A CAGINALP PHASE-FIELD SYSTEM BASED ON TYPE III HEAT
CONDUCTION WITH TWO TEMPERATURES
ALAIN MIRANVILLE
1
, RAMON QUINTANILLA
2
Abstract. Our aim in this paper is to study a generalization of the Caginalp phase-
field system based on the theory of type III thermomechanics with two temperatures
for the heat conduction. In particular, we obtain well-posedness results and study the
dissipativity of the associated solution operators. We consider here both regular and sin-
gular nonlinear terms. Furthermore, we endow the equations with two types of boundary
conditions, namely, Dirichlet and Neumann. Finally, we study the spatial behavior of
the solutions in a semi-infinite cylinder, when such solutions exist.
1. Introduction
The Caginalp phase-field system,
(1.1)
∂u
∂t
− ∆u + f(u) = T,
(1.2)
∂T
∂t
− ∆T = −
∂u
∂t
,
has been proposed in [5] to model phase transition phenomena, such as melting-solidifica-
tion phenomena. Here, u is the order parameter, T is the relative temperature (defined
as T =
˜
T − T
E
, where
˜
T is the absolute temperature and T
E
is the equilibrium melting
temperature) and f is the derivative of a double-well potential F . Furthermore, here and
below, we set all physical parameters equal to one. This system has been much studied;
we refer the reader to, e.g., [1], [2], [3], [4], [9], [10], [11], [18], [20], [21], [22], [23], [24],
[32], [39] and [45].
These equations can be derived as follows. One introduces the (total Ginzburg-Landau)
free energy
(1.3) Ψ =
Z
Ω
(
1
2
|∇u|
2
+ F (u) − uT −
1
2
T
2
) dx,
where Ω is the domain occupied by the system (we assume here that it is a bounded and
regular domain of R
n
, n = 1, 2 or 3, with boundary Γ), and the enthalpy
(1.4) H = u + T.
2010 Mathematics Subject Classification. 35K55, 35J60, 80A22.
Key words and phrases. Caginalp system, Type III thermomechanics, two temperatures, well-
posedness, dissipativity, spatial behavior, Phragm´en-Lindel¨of alternative.
1
2 A. MIRANVILLE, R. QUINTANILLA
As far as the evolution equation for the order parameter is concerned, one postulates the
relaxation dynamics (with relaxation parameter set equal to one)
(1.5)
∂u
∂t
= −
DΨ
Du
,
where
D
Du
denotes a variational derivative with respect to u, which yields (1.1). Then, we
have the energy equation
(1.6)
∂H
∂t
= −divq,
where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,
(1.7) q = −∇T,
we obtain (1.2).
Now, one essential drawback of the Fourier law is that it predicts that thermal signals
propagate at an infinite speed, which violates causality (the so-called paradox of heat
conduction, see [13]). To overcome this drawback, or at least to account for more realistic
features, several alternatives to the Fourier law, based, e.g., on the Maxwell-Cattaneo law
or recent laws from thermomechanics, have been proposed and studied, in the context of
the Caginalp phase-field system, in [29], [30], [33], [34], [35], [36] and [38].
In the late 1960’s, several authors proposed a heat conduction theory based on two
temperatures (see [6], [7] and [8]). More precisely, one now considers the conductive
temperature T and the thermodynamic temperature θ. In particular, for simple materials,
these two temperatures are shown to coincide. However, for non-simple materials, they
differ and are related as follows:
(1.8) θ = T − ∆T.
The Caginalp system, based on this two temperatures theory and the usual Fourier
law, was studied in [14].
Our aim in this paper is to study a variant of the Caginalp phase-field system based on
the type III thermomechanics theory (see [25]) with two temperatures recently proposed
in [43] (see also [15]).
In that case, the free energy reads, in terms of the (relative) thermodynamic tempera-
ture θ,
(1.9) Ψ =
Z
Ω
(
1
2
|∇u|
2
+ F (u) − uθ −
1
2
θ
2
) dx
and (1.5) yields, in view of (1.8), the following evolution equation for the order parameter:
(1.10)
∂u
∂t
− ∆u + f(u) = T − ∆T.
Furthermore, the enthalpy now reads
PHASE-FIELD SYSTEM 3
(1.11) H = u + θ = u + T − ∆T,
which yields, owing to (1.6), the energy equation
(1.12)
∂T
∂t
− ∆
∂T
∂t
+ divq = −
∂u
∂t
.
Finally, the heat flux is given, in the type III theory with two temperatures, by (see [43])
(1.13) q = −∇α − ∇T,
where
(1.14) α(t, x) =
Z
t
0
T (τ, x) dτ + α
0
(x)
is the conductive thermal displacement. Noting that T =
∂α
∂t
, we finally deduce from
(1.10) and (1.12)-(1.13) the following variant of the Caginalp phase-field system:
(1.15)
∂u
∂t
− ∆u + f(u) =
∂α
∂t
− ∆
∂α
∂t
,
(1.16)
∂
2
α
∂t
2
− ∆
∂
2
α
∂t
2
− ∆
∂α
∂t
− ∆α = −
∂u
∂t
.
We can note that we still have an infinite speed of propagation here, since (1.15) is
parabolic; actually, also (1.16) is not hyperbolic and we have not been able to prove
whether or not it exhibits a finite speed of propagation (note however that, as far as the
equation
∂
2
α
∂t
2
− ∆
∂α
∂t
− ∆α = 0,
known as the strongly damped wave equation, is concerned, one does not have a finite
speed of propagation).
Our aim in this paper is to study the well-posedness and the dissipativity of (1.15)-
(1.16). We consider here two types of boundary conditions, namely, Dirichlet and Neu-
mann. Furthermore, we consider regular nonlinear terms f (a usual choice being the
cubic nonlinear term f (s) = s
3
− s), as well as singular nonlinear terms (and, in partic-
ular, the thermodynamically relevant logarithmic nonlinear terms f(s) = k
1
s +
k
2
2
ln
1+s
1−s
,
s ∈ (−1, 1), 0 < k
2
< k
1
).
We are also interested in the study of the spatial behavior of the solutions. Spatial
decay estimates for partial differential equations are related to the Saint-Venant principle
which is both a mathematical and a thermomechanical aspect which has deserved much
attention in the last years (see [26] and the references therein). Such studies describe how
the influence of the perturbations on a part of the boundary is damped for the points which
are far away from the perturbations. Spatial decay estimates for elliptic [16], parabolic
4 A. MIRANVILLE, R. QUINTANILLA
[27], [28], hyperbolic [17] and/or combinations of such [42] have been obtained in the last
years. However, as far as nonlinear equations are concerned, such a knowledge is limited
(see [33], [34], [35], [37] and [38]). What is usual is to consider a semi-infinite cylinder
whose finite end is perturbed and see what happens when the spatial variable goes to
infinity. However, we do not study the existence of solutions to this problem; in fact,
this does not seem to be an easy task (see, e.g., [37]). We thus assume the existence of
solutions and then only study the spatial asymptotic behavior in that case. More precisely,
we obtain a Phragm´en-Lindel¨of alternative, i.e., either a growth or a decay estimate. An
upper bound on the amplitude term, when the solution decays, is also derived, in terms
of the boundary conditions.
Notation. We denote by ((·, ·)) the usual L
2
-scalar product, with associated norm k · k;
more generally, k · k
X
denotes the norm on the Banach space X.
We set, for v ∈ L
1
(Ω),
hvi =
1
Vol(Ω)
Z
Ω
v dx
and v = v − hvi.
We further note that
v 7→ (kvk
2
+ hvi
2
)
1
2
and
v 7→ (k∇vk
2
+ hvi
2
)
1
2
are norms on L
2
(Ω) and H
1
(Ω), respectively, which are equivalent to their usual norms.
Throughout the paper, the same letters c and c
0
denote (generally positive) constants
which may vary from line to line. Similarly, the same letter Q denotes (positive) monotone
increasing (with respect to each argument) functions which may vary from line to line.
2. Dirichlet boundary conditions
2.1. Setting of the problem. We consider the following initial and boundary value
problem:
(2.1)
∂u
∂t
− ∆u + f(u) =
∂α
∂t
− ∆
∂α
∂t
,
(2.2)
∂
2
α
∂t
2
− ∆
∂
2
α
∂t
2
− ∆
∂α
∂t
− ∆α = −
∂u
∂t
,
(2.3) u = α = 0 on Γ,
(2.4) u|
t=0
= u
0
, α|
t=0
= α
0
,
∂α
∂t
|
t=0
= α
1
.
As far as the nonlinear term f is concerned, we assume that
PHASE-FIELD SYSTEM 5
(2.5) f ∈ C
1
(R), f(0) = 0,
(2.6) f
0
≥ −c
0
, c
0
≥ 0,
(2.7) f(s)s ≥ c
1
F (s) − c
2
≥ −c
3
, c
1
> 0, c
2
, c
3
≥ 0, s ∈ R,
where F (s) =
R
s
0
f(ξ) dξ. In particular, the usual cubic nonlinear term f(s) = s
3
− s
satisfies these assumptions.
2.2. A priori estimates. The estimates derived in this subsection will be formal, but
they can easily be justified within a Galerkin scheme.
We multiply (2.1) by
∂u
∂t
and have, integrating over Ω and by parts,
(2.8)
1
2
d
dt
(k∇uk
2
+ 2
Z
Ω
F (u) dx) + k
∂u
∂t
k
2
= ((
∂α
∂t
− ∆
∂α
∂t
,
∂u
∂t
)).
We then multiply (2.2) by
∂α
∂t
− ∆
∂α
∂t
to obtain
(2.9)
1
2
d
dt
(k∇αk
2
+k∆αk
2
+k
∂α
∂t
−∆
∂α
∂t
k
2
)+k∇
∂α
∂t
k
2
+k∆
∂α
∂t
k
2
= −((
∂α
∂t
−∆
∂α
∂t
,
∂u
∂t
)).
Summing (2.8) and (2.9), we find the differential equality
(2.10)
dE
1
dt
+ 2k
∂u
∂t
k
2
+ 2k∇
∂α
∂t
k
2
+ 2k∆
∂α
∂t
k
2
= 0,
where
(2.11) E
1
= k∇uk
2
+ 2
Z
Ω
F (u) dx + k∇αk
2
+ k∆αk
2
+ k
∂α
∂t
− ∆
∂α
∂t
k
2
satisfies
(2.12) E
1
≥ c(kuk
2
H
1
(Ω)
+ kαk
2
H
2
(Ω)
+ k
∂α
∂t
k
2
H
2
(Ω)
) − c
0
, c > 0
(note indeed that k
∂α
∂t
− ∆
∂α
∂t
k
2
= k
∂α
∂t
k
2
+ 2k∇
∂α
∂t
k
2
+ k∆
∂α
∂t
k
2
).
Next, we multiply (2.1) by u and have, owing to (2.7),
(2.13)
d
dt
kuk
2
+ c(kuk
2
H
1
(Ω)
+ 2
Z
Ω
F (u) dx) ≤ c
0
(k
∂α
∂t
k
2
+ k∆
∂α
∂t
k
2
), c > 0.
Multiplying then (2.1) by −∆u, we obtain, owing to (2.6),
(2.14)
d
dt
k∇uk
2
+ ckuk
2
H
2
(Ω)
≤ c
0
(k∇uk
2
+ k
∂α
∂t
k
2
+ k∆
∂α
∂t
k
2
), c > 0.