B∇�1)|2p + |div(B∇�2)|2p +

1)|div(B∇�)||��/�t | dx is treated as above). Now, if we assume that �0 ∈ H 4per(�), then,

in three space dimensions, we can obtain the same results as in (i), without restrictions onp. Indeed, in that case, we have

∫� |div(B∇�m

0 )|2p+2 dx�c‖�m0 ‖2p+2

H 4(�)�c′(|A2�m

0 |2p+2 +|[1/Vol(�)] ∫� �m

0 dx|2p+2)�c′′(�0). Furthermore, we write, in (2.20)

∣∣∣∣(

g′(div(B∇�)),��

�t

)∣∣∣∣ �c

∫�(|div(B∇�)|2p+1 + 1)

∣∣∣∣��

�t

∣∣∣∣ dx

�c

∥∥∥∥��

�t

∥∥∥∥L∞(�)

∫�(|div(B∇�)|2p+1 + 1) dx

��

∣∣∣∣��

�t

∣∣∣∣2

+ �6

∣∣∣∣∇B1/2∇ ��

�t

∣∣∣∣2

+ c′(∫

�|div(B∇�)|2p+2 dx

) (2p+1)(p+1) + c′′,

∀�, �6 > 0.

Remark 2.3. We recall that � = div(B∇�) + m and � = La(��/�t) (for a proper choiceof �0). We thus deduce from Theorem 2.1 that, if �0 ∈ H 3

per(�) (which would correspond

to �0 ∈ H 1per(�)), then, � ∈ L∞(R+; H 1

per(�)) and � ∈ L2(0, T ; H 1per(�)), ∀T > 0. This

is in agreement with the results obtained by studying formulation (1.8)–(1.9) (see [19]).

Remark 2.4. We deduce from (2.37) the existence of a bounded absorbing set for ∇�in H 2

per(�)2. We can then obtain, using the higher order a priori estimates derived and

the uniform Gronwall’s lemma (see [22]), a bounded absorbing set for ∇� in H 3per(�)2.

However, we are not able to obtain uniform estimates on � and are thus not able to provethe existence of attractors for �.

2.4. The case B positive semi-definite

We assume in this subsection that the matrix B can have vanishing eigenvalues (this casecontains the classical Cahn–Hilliard setting, which corresponds to d = 0, B = I, > 0,and B = 0 in (2.7)). In that case, we have the

Theorem 2.2. (i) We assume that (2.4)–(2.6) hold and that �0 ∈ H 3per(�). Then, (2.7) and

(2.8) possesses at least one solution � such that � ∈ C([0, T ]; H 3−�(�)) ∩ L∞(0, T ;H 3

per(�)), ∇� ∈ L∞(R+; H 2per(�)2), div(B∇�) ∈ L∞(R+; L2p+2(�)) and ��/�t ∈

L2(0, T ; H 1per(�)), ∀� > 0, ∀T > 0.


(ii) If we further assume that g is of class C2 and that (2.26) (resp. (2.26) and (2.28))holds, then, we have � ∈ L2(0, T ; H 4

per(�)) (resp. � ∈ L∞(0, T ; H 4per(�)) and ��/�t ∈

L2(0, T ; H 2per(�)), if �0 ∈ H 4

per(�)), ∀T > 0.

Proof. (i) We proceed as in the proof of Theorem 2.1(i). We obtain, summing (2.19), (2.22)and (2.23) and choosing �1 and �2 small enough (for the sake of simplicity, we omit thesuperscript m here)

dE3

dt+ c

∥∥∥∥��

�t

∥∥∥∥2

H 1(�)

�c′(�0)‖�m‖2H 2(�)

+ c′′, (2.46)

where

E3 = |�|2 + (B∇�, ∇�) + |∇B1/2∇�|2 + �|∇div(B∇�)|2+ 2

∫�

g(div(B∇�)) dx, (2.47)

satisfies

E3 �c‖�‖2H 3(�)

+ c′∫�

|div(B∇�)|2p+2 dx − c′′. (2.48)

Furthermore, summing (2.23) and �7 (2.24), �7 > 0 small enough, we have

dE4

dt+ cE4 + c′

(∣∣∣∣B1/2∇ ��

�t

∣∣∣∣2

+(

B∇B1/2∇ ��

�t, ∇B

12 ∇ ��

�t

))�c′′, (2.49)

where

E4 = �|∇div(B∇�)|2 + 2∫�

g(div(B∇�)) dx + �7(|B1/2∇�|2

+ (B∇B1/2∇�, ∇B1/2∇�)). (2.50)

(ii) We proceed as in the proof of Theorem 2.1(iii), by considering now (2.32) instead of(2.29).

Remark 2.5. We are not able to obtain the uniqueness of solutions here (at least, with theregularity (i)).

Remark 2.6. We note that, when B is not positive definite (which implies in particularthat (1.13) is not satisfied), we have not been able to obtain the existence of solutionsfor formulation (1.8)–(1.9). Furthermore, the regularity obtained in (ii) implies that, for�0 ∈ H 1

per(�), then, � ∈ L2(0, T ; H 2per(�)). We have not been able to obtain this regularity

from (1.14) in that case (in general; when d = 0, we can obtain this regularity), see [16].

Remark 2.7. In three space dimensions, we cannot remove the restrictions on p (i.e. p ∈[1, 2]) in Theorem 2.2(i), even if �0 is more regular (indeed, these restrictions are necessaryto derive (2.22)). Now, when �0 ∈ H 4

per(�), we can obtain (without restrictions on p in three


space dimensions) the same regularity for �, but only the regularity L2(0, T ; H 1per(�)′ +

L(2p+2)/(2p+1)(�)) for ��/�t(with ∇��/�t ∈ L2(0, T ; L2(�)2)). To do so, we proceed asin Remark 2.2, except that we do not use (2.22). In order to obtain an estimate on ��/�t

(which is necessary for the passage to the limit in the nonlinear term), we rewrite the problemin the form

dL�

dt+ �div(B∇��) − g′(div(B∇�)) = 0, (2.51)

where

Lv = v − d.∇v − div(B∇v). (2.52)

We thus deduce from the above estimates that L��/�t is bounded in L2(0, T ; H 1per(�)′ +

L(2p+2)/(2p+1)(�)), which yields, since ∇��/�t is bounded in L2(0, T ; L2(�)2) (whichimplies that −d.∇��/�t − div(B∇��/�t) is bounded in L2(0, T ; H 1

per(�)′)), that ��/�t

is also bounded in L2(0, T ; H 1per(�)′ + L(2p+2)/(2p+1)(�)).

References

[1] J.W. Barrett, J.F. Blowey, Finite element approximation of the Cahn–Hilliard equation with concentrationdependent mobility, Math. Comput. 68 (1999) 487–517.

[2] A. Bonfoh,A. Miranville, On Cahn–Hilliard–Gurtin equations, NonlinearAnalysis, vol. 47 no. 5, Proceedingsof the third World Congress of Nonlinear Analysts, 2001, 3455–3466.

[3] J.W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961) 795–801.[4] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 2

(1958) 258–267.[5] M. Carrive, A. Miranville, A. Piétrus, The Cahn–Hilliard equation for deformable continua, Adv. Math. Sci.

Appl. 10 (2) (2000) 539–569.[6] L. Cherfils, A. Miranville, Generalized Cahn–Hilliard equations with a logarithmic free energy, Rev. Real

Acad. Ciencias 94 (1) (2000) 19–32.[7] J.W. Cholewa, T. Dlotko, Global attractors of the Cahn–Hilliard system, Bull. Austral. Math. Soc. 49 (1994)

277–302.[8] A. Debussche, L. Dettori, On the Cahn–Hilliard equation with a logarithmic free energy, Nonlinear Anal.

TMA 24 (1995) 1491–1514.[9] C.M. Elliott, H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27

(2) (1996) 404–423.[10] E. Fried, M. Gurtin, Continuum theory of thermally induced phase–transitions based on an order parameter,

Physica D 68 (1993) 326–343.[11] E. Fried, M. Gurtin, Dynamic solid–solid phase transitions with phase characterized by an order parameter,

Physica D 72 (1994) 287–308.[12] H. Garcke, Habilitation Thesis, Bonn University, 2000.[13] M. Gurtin, Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance,

Physica D 92 (1996) 178–192.[14] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.[15] D. Li, C. Zhong, Global attractor for the Cahn–Hilliard system with fast growing nonlinearity, J. Differential

Equations 149 (2) (1998) 191–210.[16] A. Miranville, Some generalizations of the Cahn–Hilliard equation, Asymptotic Anal. 22 (2000) 235–259.[17] A. Miranville, Existence of solutions for a Cahn–Hilliard–Gurtin model, C. R. Acad. Sci. Paris Ser. I 331

(10) (2000) 845–850.


[18] A. Miranville, Long-time behavior of some models of Cahn–Hilliard equations in deformable continua,Nonlinear Anal. 2 (B3) (2001) 273–304.

[19] A. Miranville, Consistent models of Cahn–Hilliard–Gurtin equations with Neumann boundary conditions,Physica D 158 (1–4) (2001) 233–257.

[20] B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of a class of pattern formationequations, Comm. Partial Differential Equations 14 (2) (1989) 245–297.

[21] A. Novick-Cohen, On the viscous Cahn–Hilliard equation, in: J.M. Ball (Ed.), Material Instabilities inContinuum and Related Problems, Oxford University Press, Oxford, 1988, pp. 329–342.

[22] R. Temam, Infinite Dimensional Dynamical Systems in Physics and Mechanics, second ed., Springer, Berlin,1997.

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A new formulation of the Cahn–Hilliard equation

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