ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

A GENERALIZED SOLUTION TO A CAHN-HILLIARD/ALLEN-CAHN SYSTEM

JOS ´E LUIZ BOLDRINI, PATR´ICIA NUNES DA SILVA

Abstract. We study a system consisting of a Cahn-Hilliard and several Allen-
Cahn type equations. This system was proposed by Fan, L.-Q. Chen, S. Chen
and Voorhees for modelling Ostwald ripening in two-phase system. We prove
the existence of a generalized solution whose concentration component is in
L^{∞}.

1. Introduction

Ostwald ripening is a phenomenon observed in a wide variety of two-phase sys- tems in which there is coarsening of one phase dispersed in the matrix of another.

Because its practical importance, this process has been extensively studied in sev- eral degrees of generality. In particular for Ostwald ripening of anisotropic crystals, Fan et al. [6] presented a model taking in consideration both the evolution of the compositional field and of the crystallographic orientations. In the work of Fan et al. [6], there are also numerical experiments used to validate the model, but there is no rigorous mathematical analysis of the model. Our objective in this paper is to do such mathematical analysis.

By defining orientation and composition field variables, the kinetics of coupled grain growth can be described by their spatial and temporal evolution, which is related with the total free energy of the system. The microstructural evolution of Ostwald ripening can be described by the Cahn-Hilliard/Allen-Cahn system

∂tc=∇ ·[D∇(∂cF −κc∆c)], (x, t)∈ΩT

∂_{t}θ_{i}=−L_{i}(∂_{θ}_{i}F −κ_{i}∆θ_{i}), (x, t)∈Ω_{T}

∂nc=∂n(∂cF −κc∆c) =∂nθi= 0, (x, t)∈ST

c(x,0) =c0(x), θi(x,0) =θi0(x), x∈Ω

(1.1)

for i = 1, . . . , p. Here, Ω is the physical region where the Ostwald process is occurring; ΩT = Ω×(0, T); ST = ∂Ω×(0, T); 0 < T < +∞; n denotes the unitary exterior normal vector and ∂n is the exterior normal derivative at the boundary; c(x, t) is the compositional field (fraction of the soluto with respect to

2000Mathematics Subject Classification. 47J35, 35K57, 35Q99.

Key words and phrases. Cahn-Hilliard and Allen-Cahn equations; Ostwald ripening;

phase transitions.

c

2004 Texas State University - San Marcos.

Submitted June 10, 2004. Published October 25, 2004.

P.N.d.S. was supported by grant 98/15946-5 from FAPESP, Brazil.

1

the mixture) which takes one value within the matrix phase, another value within a
second phase grain andc(x, t) varies between these values at the interfacial region
between the matrix phase and a second phase grain;θi(x, t), fori= 1, . . . , p, are the
crystallographic orientations fields; D,λ_{c}, L_{i}, λ_{i} are positive constants related to
the material properties. The function F =F(c, θ1, . . . , θ_{p}) is the local free energy
density which is given by

F(c, θ1, . . . , θp)

=−A

2(c−c_{m})^{2}+B

4(c−c_{m})^{4}+D_{α}

4 (c−c_{α})^{4}
+Dβ

4 (c−cβ)^{4}+

p

X

i=1

[−γ

2(c−cα)^{2}θ_{i}^{2}+δ
4θ^{4}_{i}] +

p

X

i=1 p

X

i6=j=1

εij

2 θ^{2}_{i}θ^{2}_{j}.

(1.2)

where c_{α} and c_{β} are the solubilities in the matrix phase and the second phase
respectively, and c_{m}= (c_{α}+c_{β})/2. The positive coefficientsA, B, D_{α}, D_{β}, γ, δ
andε_{ij} are phenomenological parameters.

In this paper we obtain a (p+ 2)-tuple which satisfies a variational inequal- ity related to Problem (1.1) and also satisfies the physical requirement that the concentration takes values in the closed interval [0,1].

Our approach to the problem is to analyze a three-parameter family of suitable systems which contain a logarithmic perturbation term and approximate the model presented by Fan et al. [6]. In this analysis, we show that the approximate solutions converge to a generalized solution of the original continuous model and this, in particular, will furnish a rigorous proof of the existence of generalized solutions (see the statement of Theorem 2.1). Our approach is similar to that used by Passo et al. [3] for an Cahn-Hilliard/Allen-Cahn system with degenerate mobility.

2. Existence of Solutions

Including the physical restriction on the concentration, Problem (1.1) is stated as follows:

∂_{t}c=∇[D∇(∂_{c}F −κ_{c}∆c)], (x, t)∈Ω_{T}

∂tθi=−Li(∂θiF −κi∆θi), (x, t)∈ΩT

∂nc=∂n(∂cF −κc∆c) =∂nθi= 0, (x, t)∈ST

c(x,0) =c_{0}(x), θ_{i}(x,0) =θ_{i0}(x), x∈Ω
0≤c≤1 (x, t)∈Ω_{T}

(2.1)

fori= 1, . . . , p.

Throughout this paper, standard notation will be used for the several required
functional spaces. We denote byf the mean value of f in Ω of a givenf ∈L^{1}(Ω).

The duality pairing betweenH^{1}(Ω) and its dual is denoted byh·,·iand (·,·) denotes
the inner product inL^{2}(Ω). We will prove the following:

Theorem 2.1. Let T >0 andΩ⊂R^{d},1≤d≤3 be a bounded domain withC^{3}–
boundary. For all c_{0}, θ_{i0}, i= 1, . . . , p, satisfying c_{0}, θ_{i0} ∈ H^{1}(Ω), for i= 1, . . . , p,
0≤c_{0} ≤1, there exists a unique(p+ 2)-tuple(c, w−w, θ_{1}, . . . , θ_{p})such that, for
i= 1, . . . , p,

(a) c, θi∈L^{∞}(0, T, H^{1}(Ω))∩L^{2}(0, T, H^{2}(Ω)),
(b) w∈L^{2}(0, T, H^{1}(Ω))

(c) ∂_{t}c∈L^{2}(0, T,[H^{1}(Ω)]^{0}),∂_{t}θ_{i}∈L^{2}(Ω_{T})

(d) 0≤c≤1 a.e. inΩT

(e) c(x,0) =c0(x),θi(x,0) =θi0(x)

(f) ∂cF(c, θ1, . . . , θp),∂θiF(c, θ1, . . . , θp)∈L^{2}(ΩT)
(g) ∂_{n}c_{|}

ST =∂_{n}θ_{i|}

ST = 0 inL^{2}(S_{T})
(h) (c, w, θ_{1}, . . . , θ_{p}) satisfies

Z T 0

h∂tc, φidt=− Z Z

ΩT

∇w∇φ, ∀φ∈L^{2}(0, T, H^{1}(Ω)), (2.2)

Z T 0

ξ(t){κcD(∇c,∇φ− ∇c)−(w−D∂cF(c, θ1, . . . , θp), φ−c)}dt≥0, (2.3)
for allξ∈C[0, T), ξ≥0, ∀φ∈K={η∈H^{1}(Ω), 0≤η≤1, η=c_{0}}, and

Z Z

ΩT

∂tθiψi=− Z Z

ΩT

L(∂θ_{i}F(c, θ1, . . . , θp)−κi∆θi)ψi, (2.4)
for allψi∈L^{2}(ΩT), i= 1, . . . , p, whereF is given by (1.2).

Remark 2.2. The inequality obtained (2.3) is similar to one obtained by Elliott and Luckhaus [5] in the case of the deep quench limit problem for a system of nonlinear diffusion equations.

Remark 2.3. We observe that (2.3) comes from the fact that classically w is expected to be equal toD(∂cF −κc∆c) up to a function of time.

Remark 2.4. The solution presented in Theorem 2.1 is a generalized solution of (2.1). In fact, as will be shown at the end of this article, (2.3) holds as an equality in the region where 0< c(x, t)<1 for almost all times.

We start by proving the uniqueness referred to in Theorem 2.1.

Lemma 2.5. Consider a solution of (2.2)–(2.4) as in Theorem 2.1. Under the hy- potheses (a)–(e) and (h) of Theorem 2.1, the componentsc,θ1,. . .,θpare uniquely determined; the component wis uniquely determined up to a function of time.

Proof. We argue as Elliott and Luckhaus [5]. We introduce the Green’s operator
G: givenf ∈[H^{1}(Ω)]^{0}_{null}={f ∈[H^{1}(Ω)]^{0}, hf,1i= 0}, we define Gf ∈H^{1}(Ω) as
the unique solution of

Z

Ω

∇Gf∇ψ=hf, ψi, ∀ψ∈H^{1}(Ω) and
Z

Ω

Gf = 0.

Letz^{c}=c_{1}−c_{2},z^{w}=w_{1}−w_{2} andz^{θ}^{i} =θ_{i1}−θ_{i2}, i= 1, . . . , pbe the differences
of two pair of solutions to (2.2)–(2.4) as in Theorem 2.1. Since equation (2.2)
implies that the mean value of the composition field in Ω is conserved, we have
that (z^{c},1) = 0 and we find from (2.2) that

−Gz_{t}^{c}=z^{w}.

The definition of the Green operator and the fact that (z^{c},1) = 0 give

−(∇Gz_{t}^{c},∇Gz^{c}) =−(Gz_{t}^{c}, z^{c}) = (z^{w}, z^{c}) = (z^{w}, z^{c}).

Sincec_{1}, c_{2}∈K={η∈H^{1}(Ω), 0≤η≤1, η=c_{0}}, we find from (2.3) that

−κcD|∇z^{c}|^{2}+ (z^{w}, z^{c})−D(∂_{c}F(c1, θ_{11}, . . . , θ_{p1})−∂_{c}F(c2, θ_{12}, . . . , θ_{p2}), z^{c})≥0.

Thus, we have 1 2

d

dt|∇Gz^{c}|+κ_{c}D|∇z^{c}|^{2}

≤ −D(∂cF(c1, θ11, . . . , θp1)−∂cF(c2, θ12, . . . , θp2)−κc∆z^{c}, z^{c}).

We find from (2.4) that D

2Li

d

dt|z^{θ}^{i}|^{2}+Dλi|∇z^{θ}^{i}|^{2}

+D ∂θ_{i}F(c1, θ11, . . . , θp1)−∂θ_{i}F(c2, θ12, . . . , θp2), z^{θ}^{i}

= 0.

By adding the above equations, using the convexity of [F+H](c, θ_{1}, . . . , θ_{p}),
[F+H](c, θ_{1}, . . . , θ_{p}) = Dα

4 (c−c_{α})^{4}+Dβ

4 (c−c_{β})^{4}+δ
4

p

X

i=1

θ^{4}_{i}

where

H(c, θ1, . . . , θp) =A

2(c−cm)^{2}+γ
2

p

X

i=1

c^{2}θ_{i}^{2}−

p

X

i=1 p

X

i6=j=1

ε_{ij}
2 θ_{i}^{2}θ_{j}^{2},
we obtain

1 2

d

dt|∇Gz^{c}|^{2}+κ_{c}D|∇z^{c}|^{2}+

p

X

i=1

[ D 2Li

d

dt|z^{θ}^{i}|^{2}+Dλ_{i}|∇z^{θ}^{i}|^{2}]

≤ ∇(H(c1, θ11, . . . , θp1)−H(c2, θ12, . . . , θp2))·(z^{c}, z^{θ}^{1}, . . . , z^{θ}^{p}),1

(2.5)

To estimate the right-hand side of the above inequality, we use the regularity ofci

andθik. Then

(∇(θ^{2}_{i1}θ^{2}_{j1}−θ_{i2}^{2}θ_{j2}^{2})·(z^{θ}^{i}, z^{θ}^{j}),1)

= 2((θi1θ^{2}_{j1}−θi2θ^{2}_{j2}, θ^{2}_{i1}θj1−θ_{i2}^{2}θj2)·(z^{θ}^{i}, z^{θ}^{j}),1)

= 2((z^{θ}^{i}θ_{j1}^{2} +θ_{i2}(θ^{2}_{j1}−θ_{j2}^{2}), θ^{2}_{i1}θ_{j1}−θ_{i2}^{2}θ_{j2})·(z^{θ}^{i}, z^{θ}^{j}),1)

= 2((z^{θ}^{i}θ_{j1}^{2} +θi2(θj1+θj2)z^{θ}^{j}, z^{θ}^{j}θ_{i1}^{2} +θj2(θi1+θi2)z^{θ}^{i})·(z^{θ}^{i}, z^{θ}^{j}),1)

≤C[|z^{θ}^{i}|^{2}+|z^{θ}^{j}|^{2}] + Dλ_{i}

8(p−1)|∇z^{θ}^{i}|^{2}+ Dλ_{j}

8(p−1)|∇z^{θ}^{j}|^{2}
and

γ(∇(c^{2}_{1}θ^{2}_{i1}−c^{2}_{2}θ^{2}_{i2})·(z^{c}, z^{θ}^{i}),1)≤C[|z^{c}|^{2}+|z^{θ}^{i}|^{2}] +κcD

2p |∇z^{c}|^{2}+Dλi

4 |∇z^{θ}^{i}|^{2}.
The above inequalities and (2.5) imply

1 2

d

dt|∇Gz^{c}|^{2}+κcD

2 |∇z^{c}|^{2}+

p

X

i=1

D 2Li

d

dt|z^{θ}^{i}|^{2}+Dλi

2 |∇z^{θ}^{i}|^{2}

≤C

kz^{c}k^{2}_{L}2(Ω)+

p

X

i=1

kz^{θ}^{i}k^{2}_{L}2(Ω)

.

From the definition of the Green operator,|z^{c}|^{2}= (∇Gz^{c},∇z^{c}). Using the H¨older
inequality, we rewrite the above inequality as

1 2

d

dt|∇Gz^{c}|^{2}+κ_{c}D

4 |∇z^{c}|^{2}+

p

X

i=1

D 2Li

d

dt|z^{θ}^{i}|^{2}+Dλ_{i}

2 |∇z^{θ}^{i}|^{2}

≤C

k∇Gz^{c}k^{2}_{L}2(Ω)+

p

X

i=1

kz^{θ}^{i}k^{2}_{L}2(Ω)

. A standard Gronwall argument then yields

∇Gz^{c}= 0 and z^{θ}^{i} = 0, i= 1, . . . , p,
since

Gz^{c}(0) = 0 and z^{θ}^{i}(0) = 0, i= 1, . . . , p.

The uniqueness is proved since|z^{c}|^{2}= (∇Gz^{c},∇z^{c}) = 0.

As a corollary of this, we have the following:

Lemma 2.6. Under the conditions of Theorem 2.1, when eitherc0 ≡0 orc0≡1 almost everywhere on Ω, there is a solution of (2.2)–(2.4).

Proof. In such cases, since 0 ≤c_{0}(x) ≤ 1, we have in fact that eitherc_{0}(x) = 0
or c_{0}(x) = 1. Now, take c identically zero or one, respectively. Then, equation
(2.2) is trivially satisfied and will imply that w is a constant. Otherwise, (2.3) is
also trivially satisfied and to obtain a solution of the Problem (2.2)–(2.4), we just
have to solve the nonlinear parabolic system (2.4). But this system can be solved
rather easily by standard methods, like Galerkin method, for instance, since the
nonlinearities have the right sign and thus furnish suitable estimates.

By the above lemma, we have uniqueness ofcin the cases where eitherc_{0}≡0 or
c_{0}≡1 almost everywhere on Ω. Thus, to prove Theorem 2.1, it just remain to deal
with the cases where the mean value of the initial conditionc_{0} is strictly between
to zero and one. Thus, in the following we assume that

c_{0}, θ_{i0}∈H^{1}(Ω), i= 1, . . . , p,

0≤c0≤1, c0∈(0,1), (2.6) To obtain the result in Theorem 2.1, we approximate system (2.1) by a three- parameter family of suitable systems which contain a logarithmic perturbation term and then pass to the limit. In Section 3, we use the results of Passo et al. [3]

to construct such perturbed systems and together with some ideas presented by Copetti and Elliott [2] and by Elliott and Luckhaus [5], we take the limit in these systems in the last three sections.

For sake of simplicity of exposition, without loosing generality, we develop the proof for the case of dimension one and for only one orientation field variable, that is, when Ω is a bounded open interval andpis equal to one, and thus we have just one orientation field that we denoteθ. In this case, the local free energy density is reduced to

F(c, θ) =−A

2(c−c_{m})^{2}+B

4(c−c_{m})^{4}+D_{α}

4 (c−c_{α})^{4}+D_{β}

4 (c−c_{β})^{4}
+δ

4θ^{4}−γ

2(c−c_{α})^{2}θ^{2}.

(2.7)

We remark that, even though the cross terms of (1.2) involving the orientation field variables are absent in above expression, their presences whenpis greater than one will not bring any difficulty for extending the result, as we will point out at the end of the paper.

3. Perturbed Systems

In this section we construct a three-parameter family of perturbed systems. The auxiliary parameterM controls a truncation of the local free energyF which will permit the application of an existence result of Passo et al. [3]. The parametersσ and εare associated to the logarithmic term, their introduction will enable us to guarantee that the composition field variablectakes values in the closure of the set I= (0,1).

For each positive constants σ, M andε ∈(0,1), we define the perturbed local free energy density as follows:

FσεM(c, θ) =f(c) +gM(θ) +hM(c, θ) +ε[Fσ(c) +Fσ(1−c)]. (3.1) where the first three terms give a truncation of the originalF(c, θ) given in (2.7), and the last term is a logarithmic perturbation. To obtain a truncation of the local free energy density, we introduce bounded functions whose summation coincides withF for (c, θ)∈[0,1]×[−M, M]. Letf, gM andhM be such that

f(c) =−A

2(c−c_{m})^{2}+B

4(c−c_{m})^{4}+D_{α}

4 (c−c_{α})^{4}+D_{β}

4 (c−c_{β})^{4}, 0≤c≤1,
gM(θ) =δ

4h^{2}_{2}(M;θ) and hM(c, θ) =h1(c)h2(M;θ)
with

h1(c) =−γ

2(c−cα)^{2}, 0≤c≤1,
h2(M;θ) =θ^{2}, −M ≤θ≤M.

Outside the intervals [0,1] and [−M, M], we extend the above functions to satisfy
kfk_{C}2(R)≤U0, kgMk_{C}2(R)≤V0(M), (3.2)
kh_{1}k_{C}2(R)≤W_{0}, kh_{M}k_{C}2(R^{2})≤Z_{0}(M), (3.3)

|h_{2}(M;θ)| ≤Kθ^{2}, |h^{0}_{2}(M;θ)| ≤K|θ|, ∀M >0,| ∀θ∈R, (3.4)
where U0, W0, K >0 are constants and, for each M, V0(M) and Z0(M) are also
constants.

We took the logarithmic termε[Fσ(c)−Fσ(1−c)] as in Passo et al. [3]. Let us denote

F(s) =slns.

Forσ∈(0,1/e), we chooseF_{σ}^{0}(s) such that

F_{σ}^{0}(s) =

σ

2σ−s+ lnσ, ifs < σ,

lns+ 1, ifσ≤s≤1−σ,
f_{σ}(s), if 1−σ < s <2,

1, ifs≥2,

wherefσ∈C^{1}([1−σ,2]) is chosen having the following properties:

f_{σ} ≤F^{0}, f_{σ}^{0} ≥0,

f_{σ}(1−σ) =F^{0}(1−σ), f_{σ}(2) = 1,
f_{σ}^{0}(1−σ) =F^{00}(1−σ), f_{σ}^{0}(2) = 0.

Defining

Fσ(s) =−1 e+

Z s

1 e

F_{σ}^{0}(ξ)dξ,
we haveFσ∈C^{2}(R) andF_{σ}^{00}≥0.

Clearly, FσεM has a lower bound which is independent of σ and ε. We claim that FσεM can also be bounded from below independently of M. To prove this fact, we just have to estimategM(θ) +hM(c, θ). We have

g_{M}(θ) +h_{M}(c, θ) =δ

4h^{2}_{2}(M;θ) +h_{1}(c)h_{2}(M;θ)

=δ

4h2(M;θ)

h2(M;θ) +4 δh1(c)

≥ −h^{2}_{1}(c)

δ ≥ −W_{0}^{2}
δ
Therefore,

−U0−W_{0}^{2}
δ −2

e ≤ FσεM(c, θ) inR^{2},
FσεM(c, θ)< U0+gM(θ)−hM(c, θ) in clI.

(3.5) Then, the perturbed systems we will consider are

∂tc=D(∂cFσεM(c, θ)−κccxx)xx, (x, t)∈ΩT

∂tθ=−L[∂θFσεM(c, θ)−κθxx], (x, t)∈ΩT

∂_{n}c=∂_{n}(∂_{c}FσεM(c, θ)−κ_{c}c_{xx}) =∂_{n}θ= 0 (x, t)∈S_{T}
c(x,0) =c0(x), θ(x,0) =θ0(x), x∈Ω

(3.6)

To solve the above problem, we shall use the next proposition which is an existence result stated by Passo et al. [3] for the system

∂_{t}u= [q_{1}(u, v)(f_{1}(u, v)−κ_{1}u_{xx})_{x}]_{x}, (x, t)∈Ω_{T}

∂tv=−q2(u, v)[f2(u, v)−κ2vxx], (x, t)∈ΩT

∂nu=∂nuxx=∂nv= 0 (x, t)∈ST

u(x,0) =u_{0}(x), v(x,0) =v_{0}(x), x∈Ω

(3.7)

whereqi andfi satisfy the following hypotheses:

(H1) q_{i}∈C(R^{2},R^{+}), withq_{min}≤q_{i}≤q_{max} for some 0< q_{min}≤q_{max};

(H2) f_{1} ∈C^{1}(R^{2},R) and f_{2} ∈C(R^{2},R), withkf1kC^{1}+kf2kC^{0} ≤F_{0} for some
F_{0}>0.

Proposition 3.1. Assuming (H1), (H2) and that u0, v0 ∈H^{1}(Ω), there exists a
pair of functions(u, v)such that

(1) u∈L^{∞}(0, T, H^{1}(Ω))∩L^{2}(0, T, H^{3}(Ω))∩C([0, T];H^{λ}(Ω)),λ <1
(2) v∈L^{∞}(0, T, H^{1}(Ω))∩L^{2}(0, T, H^{2}(Ω))∩C([0, T];H^{λ}(Ω)),λ <1
(3) ∂tu∈L^{2}(0, T,[H^{1}(Ω)]^{0}), ∂tv∈L^{2}(ΩT)

(4) u(0) =u_{0} andv(0) =v_{0} in L^{2}(Ω)
(5) ∂_{n}u_{|}

ST =∂_{n}v_{|}

ST = 0 inL^{2}(S_{T})

(6) (u, v)solves (3.7) in the following sense Z t

0

h∂tu, φi=− Z Z

Ω_{t}

q1(u, v)(f1(u, v)−κ1uxx)xφx, ∀φ∈L^{2}(0, T, H^{1}(Ω))
Z Z

Ωt

∂_{t}vψ=−
Z Z

Ωt

q_{2}(u, v)(f_{2}(u, v)−κ_{2}v_{xx})ψ, ∀φ∈L^{2}(Ω_{T}).

whereΩt= Ω×(0, t)andRR

Ω_{t} is the integral overΩt.

Remark 3.2. The regularity of the test functions with respect to t allow us to obtain the integrals over (0, t), instead of (0, T) as originally presented by Passo et al. [3].

Applying the above proposition, for each ε, σ, M > 0 there exists a solution (cσεM, θσεM) of Problem (3.6) in the following sense

Z t 0

h∂tcσεM, φi=− Z Z

Ω_{t}

D(∂cFσεM(cσεM, θσεM)−κc[cσεM]xx)xφx, (3.8)
forφ∈L^{2}(0, T, H^{1}(Ω)) and

Z Z

Ω_{t}

∂tθσεMψ=− Z Z

Ω_{t}

L(∂θFσεM(cσεM, θσεM)−κ[θσεM]xx)ψ, (3.9)
forψ∈L^{2}(Ω_{T}). Let us observe that equation forc in equation (3.8) implies that
the mean value ofc_{σεM} in Ω is

cσεM(t) =c0∈(0,1) (3.10)

4. Limit as M → ∞

In this section we obtain some a priori estimates that will allow us to take the limit in the parameter M. Actually, some of these estimates are also independent of the parametersσandεand will be useful in next sections.

Lemma 4.1. There exists a constant C1 independent of M (sufficiently large), σ (sufficiently small) andε such that

(1) kcσεMk_{L}^{∞}_{(0,T ,H}1(Ω))≤C1

(2) kθσεMk_{L}^{∞}_{(0,T ,H}1(Ω))≤C1

(3) k(∂cFσεM −κc(cσεM)xx)xk_{L}2(ΩT)≤C1

(4) k∂θFσεM−κ(θσεM)xxk_{L}2(ΩT)≤C1

(5) k∂tcσεMk_{L}^{∞}_{(0,T ,[H}1(Ω)]^{0})≤C1

(6) k∂tθσεMk_{L}2(ΩT)≤C1

(7) kFσεM(cσεM, θσεM)k_{L}^{∞}_{(0,T ,L}1(Ω))≤C1

Proof. To obtain items 3, 4 and 7, we argue as Passo et al. [3] and Elliott and Garcke [4]. First, we observe that by the regularity of cσεM and θσεM, we could take

∂_{c}FσεM−κ_{c}(c_{σεM})_{xx} and ∂_{θ}FσεM−κ(θ_{σεM})_{xx}

as test functions in the equations (3.8) and (3.9), respectively. By adding the resulting identities, we obtain

Z t 0

h∂tc_{σεM}, ∂_{c}FσεM −κ_{c}(c_{σεM})_{xx}i+
Z Z

Ω_{t}

∂_{t}θ_{σεM}∂_{θ}FσεM−κ(θ_{σεM})_{xx}

=− Z Z

Ωt

D[(∂cFσεM−κc(cσεM)xx)x]^{2}−
Z Z

Ωt

L[∂θFσεM−κ(θσεM)xx]^{2}.
(4.1)

Now, given a smallh >0, we consider the functions cσεM h(x, t) = 1

h Z t

t−h

cσεM(τ, x)dτ

where we setcσεM(x, t) =c0(x) fort≤0. Since∂tcσεM h(x, t)∈L^{2}(ΩT), we have
Z T

0

h(cσεM h)t,[∂cFσεM h−κc(cσεM h)xx]idt +

Z Z

ΩT

(θσεM)t[∂θFσεM h−κ(θσεM)xx]

= Z

Ω

[κ_{c}

2 |[cσεM h(t)]_{x}|^{2}+κ

2|[θσεM]_{x}(t)|^{2}+FσεM h(t)]

− Z

Ω

[κ_{c}

2 |[c0]_{x}|^{2}+κ

2|[θ0]_{x}|^{2}+FσεM(c_{0}, θ_{0})].

Taking the limit ashtends to zero in the above expression and using the result in (4.1), we obtain

Z Z

Ωt

D[(∂cFσεM −κc(cσεM)xx)x]^{2}+
Z Z

Ωt

L[∂θFσεM−κ(θσεM)xx]^{2}
+κ_{c}

2 k[cσεM]_{x}(t)k^{2}_{L}2(Ω)+κ

2k[θσεM]_{x}(t)k^{2}_{L}2(Ω)+
Z

Ω

FσεM(t)

=κ_{c}

2 k[c0]_{x}k^{2}_{L}2(Ω)+κ

2k[θ0]_{x}k^{2}_{L}2(Ω)+
Z

Ω

FσεM(c_{0}, θ_{0})

for almost every t ∈ (0, T]. Using (2.6) and (3.5), we could choose M0 and σ0, depending only on the initial conditions, to obtain for allM > M0and allσ < σ0

Z Z

ΩT

D[(∂cFσεM −κc(cσεM)xx)x]^{2}+
Z Z

ΩT

L[∂θFσεM−κ(θσεM)xx]^{2}
+κ_{c}

2k[cσεM]_{x}(t)k^{2}_{L}2(Ω)+κ

2k[θσεM]_{x}(t)k^{2}_{L}2(Ω)+
Z

Ω

FσεM(t)≤C_{1}

(4.2)

which implies items 3, 4 and 7 since we have (3.5). Using the Poincar´e’s inequality, (3.5) and (3.10), Item 1 is also verified.

To prove Item 6, we chooseψ=∂tθσεM as a test function in (3.9), which yields Z Z

ΩT

[∂tθσεM]^{2}=−
Z Z

ΩT

(∂θFσεM −κ(θσεM)xx)∂tθσεM

≤Z Z

Ω_{T}

(∂_{θ}F_{σεM} −κ(θ_{σεM})_{xx})^{2}1/2Z Z

Ω_{T}

[∂_{t}θ_{σεM}]^{2}1/2

. Since

Z

Ω

θ_{σεM}^{2} ≤2
Z

Ω

|θ_{0}|^{2}+ 2t
Z Z

Ω_{T}

(∂_{t}θ_{σεM})^{2}dτ ≤C_{2},

Item 6 and (4.2), it follows that Item 2 is verified. Finally, Item 5 follows since

Z T

0

h∂tcσεM, φi ≤Z Z

ΩT

[(∂cFσεM −κc(cσεM)xx)x]^{2}1/2Z Z

ΩT

(φx)^{2}1/2

for allφ∈L^{2}(0, T, H^{1}(Ω)).

Remark 4.2. From (4.2), using (3.5), we obtain Z Z

ΩT

D[(∂cFσεM −κc(cσεM)xx)x]^{2}+
Z Z

ΩT

L[∂θFσεM −κ(θσεM)xx]^{2}
+κc

2 k[cσεM(t)]xk^{2}_{L}2(Ω)+κ

2k[θσεM]x(t)k^{2}_{L}2(Ω)≤C1.

(4.3)

Lemma 4.3. ForM sufficiently large and σ sufficiently small, there exist a con-
stantC_{3} independent ofσ, M andεand a constantC_{3}^{0}(σ)independent ofM andε
such that

(1) k∂cFσεMkL^{2}(0,T ,H^{1}(Ω))≤C_{3}^{0},
(2) k∂θFσεMkL^{2}(Ω_{T})≤C_{3},
(3) k[cσεM]_{xx}kL^{2}(Ω_{T})≤C_{3},
(4) k[θσεM]_{xx}kL^{2}(Ω_{T})≤C_{3},

Proof. First, we prove items 2 and 4. From Item 4 of Lemma 4.1, we have Z Z

Ω_{T}

(∂_{θ}F_{σεM})^{2}−2κ
Z Z

Ω_{T}

∂_{θ}F_{σεM}[θ_{σεM}]_{xx}+κ^{2}
Z Z

Ω_{T}

[θ_{σεM}]^{2}_{xx}≤C_{3}. (4.4)
Since

∂θFσεM[θσεM]xx= [g^{0}_{M}(θσεM) +∂θhM(cσεM, θσεM)][θσεM]xx

= [g^{0}_{M}(θσεM) +h1(cσεM)h^{0}_{2}(M;θσεM)][θσεM]xx,
using (3.3) and (3.4), we obtain

2κ∂_{θ}F_{σεM}[θ_{σεM}]_{xx}≤κ^{2}

2 [θ_{σεM}]^{2}_{xx}+C_{3}[θ^{6}_{σεM} +W_{0}^{2}θ_{σεM}^{2} ].

Thus, from Item 2 of Lemma 4.1, it follows from (4.4) that Z Z

Ω_{T}

(∂θFσεM)^{2}+κ^{2}
2

Z Z

Ω_{T}

[θσεM]^{2}_{xx}≤C3. (4.5)
Now, we prove Item 3. Defining,HσεM =∂cFσεM−κc[cσεM]xx, since [cσεM]x|ST =
0, we have

Z Z

ΩT

HσεM = Z Z

ΩT

∂cFσεM, and from Item 3 of Lemma 4.1,

Z Z

Ω_{T}

[H_{σεM}]^{2}_{x}≤C_{1}.

Using the definition ofFσεM, given in (3.1), and an integration by parts, we obtain Z Z

ΩT

H_{σεM}^{2}

= Z Z

Ω_{T}

(∂_{c}F_{σεM})^{2}+ 2κ_{c}ε
Z Z

Ω_{T}

[F_{σ}^{00}(c_{σεM}) +F_{σ}^{00}(1−c_{σεM})][c_{σεM}]^{2}_{x}

−2κcL Z Z

Ω_{T}

(f^{0}(cσεM) +h^{0}_{1}(cσεM)h2(M;θσεM))[cσεM]xx+κ^{2}_{c}
Z Z

Ω_{T}

[cσεM]^{2}_{xx}.
On the other hand, we can write

Z Z

Ω_{T}

H_{σεM}^{2} =
Z Z

Ω_{T}

[HσεM−HσεM]^{2}+
Z Z

Ω_{T}

HσεM 2

≤CP

Z Z

ΩT

[HσεM]^{2}_{x}+
Z Z

ΩT

(∂cFσεM)^{2}

whereCP denotes the constant appearing in Poincar´e’s inequality. From these two
last results, Item 3 follows recalling that [F_{σ}^{00}(cσεM) +F_{σ}^{00}(1−cσεM)]≥0 and using
(3.2), (3.3), (3.4) and Item 2 of Lemma 4.1.

Finally, recalling that for eachσ,F_{σ}^{0}(s) is bounded inR, using again the defini-
tion off andh_{M} and Item 2 of Lemma 4.1, we obtain

k∂cFσεMk^{2}_{L}2(Ω_{T})≤C
Z Z

ΩT

{[f^{0}(cσεM)]^{2}+ [h^{0}_{1}(cσεM)]^{2}[h2(M;θσεM)]^{2}
+ε^{2}[F_{σ}^{0}(cσεM)−F_{σ}^{0}(1−cσεM)]}

≤C{[U_{0}^{2}|ΩT|+W_{0}^{2}kθσεMk^{4}_{L}4] +C(σ)} ≤C_{3}^{0}(σ).

A similar argument shows thatk[∂_{c}F_{σεM}]_{x}k^{2}_{L}2(Ω_{T}) is also bounded by a constant
which depends only onσ. Thus, we have proved the Item 1.

We can now state the following result.

Proposition 4.4. Forσ(sufficiently small), there exists a pair(cσε, θσε)such that:

(1) c_{σε}∈L^{∞}(0, T, H^{1}(Ω))∩L^{2}(0, T, H^{3}(Ω))
(2) θ_{σε}∈L^{∞}(0, T, H^{1}(Ω))∩L^{2}(0, T, H^{2}(Ω))
(3) ∂_{t}c_{σε}∈L^{2}(0, T,[H^{1}(Ω)]^{0}), ∂_{t}θ_{σε}∈L^{2}(Ω_{T})
(4) ∂cFσε(cσε, θσε), ∂θFσε(cσε, θσε)∈L^{2}(ΩT)
(5) cσε(0) =c0 andθσε(0) =θ0 in L^{2}(Ω)
(6) [cσε]x|_{ST} = [θσε]x|_{ST} = 0 inL^{2}(ST)

(7) (c_{σε}, θ_{σε})solves the perturbed system (3.6) in the following sense:

Z T 0

h∂tcσε, φi=− Z Z

ΩT

D[∂cFσε(cσε, θσε)−κc(cσε)xx]xφx (4.6)
for allφ∈L^{2}(0, T, H^{1}(Ω)), and

Z Z

Ω_{T}

∂_{t}θ_{σε}ψ=−
Z Z

Ω_{T}

L(∂_{θ}Fσε(c_{σε}, θ_{σε})−κ(θ_{σε})_{xx})ψ (4.7)
for allψ∈L^{2}(ΩT), andFσε is given by

F_{σε}(c, θ) =f(c) +δ

4θ^{4}+h_{1}(c)θ^{2}+ε[F_{σ}(c) +F_{σ}(1−c)].

Proof. First, let us observe that from Item 3 of Lemma 4.1 and Item 1 of Lemma 4.3,
the norm of [cσεM]xxx in L^{2}(ΩT) is bounded by a constant which does not depend
onM. This fact, the estimates of Lemmas 4.1 and 4.3 together with a compactness
argument imply that there exists a subsequence (still denoted by{(cσεM, θ_{σεM})})
that satisfies (asM goes to infinity)

cσεM, θσεM converge weakly-* to cσε, θσε in L^{∞}(0, T, H^{1}(Ω)),
cσεM, converges weakly to cσε in L^{2}(0, T, H^{3}(Ω)),
θσεM, converges weakly to θσε in L^{2}(0, T, H^{2}(Ω)),

∂tcσεM, converges weakly to ∂tcσε in L^{2}(0, T,[H^{1}(Ω)]^{0}),

∂tθσεM, converges weakly to ∂tθσε in L^{2}(ΩT)
cσεM, θσεM converge to cσε, θσε in L^{2}(ΩT).

By recalling Lemmas 4.1 and 4.3, items 1–3 now follow. Now, items 1 and 2 of Lemma 4.3 imply that

∂cFσεM(cσεM, θσεM) converges weakly to G in L^{2}(ΩT),

∂θFσεM(cσεM, θσεM) converges weakly to H in L^{2}(ΩT).

Since the strong convergence of the sequence (cσεM) implies that (at least for a subsequence)∂cFσεM(cσεM, θσεM) converges pointwise in ΩT, it follows from Lions [7, Lemma 1.3], that G=∂cFσε(cσε, θσε). Similarly, we haveH=∂θFσε(cσε, θσε).

Thus Item 4 is proved.

Item 5 is straightforward. Now, by compactness we have that
cσεM converges to cσε in L^{2}(0, T, H^{2−λ}(Ω)), λ >0,
θσεM converges to θσε in L^{2}(0, T, H^{2−λ}(Ω)), λ >0,

which imply Item 6. To prove Item 7, by using the previous convergences, we pass to the limit asM goes to infinity in the equations (3.8) and (3.9).

5. Limit as σ→0^{+}

In this section we obtain some a priori estimates that allow taking the limit in the parameterσ.

First, let us note that (4.6) implies that the mean value ofc_{σε} in Ω is given by

cσε(t) =c0∈(0,1), (5.1)

We start with the following Lemma.

Lemma 5.1. There exists a constantC1independent ofεandσ(sufficiently small) such that

(1) kcσεk_{L}^{∞}_{(0,T ,H}1(Ω))≤C1

(2) kθσεk_{L}∞(0,T ,H^{1}(Ω))≤C1

(3) k[∂cFσε−κc(cσε)xx]xk_{L}2(ΩT)≤C1

(4) k∂θFσε−κ(θσε)xxk_{L}2(ΩT)≤C1

(5) k∂tcσεk_{L}∞(0,T ,[H^{1}(Ω)]^{0})≤C1

(6) k∂tθσεk_{L}2(ΩT)≤C1

(7) kFσε(cσε, θσε)k_{L}∞(0,T ,L^{1}(Ω))≤C1

Proof. Let us observe that in the proof of Proposition 4.4, we have identified the weak limits, when M goes to infinity, of the sequences ∂cFσεM and ∂θFσεM as

∂cFσε and ∂θFσε, respectively. Thus, by taking the inferior limit as M goes to infinity, of estimate (4.3), we obtain

κc

2 k(cσε)xk^{2}_{L}∞(0,T ,L^{2}(Ω))+κ

2k(θσε)xk^{2}_{L}∞(0,T ,L^{2}(Ω))

+Dk[∂cFσε−κc(cσε)xx]xk^{2}_{L}2(Ω_{T})+Lk∂θFσε−κ(θσε)xxk^{2}_{L}2(Ω_{T})≤C1.

(5.2)
The items 3 and 4 follow from (5.2). Using (5.2), Poincar´e’s inequality and (5.1),
we obtain Item 1. To prove items 2, 5 and 6, we just take the inferior limit of
items 2, 5 and 6 of Lemma 4.1. Finally, using (3.2), (3.3), (3.4) and the esti-
mates of Lemma 4.1, we can estimate kf(c) + ^{δ}_{4}θ^{4}+h_{1}(c)θ^{2}kL^{∞}(0,T ,L^{1}(Ω)). Also
with the estimates of Lemma 4.1, we get the strong convergence of a subsequence
of (c_{σε}). Using this convergence and the Fatou’s Lemma, we get a bound for
kε[F_{σ}(c) +F_{σ}(1−c)]k_{L}∞(0,T ,L^{1}(Ω))which together with the previous estimate yield

Item 7.

As Passo et al. [3], by arguing in a standard way (see Bernis and Friedman [1]

for a proof, p. 183), we obtain

Corolary 5.2. There exists a constant C2 independent of ε and σ (sufficiently small) such that

kcσεk

C^{0,}^{1}2,1

8(cl Ω_{T})≤C2 and kθσεk

C^{0,}^{1}2,1

8(cl Ω_{T})≤C2 (5.3)
By Corollary 5.2, we can extract a subsequence (still denoted by (c_{σε}, θ_{σε})) such
that

(cσε, θσε) converges uniformly to (cε, θε) in cl ΩT asσapproaches zero,

cε∈C^{0,}^{1}^{2}^{,}^{1}^{8}(cl ΩT) and θε∈C^{0,}^{1}^{2}^{,}^{1}^{8}(cl ΩT). (5.4)
We now demonstrate that the limitcεlies within the interval

I={c∈R,0< c <1}.

Lemma 5.3. |ΩT \ B(cε)|= 0with B(c) ={(x, t)∈cl ΩT, c(x, t)∈I}.

Proof. Arguing as Passo et al. [3], letN denote the operator defined as minus the
inverse of the Laplacian with zero Neumann boundary conditions. That is, given
f ∈[H^{1}(Ω)]^{0}_{null}={f ∈[H^{1}(Ω)]^{0}, hf,1i= 0}, we defineN f ∈H^{1}(Ω) as the unique
solution of

Z

Ω

(N f)^{0}ψ^{0}=hf, ψi, ∀ψ∈H^{1}(Ω) and
Z

Ω

N f = 0. (5.5)
By (5.1) and Item 1 of Lemma 5.1, N(c_{σε} −c_{σε}) is well defined. Choosing
φ=N(c_{σε}−c_{σε}) as a test function in the equation (4.6), we have

Z T 0

h∂tcσε, N(cσε−cσε)idt

=− Z Z

ΩT

D[∂cFσε−κc(cσε)xx]x[N(cσε−cσε)]x

=− Z Z

Ω_{T}

D(c_{σε}−c_{σε})∂_{c}F_{σε}(c_{σε}, θ_{σε})−Dκ_{c}
Z Z

Ω_{T}

[(c_{σε})_{x}]^{2}

Now, estimates in Lemma 5.1 and the definition ofN imply Z Z

Ω_{T}

(cσε−cσε)∂cFσε(cσε, θσε)≤C4. (5.6) We observe that the following identity holds for anym∈R,

(c−m)∂cFσε(c, θ)

= (c−m)[f^{0}(c) +h^{0}_{1}(c)θ^{2}+ε(F_{σ}^{0}(c)−F_{σ}^{0}(1−c))]

=ε{c[F_{σ}^{0}(c)−1] + (1−c)[F_{σ}^{0}(1−c)−1] + 2}

+ (c−m)[f^{0}(c) +h^{0}_{1}(c)θ^{2}]−ε−εmF_{σ}^{0}(c)−ε(1−m)F_{σ}^{0}(1−c)

(5.7)

We observe that the terms inside the braces are bounded from below since for any σ∈(0,1/e), we have

−1/e≤σlnσ≤s[F_{σ}^{0}(s)−1]≤0, s≤σ,

−1/e≤slns=s[F_{σ}^{0}(s)−1]≤0, σ≤s≤1−σ,

−2≤s[F_{σ}^{0}(s)−1]≤0, 1−σ≤s≤2,
0 =s[F_{σ}^{0}(s)−1], s≥2.

We now recall that the mean value ofcσεin Ω is conserved and is equal toc0which
belongs to the interval (0,1). Thus, sincef^{0}, h^{0}_{1} are uniformly bounded, using the
estimates in Lemma 5.1, by setting m =c_{σε} =c_{0} in (5.7) and noting F_{σ}^{0} ≤1, it
follows from (5.6) that

−ε Z Z

Ω_{T}

[F_{σ}^{0}(c_{σε}) +F_{σ}^{0}(1−c_{σε})]≤C_{4} (5.8)
To complete the proof, suppose by contradiction that the set Ω_{T} \ B(cε) has a
positive measure. We start supposing that

A={(x, t)∈ΩT, cε≤0}

has positive measure. SinceF_{σ}^{0} ≤1, the estimate (5.8) gives

−ε Z Z

A

F_{σ}^{0}(cσε)≤C4.

Note, however, that the uniform convergence ofcσε implies that

∀λ >0, ∃σ_{λ}, c_{σε}≤λ, ∀(x, t)∈A, σ < σ_{λ}
therefore, due to the convexity ofF_{σ}, we haveF_{σ}^{0}(c_{σε})≤F_{σ}^{0}(λ). Hence

−ε|A|(lnλ+ 1) =−ε lim

σ→0^{+}

Z Z

A

F_{σ}^{0}(λ)≤C_{4}

which leads to a contradiction forλ∈(0,1) sufficiently small. The same argument shows thatB ={(x, t)∈ΩT, cε≥1}has zero measure.

In the next lemma we derive additional estimates which allow us to pass to the limit asσtends to zero. Its proof follows directly from the estimates of Lemma 5.1.

Lemma 5.4. There exists a constant C3 which is independent of ε and σ (suffi- ciently small) such that

(1) k∂θFσεk_{L}2(ΩT)≤C3,
(2) k[cσε]xxk_{L}2(ΩT)≤C3,
(3) k[θσε]xxk_{L}2(Ω_{T})≤C3,

To pass to the limit as σ goes to zero, we need an estimate of ∂cFσε that is independent of σ. We cannot repeat the argument that we used in Lemma 4.3 because there we obtained with a constant that depends onσ. The desired estimate will be obtained by using the next lemma, presented by Copetti and Elliott [2, p. 48], and by Elliott and Luckhaus [5, p. 23].

Lemma 5.5. Let v ∈ L^{1}(Ω) such that there exist positive constants δ_{1} and δ_{1}^{0}
satisfying

8δ_{1}< 1

|Ω|

Z

Ω

vdx <1−8δ_{1}, (5.9)

1

|Ω|

Z

Ω

([v−1]_{+}+ [−v]+)dx < δ_{1}^{0}. (5.10)
If 16δ_{1}^{0} < δ^{2}_{1} then

|Ω^{+}_{δ}

1|=|{x∈Ω, v(x)>1−2δ1}|<(1−δ1)|Ω|

and

|Ω^{−}_{δ}

1|=|{x∈Ω, v(x)<2δ1}|<(1−δ1)|Ω|.

Our task is now to verify the hypothesis of this lemma for the functionscσε. To obtain (5.10), we note that items 2 and 7 of Lemma 5.1, (3.2) and (3.3) imply that, for almost everyt∈[0, T],

ε Z

Ω

[Fσ(cσε) +Fσ(1−cσε)]dx

≤C1+

f(cσε) +δθ_{σε}^{4} /4 +h1(cσε)θ_{σε}^{2}

_{L}_{∞}_{(0,T ,L}_{1}_{(Ω))}≤C.

From the definition ofFσ(s) fors < σ (see page 6), we obtain Z

{cσε<0}

Fσ(cσε)dx

≥ |lnσ|

Z

Ω

[−cσε(·, t)]+dx−σ[|lnσ|+ 2σ]|Ω| −σkcσε(t)k_{L}2(Ω)|Ω|^{1/2}.
Hence, sinceFσ(s)≥ −1/e, we have

Z

Ω

Fσ(cσε)dx

≥ |lnσ|

Z

Ω

[−cσε(·, t)]+dx−σ[|lnσ|+ 2σ+e^{−1}]|Ω| −σkcσε(t)kL^{2}(Ω)|Ω|^{1/2}.
In the same way, we have

Z

Ω

F_{σ}(1−c_{σε})dx≥ |lnσ|

Z

Ω

[c_{σε}(·, t)−1]_{+}dx−σ[|lnσ|+ 2σ−1
+e^{−1}]|Ω| −σkcσε(t)k_{L}2(Ω)|Ω|^{1/2}.

Thus, using the above estimates and Item 1 of Lemma 5.1, we obtain Z

Ω

[c_{σε}(·, t)−1]_{+}dx+
Z

Ω

[−cσε(·, t)]+dx≤ C ε|lnσ|