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538 J. Pr¨uss and G. Simonett

appearing in the deﬁnition of G

.TheFr´echet derivative of G

at h is given by

(h)

h = −



(1+β)

(1+β)β



(∂

Δh∂

h∂

(1+β)β



2∂

hΔh∂

h +(∂



Before continuing, we note that the term 1/(1 + β) can be treated in exactly the

same way as 1/β, as a short inspection of the proof of Lemma 5.2(d) shows. It

follows then from Corollary 5.3(a)–(b) and from (5.6) that there is a constant C

such that

DG

(h)

h

(a)

≤ C

P (∇h

∞

)+Q(∇h

BC(J ;BC

)

, h

(a)

)



h

(a)

(5.26)

for all h ∈ E

(a)andalla ∈ (0,a], where P and Q are polynomials with coeﬃcients

equal to one and vanishing zero-order terms. Analogous arguments can be used

for the remaining terms (∇h|∇

h∇h)/β

and G

(h)∂

h appearing in G, yielding

the same estimate as in (5.26).

(iv) It remains to consider the nonlinear term H

(v, h):=(b − γv|∇h). The

Fr´echet derivative is given by DH

(v, h)[¯v,

h]=−(∇h|γ¯v)+(b − γv|∇

h). From

Lemma 5.5(b) with g

= ∂

h and g

= γ¯v

,where¯v

denotes the kth component of

¯v, follows (∇h|γ¯v)

(a)

≤ C

(∇h

∞

+ h

(a)

)¯v

(a)

. Lemma 5.5(c) with

g =(b − γv)

and h =

h implies

(b − γv|∇

h)

(a)

≤ C

(b − γv

∞

+ b − γv

(a)

)

h

(a)

We have, thus, shown that

DH

(v, h)≤C

(∇h

∞

+ h

(a)

+ b −γv

BC(J;BC)∩F

(a)

(5.27)

Combining the estimates in (5.16), (5.20) and (5.22)–(5.27) yields the assertions

of Proposition 4.1. 

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Jan Pr¨uss

Institut f¨ur Mathematik

Martin-Luther-Universit¨at Halle-Wittenberg

Theodor-Lieser-Str. 5

D-60120 Halle, Germany

e-mail: jan.pruess@mathematik.uni-halle.de

Gieri Simonett

Department of Mathematics

Vanderbilt University

Nashville, TN, USA

e-mail: gieri.simonett@vanderbilt.edu

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 541–576

 2011 Springer Basel AG

On Conserved Penrose-Fife Type Models

Jan Pr¨uss and Mathias Wilke

Dedicated to Herbert Amann on the oc casion of his 70th birthday

Abstract. In this paper we investigate quasilinear parabolic systems of con-

served Penrose-Fife type. We show maximal L

-regularity for this problem

with inhomogeneous boundary data. Furthermore we prove global existence

of a solution, under the assumption that the absolute temperature is bounded

from below and above. Moreover, we apply the Lojasiewicz-Simon inequality

to establish the convergence of solutions to a steady state as time tends to

inﬁnity.

Mathematics Subject Classiﬁcation (2000). 35K55, 35B38, 35B40, 35B65,

82C26.

Keywords. Conserved Penrose-Fife system, quasilinear parabolic system, max-

imal regularity, global existence, Lojasiewicz-Simon inequality, convergence to

steady states.

1. Introduction and the model

We are interested in the conserved Penrose-Fife type equations

∂

ψ =Δμ, μ = −Δψ +Φ



(ψ) − λ



(ψ)ϑ, t ∈ J, x ∈ Ω,

∂

(b(ϑ)+λ(ψ)) − Δϑ =0,t∈ J, x ∈ Ω,

(1.1)

where ϑ =1/θ and θ denotes the absolute temperature of the system, ψ is the

order parameter and Ω ⊂ R

is a bounded domain with boundary ∂Ω ∈ C

The function Φ



is the derivative of the physical potential, which characterizes

the diﬀerent phases of the system. A typical example is the double well potential

Φ(s)=(s

− 1)

with the two distinct minima s = ±1. Typically, the nonlinear

function λ is a polynomial of second order.

For an explanation of (1.1) we will follow the lines of Alt & Pawlow [2] (see

also Brokate & Sprekels [4, Section 4.4]). We start with the rescaled Landau-

542 J. Pr¨uss and M. Wilke

Ginzburg functional (total Helmholtz free energy)

F(ψ, θ)=





γ(θ)

2θ

|∇ψ|

f(ψ, θ)



dx,

where the free energy density F (ψ, θ):=

γ(θ)

|∇ψ|

+ f(ψ, θ) is rescaled by 1/θ.

The reduced chemical potential μ is given by the variational derivative of F with

respect to ψ, i.e.,

μ =

δF

δψ

(ψ, θ)=



−γ(θ)Δψ +

∂f(ψ, θ)

∂ψ



Assuming that ψ is a conserved quantity, we have the conservation law

∂

ψ +divj =0.

Here j is the ﬂux of the order parameter ψ, for which we choose the well-accepted

constitutive law j = −∇μ, i.e., the phase transition is driven by the chemical

potential μ (see [4, (4.4)]). The kinetic equation for ψ thus reads

∂

ψ =Δμ, μ =



−γ(θ)Δψ +

∂f(ψ, θ)

∂ψ



If the volume of the system is preserved, the internal energy e is given by the

variational derivative

e =

δF(ψ, θ)

δ(1/θ)

This yields the expression

e(ψ, θ)=f (ψ, θ) − θ

∂f(ψ, θ)

∂θ



γ(θ) − θ

∂γ(θ)

∂θ



|∇ψ|

It can be readily checked that the Gibbs relation

e(ψ, θ)=F (ψ, θ) − θ

∂F(ψ, θ)

∂θ

holds. If we assume that no mechanical stresses are active, the internal energy e

satisﬁes the conservation law

∂

e +divq =0,

where q denotes the heat ﬂux of the system. Following Alt & Pawlow [2], we

assume that q = ∇





, so that the kinetic equation for e reads

∂

e +Δ





=0.

Let us now assume that γ(θ)=θ and f(ψ, θ)=θΦ(ψ) − λ(ψ) − θ log θ.Inthis

case we obtain e = θ − λ(ψ)and

μ = −Δψ +Φ



(ψ) − λ



(ψ)

hence system (1.1) for ϑ =1/θ and b(s)=−1/s, s>0. Suppose (j|ν)=(q|ν)=0

on ∂Ωwithν = ν(x) being the outer unit normal in x ∈ ∂Ω. This yields the

Conserved Penrose-Fife Type Models 543

boundary conditions ∂

μ =0and∂

ϑ = 0 for the chemical potential μ and the

function ϑ, respectively. Since (1.1) is of fourth order with respect to the function ψ

we need an additional boundary condition. An appropriate and classical one from

a variational point of view is ∂

ψ = 0. Finally, this yields the initial-boundary

value problem

∂

ψ − Δμ = f

,μ= −Δψ +Φ



(ψ) − λ



(ψ)ϑ, t ∈ J, x ∈ Ω,

∂

(b(ϑ)+λ(ψ)) − Δϑ = f

,t∈ J, x ∈ Ω,

∂

μ = g

,∂

ψ = g

,∂

ϑ = g

,t∈ J, x ∈ ∂Ω,

ψ(0) = ψ

,ϑ(0) = ϑ

,t=0,x∈ Ω.

(1.2)

The functions f

,ψ

,ϑ

, Φ,λ and b are given. Note that if θ has only a small

deviation from a constant value θ

∗

> 0, then the term 1/θ can be linearized

around θ

∗

and (1.2) turns into the nonisothermal Cahn-Hilliard equation for the

order parameter ψ and the relative temperature θ −θ

∗

,providedb(s)=−1/s.

In the case of the Penrose-Fife equations, Brokate & Sprekels [4] and

Zheng [20] proved global well-posedness in an L

-setting if the spatial dimen-

sion is equal to 1. Sprekels & Z heng showed global well-posedness of the non-

conserved equations (that is ∂

ψ = −μ) in higher space dimensions in [18], a

similar result can be found in the article of Laurencot [11]. Concerning asymp-

totic behavior we refer to the articles of Kubo, Ito & Kenmochi [10], Shen &

Zheng [17], Feireisl & Schimperna [8] and Rocca & Schimperna [14]. The

last two authors studied well-posedness and qualitative behavior of solutions to

the non-conserved Penrose-Fife equations. To be precise, they proved that each

solution converges to a steady state, as time tends to inﬁnity. Shen & Zheng [17]

established the existence of attractors for the non-conserved equations, whereas

Kubo, Ito & Kenmochi [10] studied the non-conserved as well as the conserved

Penrose-Fife equations. Beside the proof of global well-posedness in the sense of

weak solutions they also showed the existence of a global attractor. Finally, we

want to mention that the physical potential Φ may also be of logarithmic type,

such that Φ



(s) has singularities at s = ±1. This forces the order parameter to

stay in the physically reasonable interval (−1, 1), provided that the initial value

ψ(0) = ψ

∈ (−1, 1). In general, such a result cannot be obtained in the case of the

double well potential, since there is no maximum principle available for the fourth-

order equation (1.2)

. For a result on global existence, uniqueness and asymptotic

behaviour of solutions of the Cahn-Hilliard equation in case of a logarithmic po-

tential, we refer the reader, e.g., to Abels & Wilke [1] and the references cited

therein. However, in this paper we will only deal with smooth potentials.

In the following sections we will prove well-posedness of (1.2) for solutions in

the maximal L

-regularity classes

ψ ∈ H

(J; L

(Ω)) ∩ L

(J; H

(Ω)),

ϑ ∈ H

(J; L

(Ω)) ∩ L

(J; H

(Ω)),

544 J. Pr¨uss and M. Wilke

where J =[0,T], T>0. In Section 2 we investigate a linearized version of (1.2)

and prove maximal L

-regularity. Section 3 is devoted to local well-posedness of

(1.2). To this end we apply the contraction mapping principle. In Section 4, we

show that the solution exists globally in time, under the unproven assumption

that the inverse of the absolute temperature is uniformly bounded from below and

above. Actually, it is a formidable task to derive such bounds for ϑ. As far as

we know, it has been proven only in some particular cases for the non-conserved

system, i.e., if (1.2)

is replaced by

∂

ψ − Δψ = λ



(ψ)ϑ − Φ



(ψ),

a semilinear heat equation for the order parameter ψ.Formoredetails,werefer

the reader to the references [9] and [16].

Finally, in Section 5, we study the asymptotic behavior of the solution to

(1.2) as t →∞. The Lojasiewicz-Simon inequality will play an important role in

the analysis.

2. The linear problem

In this section we deal with a linearized version of (1.2).

∂

u +Δ

u +Δ(η

v)=f

,t∈ J, x ∈ Ω,

∂

v − a

Δv + η

∂

u = f

,t∈ J, x ∈ Ω,

∂

Δu + ∂

(η

v)=g

,t∈ J, x ∈ ∂Ω,

∂

u = g

,∂

v = g

,t∈ J, x ∈ ∂Ω,

u(0) = u

,v(0) = v

,t=0,x∈ Ω.

(2.1)

Here η

= η

(x),η

= η

(x),a

= a

(x) are given functions such that

∈ B

4−4/p

(Ω),η

∈ B

2−2/p

(Ω) and a

∈ C(Ω). (2.2)

We assume furthermore that a

(x) ≥ σ>0 for all x ∈ Ω and some constant σ>0.

Hence equation (2.1)

does not degenerate. We are interested in solutions

u ∈ H

(J; L

(Ω)) ∩ L

(J; H

(Ω)) =: E

(T )

and

v ∈ H

(J; L

(Ω)) ∩ L

(J; H

(Ω)) =: E

(T )

of (2.1). By the well-known trace theorems (cf. [3, Theorem 4.10.2])

(T ) → C(J; B

4−4/p

(Ω)) and E

(T ) → C(J; B

2−2/p

(Ω)), (2.3)

we necessarily have u

∈ B

4−4/p

(Ω) =: X

, v

∈ B

2−2/p

(Ω) =: X

and the comp-

atibility conditions

∂

Δu

+ ∂

(η

)=g

t=0

,∂

= g

t=0

, as well as ∂

= g

t=0

Conserved Penrose-Fife Type Models 545

whenever p>5, p>5/3andp>3, respectively (cf. [6, Theorem 2.1]). In the

sequel we will assume that p>(n +2)/2andp ≥ 2. This yields the embeddings

4−4/p

(Ω) → H

(Ω) ∩ C

(

Ω) and B

2−2/p

(Ω) → H

(Ω) ∩C(

Ω).

We are going to prove the following theorem.

Theorem 2.1. Let n ∈ N, Ω ⊂ R

a bounded domain with boundary ∂Ω ∈ C

and let p>(n +2)/2, p ≥ 2, p =3, 5. Assume in addition that η

∈ B

4−4/p

(Ω),

∈ B

2−2/p

(Ω) and a

∈ C(

Ω), a

(x) ≥ σ>0 for all x ∈

Ω. Then the linear

problem (2.1) admits a unique solution

(u, v) ∈ H

; L

(Ω)

) ∩ L

;(H

(Ω) × H

(Ω))),

if and only if the data are subject to the following conditions.

1. f

∈ L

; L

(Ω)) = X(J

2. g

∈ W

1/4−1/4p

; L

(∂Ω)) ∩ L

; W

1−1/p

(∂Ω)) = Y

3. g

∈ W

3/4−1/4p

; L

(∂Ω)) ∩ L

; W

3−1/p

(∂Ω)) = Y

4. g

∈ W

1/2−1/2p

; L

(∂Ω)) ∩ L

; W

1−1/p

(∂Ω)) = Y

5. u

∈ B

4−4/p

(Ω) = X

, v

∈ B

2−2/p

(Ω) = X

6. ∂

Δu

+ ∂

(η

)=g

t=0

,p>5,

7. ∂

= g

t=0

, ∂

= g

t=0

,p>3.

Proof. Suppose that the function u ∈ E

(T ) in (2.1) is already known. Then in a

ﬁrst step we will solve the linear heat equation

∂

v − a

Δv = f

−η

∂

u, (2.4)

subject to the boundary and initial conditions ∂

v = g

and v(0) = v

.Bythe

properties of the function a

we may apply [6, Theorem 2.1] to obtain a unique

solution v ∈ E

(T ) of (2.4), provided that f

∈ L

(J × Ω), v

∈ B

2−2/p

(Ω),

∈ W

1/2−1/2p

(J; L

(∂Ω)) ∩L

(J; W

1−1/p

(∂Ω)) =: Y

(J),

and the compatibility condition ∂

= g

t=0

if p>3 is valid. The solution may

then be represented by the variation of parameters formula

v(t)=v

(t) −



−A(t−s)

∂

u(s) ds, (2.5)

where A denotes the L

-realization of the diﬀerential operator A(x)=−a

(x)Δ

means the Neumann-Laplacian and e

−At

stands for the bounded analytic

semigroup, which is generated by −A in L

(Ω). Furthermore the function v

∈

(T ) solves the linear problem

∂

− a

Δv

= f

,∂

= g

(0) = v

We ﬁx a function w

∗

∈ E

(T ) such that w

∗

t=0

= u

and make use of (2.5) and

thefactthat(u − w

∗

t=0

= 0 to obtain

v(t)=v

(t)+v

(t) − (∂

+ A)

−1

∂

(u −w

∗

)

546 J. Pr¨uss and M. Wilke

with v

(t):=−



−A(t−s)

∂

∗

.Setv

∗

= v

+ v

∈ E

(T )and

F (u)=−(∂

+ A)

−1

∂

(u − w

∗

Then we may reduce (2.1) to the problem

∂

u +Δ

u =ΔG(u)+f

,t∈ J, x ∈ Ω,

∂

Δu = ∂

G(u)+g

,t∈ J, x ∈ ∂Ω,

∂

u = g

,t∈ J, x ∈ ∂Ω,

u(0) = u

,t=0,x∈ Ω,

(2.6)

where G(u):=−η

(F (u)+v

∗

). For a given T ∈ (0,T

]weset

(T )={u ∈ E

(T ):u|

t=0

=0}

and

(T ):=X(T ) × Y

(T ) × Y

(T )

(T ):={(f,g,h) ∈ E

(T ):g|

t=0

= h|

t=0

=0},

where X(T ):=L

((0,T) × Ω),

(T ):=W

1/4−1/4p

(0,T; L

(∂Ω)) ∩ L

(0,T; W

1−1/p

(∂Ω)),

and

(T ):=W

3/4−1/4p

(0,T; L

(∂Ω)) ∩ L

(0,T; W

3−1/p

(∂Ω)).

The spaces E

(T )andE

(T ) are endowed with the canonical norms |·|

and |·|

respectively. We introduce the new function ˜u := u −w

∗

∈

(T )andweset

F (˜u):=−(∂

+ A)

−1

∂

˜u

as well as

G(˜u):=−η

F (˜u). If u ∈ E

(T ) is a solution of (2.6), then the function

˜u ∈

(T ) solves the problem

∂

˜u +Δ

˜u =Δ

G(˜u)+

,t∈ J, x ∈ Ω,

∂

Δ˜u = ∂

G(˜u)+˜g

,t∈ J, x ∈ ∂Ω,

∂

˜u =˜g

,t∈ J, x ∈ ∂Ω,

˜u(0) = 0,t=0,x∈ Ω,

(2.7)

with the modiﬁed data

:= f

− Δ(η

∗

) − ∂

∗

−Δ

∗

∈ X(T ),

˜g

:= g

−∂

(ηv

∗

) − ∂

Δw

∗

∈

(T ),

and

˜g

:= g

−∂

∗

∈

(T ).

Observe that by construction we have ˜g

t=0

=0and˜g

t=0

=0ifp>5and

p>5/3, respectively.

Conserved Penrose-Fife Type Models 547

Let us estimate the term Δ

G(u)inL

(J; L

(Ω)), where u ∈

(T ). We

compute

|Δ

G(u)|

(J;L

(Ω))

≤|

F (u)Δη

(J;L

(Ω))

+2|(∇

F (u)|∇η

(J;L

(Ω))

+ |η

F (u)|

(J;L

(Ω))

Since η

∈ B

4−4/p

(Ω) does not depend on the variable t,weobtain

F (u)Δη

(J;L

(Ω))

≤|Δη

(Ω)

F (u)|

(J;L

∞

(Ω))

|(∇

F (u)|∇η

(J;L

(Ω))

≤|∇η

∞

(Ω)

|∇

F (u)|

(J;L

(Ω))

and

|η

F (u)|

(J;L

(Ω))

≤|η

∞

(Ω)

|Δ

F (u)|

(J;L

(Ω))

Therefore we have to estimate

F (u)foreachu ∈

(T ) in the topology of the

spaces L

(J; L

∞

(Ω)) and L

(J; H

(Ω)). Let u ∈

and recall that

F (u) is deﬁned

F (u)=−(∂

+A)

−1

∂

u. The operator (∂

+A)

−1

is a bounded linear operator

from L

(J; L

(Ω)) to

(J; L

(Ω)) ∩ L

(J; H

(Ω)) =

(T ). Moreover, by the

trace theorem and by Sobolev embedding, it holds that

(J; L

(Ω)) ∩ L

(J; H

(Ω)) → C(J; B

2−2/p

(Ω)) → C(J; C(

Ω)).

Note that the bound of (∂

+ A)

−1

as well as the embedding constant do not

depend on the length of the interval J =[0,T] ⊂ [0,T

]=J

, since the time trace

at t = 0 vanishes. With these facts, we obtain

|(∂

+ A)

−1

∂

(J;L

∞

(Ω))

≤ T

1/p

|(∂

+ A)

−1

∂

∞

(J;L

∞

(Ω))

≤ T

1/p

C|(∂

+ A)

−1

∂

(T )

≤ T

1/p

C|η

∂

(J;L

(Ω))

≤ T

1/p

C|η

∞

(Ω)

|u|

(T )

To estimate

F (u)inL

(J; H

(Ω)) we need another representation of

F (u). To be

precise, we rewrite

F (u) as follows

F (u)=−(∂

+ A)

−1

∂

u = −∂

1/2

(∂

+ A)

−1

∂

1/2

(η

u).

This is possible, since u ∈

(T ). Now observe that for each u ∈

it holds that

u ∈

3/4

(J; H

(Ω)). This can be seen as follows. First of all, it suﬃces to show

that η

u ∈ L

(J; H

(Ω)), since η

does not depend on the variable t.But

|η

(J;H

(Ω))

≤|η

∇u|

(J;L

(Ω))

+ |u∇η

(J;L

(Ω))

≤ C



|η

∞

(Ω)

|u|

(T )

+ |u|

(J;L

∞

(Ω))

|η

(Ω)



≤ C|u|

(T )

|η

2−2/p

(Ω)

and this yields the claim, since

u ∈

(J; L

(Ω)) ∩ L

(J; H

(Ω)) →

3/4

(J; H

(Ω)),