Content uploaded by Antonio Segatti
Author content
All content in this area was uploaded by Antonio Segatti
Content may be subject to copyright.
arXiv:1301.5563v1 [math.AP] 23 Jan 2013
The Penrose-Fife phase-field model
with coupled dynamic boundary conditions
Alain Miranville
∗
Laboratoire de Math´ematiques et Applications,
UMR CNRS 7348, Universit´e de Poitiers - SP2MI,
Boulevard Marie et Pierre Curie,
F-86962 Chasseneuil Futurosco pe Cedex, France
E-mail: Alain.Miranville@math.univ-poitiers.fr
Elisabetta Rocca
†
Dipartimento di Matematica, Universit`a di Milano,
Via Saldini 50, 20133 Milano, Italy
E-mail: elisabetta.rocca@unimi.it
Giulio Schimperna
‡
Dipartimento di Matematica, Universit`a di Pavia,
Via Ferrata 1, I-27100 Pavia, Italy
E-mail: giusch04@unipv.it
Antonio Segatti
§
Dipartimento di Matematica, Universit`a di Pavia,
Via Ferrata 1, I-27100 Pavia, Italy
E-mail: antonio.segatti@unipv.it
January 24, 2013
Abstract
In this paper we derive, starting from the basic principles of thermodyn amics, an extended version
of the nonconserved Penrose-Fife phase transition model, in which dynamic boundary conditions
are considered in order to take into account interactions with walls. Moreover, we study the well-
posedness and the asymptotic behavior of the Cauchy problem for the PDE system associated to the
mod el, allowing the p hase configuration of the material to be described by a singular function.
∗
The work of A.M. was partially supported by the FP7-IDEAS-ERC-StG #256872 (EntroPhase). Part of the work has
been developed while A.M. was visiting the University of Milano from February 27 to March 4, 2012.
†
The work of E.R. was supported by the FP7-IDEAS-ERC-StG #256872 (EntroPhase).
‡
The work of G.S. was supported by the MIUR-PRIN Grant 2008ZKHAHN “Phase transitions, hysteresis and multi-
scaling” and by the FP7-IDEAS-ERC-StG #256872 (EntroPhase).
§
The work of A.S. was supported by the MIUR-PRIN Grant 2008ZKHAHN “Phase transitions, hysteresis and multi-
scaling” and by the FP7-IDEAS-ERC-StG #256872 (EntroPhase).
1
Key words: Penrose-Fife system, weak solution, singular PDEs, ω-limit, stationary states.
AMS (MOS) subject classification: 35K61, 35D30, 34B16 , 74H40, 34K2 1, 80A22.
1 Introduction
In this paper we derive a model for phase transitions of Penrose-Fife type settled in a bo unded domain
Ω ⊂ R
3
. The peculiarity of our approach consists in the fact that we take into account the relations
betwe en Ω and its exterior, including the effects of the interactions w ith the boundary into the free
energy and entropy functionals Ψ and S (cf. (24)-(25) below) which drive the evolution of the system.
A detailed derivatio n of the model is c arried out in Section 2. The resulting PDEs system couples four
nonlinear and singular evolution equations: two for the absolute temperature ϑ (one in the bulk Ω and
the other on the boundary Γ) and two for the phase parameter
χ
, which repre sents the local proportion
of one of the two phases:
∂ϑ
∂t
− ∆
−
1
ϑ
+ λ
′
b
(
χ
)
∂
χ
∂t
= h
b
in Ω, (1)
∂ϑ
∂t
− ∆
Γ
−
1
ϑ
+ λ
′
Γ
(
χ
)
∂
χ
∂t
+ ∂
ν
−
1
ϑ
= h
Γ
on Γ , (2)
∂
χ
∂t
− ∆
χ
− s
′
0,b
(
χ
) = −
λ
′
b
(
χ
)
ϑ
in Ω, (3)
∂
χ
∂t
− ∆
Γ
χ
− s
′
0,Γ
(
χ
) + ∂
ν
χ
= −
λ
′
Γ
(
χ
)
ϑ
on Γ. (4)
Here, ∆ stands for the Laplacian with respect to the space variables in Ω, ∆
Γ
denotes the La place-Beltrami
operator on Γ, and ν is the unit outer normal vector to Γ. Moreover, λ
b
and λ
Γ
are two quadratic functions
of
χ
related to the latent heat of the pr ocess, h
b
and h
Γ
are two heat sources, respectively in the bulk and
on the boundary, and −s
′
0,b
and −s
′
0,Γ
are two nonlinear functions whose antiderivatives −s
0,b
and −s
0,Γ
corres pond to the configuration potentials of the phase variable. We admit the case when the domains of
−s
0,b
and −s
0,Γ
are bounded, with the purpose of excluding the unphysical values of the variable
χ
. In
this case, we shall speak of singular potentials. A relevant example is the so-called logarithmic potential,
given by
− s(r) = (1 + r) log(1 + r) + (1 − r) log(1 − r) −
δ
2
r
2
, δ ≥ 0. (5)
With the choice (5), ±1 denote the pure states and the values
χ
6∈ [−1, 1] are penalized by identically
assigning the value −∞ to s outside [−1, 1]. In what follows, the potentials −s
0,b
and −s
0,Γ
will be
split into the sum of (dominating) monotone parts f , and f
Γ
respectively, a nd of quadratic perturbations
(cf. the last term in (5)). The literature devoted to the mathematical features of phase transition models
endowed with singular potentials is rather wide, also in spec ific relation with the Penrose-Fife model.
Among the various contributions we quote [31] (for the Cahn-Hillia rd equation), [21] (for the Cagina lp
phase-field model), [40] (for the Penrose-Fife system), and the references in these papers.
A notable feature of system (1)-(4) is the occurrence of dynamic boundary conditions. This ty pe of
conditions has been proposed in the literature in different contexts (for instanc e , in the framework of the
Allen-Cahn and Cahn-Hilliard models) with the aim of describing the interactions betwe en the interior
of a domain and the walls (cf., e.g. [13], [14], [15], [32], and references therein). In particular, the case of
2
singular potentials in the context of the Cahn-Hilliard evolution has been recently analyzed in [17], [18]
and [32], whereas the Caginalp phase-field system (cf. [6]) with dynamic boundary conditions on the phase
parameter has been considered in several papers (cf., e.g. [7 ], [8], [9], [16]), while only recently it has been
coupled with dyna mic boundary conditions for both the phase parameter and the temperature. Such a
problem has been considered in [20] and [11], wher e the well-posedness and the asymptotic behavior (in
terms of attractors) of solutions have been studied, also in the case of singula r potentials.
As far as the Penrose-Fife model (cf. [34]) is concerned, a vast literature is devoted to the well-
posedness (cf., e.g. [10], [23], [28]) and to the long-time behavior of solutions both in term of attractors
(cf., e.g. [25], [35], [39]) and of convergence of single trajectories to stationary states (cf. [12]). Most of
these contributions deal with Robin boundary conditions for the temperature, i.e.,
∂(1 /ϑ)
∂ν
= ξ(ϑ − ϑ
s
) on Γ , (6)
and no-flux conditions for the phase parameter:
∂
χ
∂ν
= 0 on Γ. (7)
Relation (6) establishes that the heat flux through the boundary Γ is proportional to the difference
betwe en the internal and external temperature ϑ
s
> 0 by a positive coefficient ξ, while (7 ) prescribes
that the boundary has no influence on the phase change process. Finally, we would like to quote the
recent paper [26] where a phase separation model of Penrose-Fife type with Signorini boundary condition
has bee n studied.
The main novelties of our contribution stand in the fact that we can consider dynamic boundary
conditions for a coupled pha se-field system which may display a singular character both in ϑ and in
χ
in
view of the terms 1/ϑ and s
′
0,Γ
(
χ
), s
′
0,b
(
χ
) in (1)-(4). Physically spea king, the main differ e nc e between
the dynamic boundary conditions (2), (4) and the “standard” ones (6)-(7) stands in the fact that, in
the case of dynamic b.c.’s, the walls of the the container have a significant effect on the phase trans ition
process. This may include the case of two phase changing substances in contact, w he re one occupies a
small layer surrounding the other one and this layer is approximated by a surface in the mathematical
model (concentra ted capa city phenomenon, see, e.g., [37], [38] for more details).
A rigorous derivatio n of s ystem (1)-(4) starting from the basic laws of Thermodynamics is c arried out
in the first part of the paper. Namely, the equations are obtained by combining the free energy balance
with the entropy dissipation inequality and imposing physically realistic constitutive expressions for the
energy and entropy functionals.
The second part of the paper is devoted to the mathematical analysis of the system in the framework
of weak solutions. In comparison with s imilar models, the ma in mathematical difficulty consists here in
the coupling b e twe en
1. the singularity of the model, in particular due to the presence of
(a) the term 1/ϑ both in the bulk equa tions (1) and (3) and in the boundary equations (2) and (4),
(b) the (possibly) singular functions s
′
0,b
in the phase equation (3) in the bulk, and s
′
0,Γ
in the
phase equation (4) on the boundary,
2. and the occurrence of dynamic boundary conditions.
3
Actually, as already noted in the case of the Caginalp model with dynamic boundary co nditions (see
[20]), if we are in presence of singular potentials, a particular care is needed. For instanc e , in order for
equation (3) to ma ke sense a .e . in the space-time domain, the monotone part f of −s
′
0,b
needs to be
controlled fro m above by the monotone part f
Γ
of −s
′
0,Γ
(cf. (83) below). In particular, this happens
when the potentials −s
′
0,b
and −s
′
0,Γ
have the same effective domain (e.g., the set [−1, 1], as in the case
(5)) and |s
′
0,b
| explodes at most as fast as |s
′
0,Γ
| as
χ
approaches the boundary of this domain (the values
±1 in the specific case).
It is worth noticing that, in the case of completely ge neral singular potentials (i.e., without any
compatibility condition assumed), a weak solution is still expected to exist; however, the equations (3)-
(4) ruling the b ehavior of
χ
need to be interpreted in a weaker sense either by means of duality arguments
or of variational inequality techniques (cf. [20], [32] for more details). This issue will be addresse d in a
forthcoming paper, where we also plan to weaken the regularity assumptions on the initial temperature
ϑ
0
. Actually, in additio n to the natural co nditions represented by the finiteness of the initial energy and
entropy, we will as sume here that ϑ
0
has the L
2
-regularity. This ass umption, albeit meaningful, is not
required in the physical derivation of the model and can be then considered to be somehow artificial. In
other similar contexts (cf., e.g. [40]), the L
2
-regularity of the initial temper ature has been shown to be
not necessary and, for this reason, we will try to remove it a lso for this model.
After proving well-posedness of system (1)-(4), we shall analyze further properties of solutions. Pro-
ceeding along the lines of [40] where the case of standa rd b.c.’s is treated, we shall prove, by using a Moser
iteration argument, that the temperature is uniformly bounded from below fo r strictly positive times.
Moreove r, if ϑ
0
is slightly more summable (cf. (111) below), we also have a uniform upper bound. This
kind of behavior occurs commonly in the framework of parabolic equations with very-fast diffusion terms
(see [42] and [3] for the Cauchy problem in the whole space and the recent contributions [40] and [41] for
the bounded do main cas e with Neumann and dynamic boundary conditions, respectively). In par ticular,
in dimension three the exponent p = 3 appears to be critical in the sense that solutions s tarting with
initial conditions in L
p
with p > 3 become L
∞
for strictly po sitive times. On the other hand, for p < 3
the situation is drastica lly different as the self-similar so lution in R
3
((·)
+
denoting the positive part)
Θ(x, t) :=
2(T − t)
1/2
+
|x|
(8)
shows. The smoothing effect for p = 3 is currently an open problem (see [42] for further details). Moreover,
it is worth noting that, at least when no external source is pr esent, the regula rization estimates are also
uniform with respect to time and give rise to additional r egularity properties for both components of the
solution.
Taking advantage of the reg ularization estimates, in the last part of the paper we finally investigate
the long- time behavior of trajectories. Namely, we are able to prove that, at least in case (111) holds,
any weak solution admits a no n-empty ω-limit set which only contains stationar y solutions o f the system.
A more precise characterization of the long-time behavior relies on the structure of the set of s teady
state solutions, which requires some further explanation. Actually, integra ting (1) in space , using the
boundary condition (2) , and assuming zero external sourc e , one readily sees that the value
µ =
Z
Ω
ϑ + λ
b
(
χ
)
dx +
Z
Γ
ϑ + λ
Γ
(
χ
)
dS, (9)
representing the “total mass” of the internal energy, is conserved in the evolution of the system. Conse-
quently, any limit point of a given weak solution has also to respect the constraint (9), with µ depending
4
only on the initial datum. However, when µ is small, we are not able to exclude that the set of stationary
states satisfying (9) might contain temperatures ϑ being arbitrarily close to 0 (note that the stationary
formulation o f (1)-(2) simply prescribes that ϑ is a constant function). Consequently, the right-hand side s
of the steady state system associated to (3)-(4) might be arbitrarily large, which considerably weakens
the regularity proper ties of the set of its solutions.
On the contrary, it is easy to show tha t this situation cannot occur when either µ is large enough or
λ
′
satisfies a suitable sign condition (cf. (214) below). If either pr operty holds, the set of stationary states
is bounded in a very strong norm, independently of the magnitude of the initial data (but depending on
the value of µ). This fact, together w ith the precompactness of solution trajectories in the natural energy
space and with the existence of a coercive Liapounov functional (namely, the energy Ψ), implies that the
system (1)-(4) admits a smooth global attractor, which is the la st result we prove.
We conclude by giving the plan of the paper: in the next Section 2 we deta il the physical derivation o f
the model from the basic thermodyna mical principles. In Section 3 we introduce our precise concept of
solutions and s tate our main mathematical results re lated to well-posedness and regularization properties
of weak solutions. The proofs are given in the subsequent Section 4. Finally, the long-time behavior of
solutions is separately analyzed in the last Section 5.
2 Derivation of the model
The Penrose-Fife model is derived by considering the fre e energy density w and the entropy density s,
assuming that these quantities depend both on the order parameter
χ
and the absolute temperature ϑ
(as in the origina l pape r by Penrose and Fife [34]), and imposing that the basic laws of Thermodynamics
are satisfied. Assuming that we always have sufficient regularity of the involved variables, we can then
write
w = e − ϑs (Gibbs
′
relation), (10)
where
s = −
∂w
∂ϑ
, (11)
and the internal energy density e is defined by
e =
∂(
w
ϑ
)
∂(
1
ϑ
)
. (12)
Moreove r, we as sume that the total free energy has the expression
Ψ(
χ
, ∇
χ
, ϑ) =
Z
Ω
κϑ
2
|∇
χ
|
2
+ w(ϑ,
χ
)
dx, (13)
whereas the entropy functional is given by
S(
χ
, ∇
χ
, e) =
Z
Ω
−
κ
2
|∇
χ
|
2
+ s(e,
χ
)
dx, (14)
where Ω is the domain occupied by the system and κ > 0 denotes an interfacial energy coefficient.
The evolution equations for
χ
and ϑ are then obtained by stating the relatio ns
∂
χ
∂t
= K
⋆
δS
δ
χ
, K
⋆
> 0, (15)
5
∂e
∂t
= K∆
δS
δe
+ h, K > 0, (16)
where δ denotes a variational derivative and h is a source term.
Taking finally
e = c
0
ϑ + λ(
χ
), (17)
where c
0
> 0 stands for the specific heat of the system (assumed constant) and λ is the latent heat
density, typically (cf. [34, p. 53]) given by
λ(r) = −ar
2
+ br + c, a > 0, (18)
we find (see, e.g., [30]; see also below)
s(e,
χ
) = c
0
ln ϑ + s
0
(
χ
) + c
1
. (1 9)
Here, −s
0
denotes a configuration potential, typically having a double-well character (one can also con-
sider a logarithmic double-well potential of the form (5)), and such that s
′′
0
≤ δ, δ ≥ 0, and c
1
is a
constant. These choices give rise to the Penrose-Fife system:
∂
χ
∂t
= K
⋆
κ∆
χ
+ s
′
0
(
χ
) −
c
0
λ
′
(
χ
)
ϑ
, (20)
c
0
∂ϑ
∂t
= −K∆
c
0
ϑ
− λ
′
(
χ
)
∂
χ
∂t
+ h. (21)
These equations are usually endowed with the boundary conditions (cf. the Introduction)
∂
χ
∂ν
= 0 on Γ, (22)
∂ϑ
∂ν
= −ξ(ϑ − ϑ
s
) on Γ , ξ, ϑ
s
> 0, (23)
where Γ = ∂Ω and ν is the unit outer normal vector to Γ.
Now, in order to take into account the interactions with the exterior of Ω (e.g., the walls), it is natural,
following [27] (see also [11] and [19]), to add a boundary contribution to the total free energy and to take,
in place of (13),
Ψ = Ψ(
χ
, ∇
χ
, ∇
Γ
χ
, ϑ) =
Z
Ω
κ
b
ϑ
2
|∇
χ
|
2
+ w
b
(ϑ,
χ
)
dx (24)
+
Z
Γ
κ
Γ
ϑ
2
|∇
Γ
χ
|
2
+ w
Γ
(ϑ,
χ
)
dS, κ
b
, κ
Γ
> 0,
where ∇
Γ
is the surface gradient and w
b
and w
Γ
are the bulk and surface free energy densities, res pectively.
Similarly, it is reasonable, in view of (24), to introduce the generalized entropy functional
S = S(
χ
, ∇
χ
, ∇
Γ
χ
, e) =
Z
Ω
−
κ
b
2
|∇
χ
|
2
+ s
b
(e,
χ
)
dx +
Z
Γ
−
κ
Γ
2
|∇
Γ
χ
|
2
+ s
Γ
(e,
χ
)
dS, (25)
where s
b
and s
Γ
are the bulk and surface entropy densities, respectively.
6
As b efore, we as sume that Gibbs’ relation holds, namely,
w
b
= e − ϑs
b
in Ω, (26)
w
Γ
= e − ϑs
Γ
on Γ , (27)
and that
s
b
= −
∂w
b
∂ϑ
in Ω, (28)
s
Γ
= −
∂w
Γ
∂ϑ
on Γ , (29)
e =
∂(
w
b
ϑ
)
∂(
1
ϑ
)
in Ω, (30)
e =
∂(
w
Γ
ϑ
)
∂(
1
ϑ
)
on Γ. (31)
We now note that, in view of (25),
δS
δ
χ
= κ
b
∆
χ
+
∂s
b
∂
χ
in Ω, (32)
δS
δ
χ
= κ
Γ
∆
Γ
χ
+
∂s
Γ
∂
χ
− κ
b
∂
χ
∂ν
on Γ, (33)
where ∆
Γ
is the Laplace-Beltrami op e rator.
Then, assuming that, as in the classical model,
∂
χ
∂t
= K
⋆
δS
δ
χ
, K
⋆
> 0, (34)
we obtain the equations
∂
χ
∂t
= K
⋆
κ
b
∆
χ
+
∂s
b
∂
χ
in Ω, (35)
∂
χ
∂t
= K
⋆
κ
Γ
∆
Γ
χ
+
∂s
Γ
∂
χ
− κ
b
∂
χ
∂ν
on Γ . (36)
Next, in order to describe the evolution of the temperature, we generalize (16) as follows. We intro-
duce, for U =
u|
Ω
v|
Γ
(regular enough at this stage), the linear operator A defined by
AU =
−∆u|
Ω
−∆
Γ
v|
Γ
+
∂u|
Ω
∂ν
|
Γ
(37)
and write that
∂
∂t
e|
Ω
e|
Γ
= −KA
δS
δe
|
Ω
δS
δe
|
Γ
+ H, (38)
where H =
h
b
h
Γ
is a forcing term, with h
b
and h
Γ
standing for the bulk and boundary heat sources,
respectively. Introducing the linear operator A is natural when considering dynamic boundary conditions
(see [11] and [19]). Noting that
δS
δe
=
∂s
b
∂e
in Ω, (39)
7
δS
δe
=
∂s
Γ
∂e
on Γ, (40)
we deduce the equations
∂e
∂t
= K∆
∂s
b
∂e
+ h
b
in Ω, (41)
∂e
∂t
= K∆
Γ
∂s
Γ
∂e
+ h
Γ
− K
∂
∂ν
∂s
b
∂e
on Γ . (42)
We then assume that (see (17))
e = c
0,b
ϑ + λ
b
(
χ
) in Ω, c
0,b
> 0, (43)
e = c
0,Γ
ϑ + λ
Γ
(
χ
) on Γ, c
0,Γ
> 0. (44)
Noting that it follows from (30) and (31) that
e = −ϑ
2
∂(
w
b
ϑ
)
∂ϑ
in Ω, (45)
e = −ϑ
2
∂(
w
Γ
ϑ
)
∂ϑ
on Γ , (46)
we have, freezing
χ
and integrating between ϑ
0
and ϑ, ϑ
0
, ϑ > 0,
w
b
(ϑ,
χ
)
ϑ
−
w
b
(ϑ
0
,
χ
)
ϑ
0
= −
Z
ϑ
ϑ
0
(
c
0,b
τ
+
λ
b
(
χ
)
τ
2
) dτ in Ω, (47)
w
Γ
(ϑ,
χ
)
ϑ
−
w
Γ
(ϑ
0
,
χ
)
ϑ
0
= −
Z
ϑ
ϑ
0
(
c
0,Γ
τ
+
λ
Γ
(
χ
)
τ
2
) dτ on Γ, (48)
which we rewrite in the form
w
b
(ϑ,
χ
) = −c
0,b
ϑ ln ϑ + c
0,b
ϑ ln ϑ
0
+ λ
b
(
χ
) − ϑs
0,b
(
χ
) in Ω, (49)
w
Γ
(ϑ,
χ
) = −c
0,Γ
ϑ ln ϑ + c
0,Γ
ϑ ln ϑ
0
+ λ
Γ
(
χ
) − ϑs
0,Γ
(
χ
) on Γ , (50)
where we have defined s
0,b
:= (w
b
(ϑ
0
,
χ
) − λ
b
(
χ
))/ϑ
0
and s
0,Γ
:= (w
Γ
(ϑ
0
,
χ
) − λ
Γ
(
χ
))/ϑ
0
.
We finally deduce from (28)-(29) and (49)-(50) that
s
b
(e,
χ
) = c
0,b
ln ϑ + s
0,b
(
χ
) + c
1,b
, c
1,b
= −c
0,b
ln ϑ
0
+ c
0,b
in Ω, (51)
s
Γ
(e,
χ
) = c
0,Γ
ln ϑ + s
0,Γ
(
χ
) + c
1,Γ
, c
1,Γ
= −c
0,Γ
ln ϑ
0
+ c
0,Γ
on Γ. (52)
We now note that it follows from (43)-(44) and (51)-(52) that
s
b
(e,
χ
) = c
0,b
ln
e − λ
b
(
χ
)
c
0,b
+ s
0,b
(
χ
) + c
1,b
in Ω, (53)
s
Γ
(e,
χ
) = c
0,Γ
ln
e − λ
Γ
(
χ
)
c
0,Γ
+ s
0,Γ
(
χ
) + c
1,Γ
on Γ, (54)
8
which yields
∂s
b
∂e
=
1
ϑ
,
∂s
b
∂
χ
= −
λ
′
b
(
χ
)
ϑ
+ s
′
0,b
(
χ
) in Ω, (55)
∂s
Γ
∂e
=
1
ϑ
,
∂s
Γ
∂
χ
= −
λ
′
Γ
(
χ
)
ϑ
+ s
′
0,Γ
(
χ
) on Γ . (56)
We finally deduce from (39)-(42) and (55)-(5 6) the following Penrose-Fife system with dynamic bound-
ary conditions:
∂
χ
∂t
= K
⋆
κ
b
∆
χ
+ s
′
0,b
(
χ
) −
λ
′
b
(
χ
)
ϑ
in Ω, (57)
∂
χ
∂t
= K
⋆
κ
Γ
∆
Γ
χ
+ s
′
0,Γ
(
χ
) −
λ
′
Γ
(
χ
)
ϑ
− κ
b
∂
χ
∂ν
on Γ , (58)
c
0,b
∂ϑ
∂t
= −K∆
1
ϑ
− λ
′
b
(
χ
)
∂
χ
∂t
+ h
b
in Ω, (59)
c
0,Γ
∂ϑ
∂t
= −K∆
Γ
1
ϑ
− λ
′
Γ
(
χ
)
∂
χ
∂t
+ h
Γ
+ K
∂
∂ν
1
ϑ
on Γ. (60)
Taking, for simplicity, all constants equal to one, (57)-(60) reduces to
∂
χ
∂t
= ∆
χ
+ s
′
0,b
(
χ
) −
λ
′
b
(
χ
)
ϑ
in Ω, (61)
∂
χ
∂t
= ∆
Γ
χ
+ s
′
0,Γ
(
χ
) −
λ
′
Γ
(
χ
)
ϑ
−
∂
χ
∂ν
on Γ , (62)
∂ϑ
∂t
= −∆
1
ϑ
− λ
′
b
(
χ
)
∂
χ
∂t
+ h
b
in Ω, (63)
∂ϑ
∂t
= −∆
Γ
1
ϑ
− λ
′
Γ
(
χ
)
∂
χ
∂t
+ h
Γ
+
∂
∂ν
1
ϑ
on Γ , (64)
which can also be rewritten in the following compact form:
∂
∂t
χ
|
Ω
χ
|
Γ
= −A
χ
|
Ω
χ
|
Γ
+
s
′
0,b
(
χ
)
s
′
0,Γ
(
χ
)
−
λ
′
b
(
χ
)
ϑ
λ
′
Γ
(
χ
)
ϑ
!
, (65)
∂
∂t
ϑ|
Ω
ϑ|
Γ
= A
1
ϑ
|
Ω
1
ϑ
|
Γ
−
λ
′
b
(
χ
)
∂
χ
∂t
λ
′
Γ
(
χ
)
∂
χ
∂t
!
+ H. (66)
The remainder of the pa per is devoted to the mathematical analysis of system (65)-(66) in the framework
of weak solutions.
9
3 Main assumptions and preliminary results
We introduce he re our main assumptions, together with several mathematical tools which are needed in
order to provide a precise analytical statement of our results.
We let Ω be a sufficiently smooth, bounded, and connected domain in R
3
with boundary Γ. We set
Ω := Ω ∪ Γ, set H := L
2
(Ω), and denote by (·, ·) the scalar pr oduct both in H and in H
3
and by k · k
the related norm. Next, we set V := H
1
(Ω) and denote by V
′
the (to pological) dual of V . The duality
betwe en V
′
and V will be indicated by h·, ·i. Identifying H with H
′
through the scalar product of H, it
is then well known that V ⊂ H ⊂ V
′
with continuous and dense inclusions. In other words, (V, H, V
′
)
constitutes a Hilbert triplet (see, e.g., [29]). Such a triplet is usually used for stating weak formulations
of elliptic or parabolic problems defined on Ω.
However, since system (65)-(66) also includes equatio ns defined on Γ, we need to introduce some
further spaces taking also boundary contributions into account. Thus, we set H
Γ
:= L
2
(Γ), V
Γ
:= H
1
(Γ),
and denote by (·, ·)
Γ
the scalar product in H
Γ
, by k · k
Γ
the c orresponding norm, and by h·, ·i
Γ
the duality
betwe en V
′
Γ
and V
Γ
. In g eneral, the s ymbol k · k
X
indicates the norm in the generic (real) Banach space
X and h·, ·i
X
stands for the duality between X
′
and X. We also denote by ∇
Γ
the tangential gradient
on Γ and by ∆
Γ
the Laplace-Beltrami operator. Thus, we can define the spaces
H := H × H
Γ
and V :=
z ∈ V : z|
Γ
∈ V
Γ
. (67)
Here and in the following, z|
Γ
, or als o z
Γ
, will denote the trace of z in the sense of a suitable trace
theorem. Next, we introduce the H -scalar product in the following natural way:
(k, κ), (s, σ)
H
:= (k, s) + (κ, σ)
Γ
. (68)
Then, we set, respectively on V and on V ,
((z, w))
V
:=
Z
Ω
(∇z · ∇w) dx +
Z
Γ
z|
Γ
w|
Γ
+ ∇
Γ
z|
Γ
· ∇
Γ
w|
Γ
dS, (69)
((z, w))
V
:=
Z
Ω
(∇z · ∇w) dx +
Z
Γ
z|
Γ
w|
Γ
dS, (70)
together with the related norms k · k
V
, k · k
V
. It is not difficult to prove (s ee, e.g., [33, Lemma 2.1]) that
the space V is dense in H. Conc e rning the s calar product in (70), we notice that the related norm k · k
V
is obviously equivalent to the usual one. We also set
W :=
z ∈ V : z ∈ H
2
(Ω), z|
Γ
∈ H
2
(Γ)
(71)
and endow this space with the graph norm, so that W ⊂ V, continuously and compactly.
The above defined functional spaces allow to introduce some elliptic operators. We set:
A : V → V
′
, hAz
1
, z
2
i :=
Z
Ω
∇z
1
· ∇z
2
dx, (72)
A
Γ
: V
Γ
→ V
′
Γ
, hA
Γ
ζ
1
, ζ
2
i
Γ
:=
Z
Γ
∇
Γ
ζ
1
· ∇
Γ
ζ
2
dS, (73)
A : V → V
′
, hAz
1
, z
2
i
V
:= hAz
1
, z
2
i + hA
Γ
z
1,Γ
, z
2,Γ
i
Γ
. (74)
We sha ll also use the operator A defined in (37). Note that A can be interpreted as a n operator defined
on W a nd taking values in H, thanks to the trace theore m for normal derivatives.
10
In what follows, we will use the following convention: as far as eq uations on Ω are concer ned, the
elements of V will be interpreted as functions defined on Ω with the proper regularity. When, instead,
as in most cas es in the paper, a system defined on Ω × Γ is considered, then the elements of V will be
considered as pairs of functions (z, z|
Γ
). In other words, V will be identified with a (closed) subspace of
the product space H
1
(Ω)× H
1
(Γ). Analogously, V will be identified w ith a subspace of H
1
(Ω)× H
1/2
(Γ),
in view o f the trace theorem. If we have, instead, h ∈ H, h will be often thought as a pair of functions
belonging, respectively, to H and to H
Γ
, a nd both deno ted by the sa me letter h. Of course, if we do not
have additional regularity, the second component of h needs not be the trace of the first one. Identifying
H with H
′
through the s calar product (68), we obtain the chain of continuous and de nse (thanks to
the dens ity of V into H, to the density of H
2
(Ω) into V , and to the continuous inclusion H
2
(Ω) ⊂ V)
embeddings
V ⊂ V ⊂ H ⊂ V
′
⊂ V
′
. (75)
In particular, the relation
(k, κ), (z, z|
Γ
)
H
=
Z
Ω
kz dx +
Z
Γ
κz|
Γ
dS = h(k, κ), zi
V
(76)
holds for any z ∈ V and (k, κ) ∈ H. Of course, an analog ous relation could be stated for z ∈ V .
Next, for any function, or functional z, defined on Ω, we set
m
Ω
(z) :=
1
|Ω|
Z
Ω
z dx, (77)
where the integral is substituted with the duality hz, 1i in case, e.g., z ∈ V
′
. We also define the measure
dm given by
Z
Ω
v dm :=
Z
Ω
v dx +
Z
Γ
v
Γ
dS, (78)
where v represents a generic function in L
1
(Ω) × L
1
(Γ). With some abuse of notation, we will also w rite
m(v) :=
1
|Ω| + |Γ|
Z
Ω
v dm, (79)
i.e., the “mean value” of v w.r.t. the measure dm. Here |Γ| repr esents the surface mea sure of Γ.
With these functional spaces at our disposal, we can now state our hypotheses on the nonlinear terms.
For convenience, we split − s
′
0,b
(respectively, −s
′
0,Γ
) into a sum of a (dominating) monotone part f
(respectively, f
Γ
) and a linear perturbation. More precisely, we assume that
− s
′
0,b
= f(r) − δr ∀ r ∈ dom f, −s
′
0,Γ
(r) = f
Γ
(r) − δr ∀ r ∈ dom f
Γ
(80)
and for some δ ≥ 0, with
f ∈ C
0
(dom f, R), f
Γ
∈ C
0
(dom f
Γ
, R) monotone, f(0) = f
Γ
(0) = 0, (81)
where dom f and dom f
Γ
(i.e., the domains of f and f
Γ
) are open intervals of R containing 0. We will
say that, for instance, −s
0,b
is a “ singular” potential if its domain does not coincide with the whole real
line (and we will say that it is a “regular” potential otherwise). In both cases, we will assume that
lim
r→∂ dom f
(f(r) − δr) sign r = lim
r→∂ dom f
Γ
(f
Γ
(r) − δr) sign r = + ∞. (82)
11
The key assumption that will allow us to obtain a pointwise estimate of the nonlinear terms f(
χ
) and
f
Γ
(
χ
) is the following compatibility c ondition: we ask that there exist two c onstants c
s
> 0 and C
s
≥ 0
such that
dom(f
Γ
) ⊆ dom(f), f (r)f
Γ
(r) ≥ c
s
|f(r)|
2
− C
s
∀ r ∈ dom(f
Γ
). (83)
In other words, the boundary nonlinear term f
Γ
(r), up to the sign, has to be larger than the bulk nonlinear
term f(r), at least for r far from 0. We also introduce, whenever they make sense, the antiderivative s
F (r) :=
Z
r
0
f(s) ds, F
Γ
(r) :=
Z
r
0
f
Γ
(s) ds. (84)
Notice that, in case (for instance) dom f is bounded, but f is globally summable on dom f , F can
(and will) be extended by continuity to
dom f. This is the case, e.g., of the log arithmic potential (5).
Moreove r, F (and, analog ously, F
Γ
) will be thought to be further extended to the whole real line by
assuming the value +∞ outside its effective domain. Then, identifying f a nd f
Γ
with maximal monotone
graphs in R × R, we have f = ∂F and f
Γ
= ∂F
Γ
, ∂ representing the subdifferential of convex analys is
(here in R). We refer to the monographs [1, 2, 4] for an extensive presentation of the theory of maximal
monotone operators and of subdifferentials.
To obtain an estimate of the full V -norm of u , we will also need a proper form of Poincar´e’s inequality
(see, e.g., [40, Lemma 3.2]):
Lemma 3.1. Assume that Ω is a bounded open subset of R
d
. Suppose v ∈ W
1,1
(Ω) and v ≥ 0 a.e. in
Ω. Then, s e tting K :=
R
Ω
(log v)
+
dx, the following estimate holds:
kvk
L
1
(Ω)
≤ |Ω|e
C
1
K
+
C
2
|Ω|
k∇vk
L
1
(Ω)
, (85)
the constants C
1
and C
2
depe nding only o n Ω.
Besides assumptions (80)-(83), we shall analyze system (61)-(64) under the following hypotheses :
λ
b
∈ C
2
(dom f) with λ
′′
b
∈ L
∞
(dom f), λ
Γ
∈ C
2
(dom f
Γ
) with λ
′′
Γ
∈ L
∞
(dom f
Γ
), (86)
(ϑ
0
, η
0
) ∈ H, ϑ
0
, η
0
> 0 a.e., (log ϑ
0
, log η
0
) ∈ L
1
(Ω) × L
1
(Γ), (87)
χ
0
∈ V with s
0,b
(
χ
0
) ∈ L
1
(Γ) and s
0,Γ
(
χ
0
Γ
) ∈ L
1
(Γ), (88)
H ∈ L
2
(0, T ; H), m(H) = 0 a.e. in (0, T ). (89)
Let us define the energy functional as
E[(ϑ, η), (
χ
,
χ
Γ
)] :=
Z
Ω
ϑ − log ϑ + λ
b
(
χ
) +
|∇
χ
|
2
2
− s
0,b
(
χ
)
dx
+
Z
Γ
η − log η + λ
Γ
(
χ
Γ
) +
|∇
Γ
χ
Γ
|
2
2
− s
0,Γ
(
χ
Γ
)
dS, (90)
whenever it makes sense. We will often just write E[ϑ,
χ
] for brevity. Note that we admit E to take the
value +∞. However, we can readily check that the above assumptions (87)-(88) make the initial energy
finite. Namely, we have
E
0
:= E[(ϑ
0
, η
0
), (
χ
0
,
χ
0
Γ
)] < ∞. (91)
12
Together with (86)-(89), we will also need the energy functional E to be coercive with respect to (
χ
,
χ
Γ
).
This is obtained asking that there exist c
V
> 0 and c ∈ R such that
E[ϑ,
χ
] ≥ c
V
k
χ
k
2
V
− c for all pairs (ϑ,
χ
). (92)
Such an assumption is satisfied for instance whenever
λ
b
(r) − s
0,b
(r) ≥ c
1
r
2
− c
2
for all r ∈ dom f and λ
Γ
(r) − s
0,Γ
(r) ≥ c
1
r
2
− c
2
for all r ∈ dom f
Γ
, (93)
and for some c
1
> 0 and c
2
∈ R. Of course, (92) holds trivially if −s
0,b
, or −s
0,Γ
, or both, are singular
potentials, or also in the case of standard double-well p otentials.
It is worth noting that the first of assumptions (87) on (ϑ
0
, η
0
) is s omehow artificial, in the sense that
it is stronger than what would be required in order for the initial energy to be finite. Actually, one could
relax (87) by taking just (ϑ
0
, η
0
) ∈ L
1
(Ω) × L
1
(Γ) or even (ϑ
0
, η
0
) ∈ V
′
, paying the price of having a
weaker (and more delicate to deal with) notion of solution. The Neumann-Neumann Penrose-Fife model
with L
1
or V
′
initial temperature has be en recently studied in [40]. In a fo rthcoming paper, we intend
to analyze also the present model in a similar regularity setting.
3.1 Weak solutions
We use the functional framework introduced above to specify a rigorous concept of weak solution to the
initial value problem for (65)-(66), named Problem (P) in what follows.
Definition 3.2. We say that a quadruplet (ϑ, η, u,
χ
) is a weak solution to Problem (P) if there hold
(ϑ, η) ∈ H
1
(0, T ; V
′
) ∩ L
∞
(0, T ; H), (94)
(log ϑ, log η) ∈ L
∞
(0, T ; L
1
(Ω) × L
1
(Γ)), ϑ, η > 0 almost everywhere, (95)
u ∈ L
2
(0, T ; V), (96)
χ
∈ H
1
(0, T ; H) ∩ L
∞
(0, T ; V) ∩ L
2
(0, T ; W), (97)
f(
χ
) ∈ L
2
(0, T ; H), f
Γ
(
χ
Γ
) ∈ L
2
(0, T ; H
Γ
), (98)
together with, a.e. in (0, T ), the eq uations
∂
∂t
ϑ
η
= −A
u
u
Γ
−
λ
b
(
χ
)
t
λ
Γ
(
χ
Γ
)
t
+
h
b
h
Γ
in V
′
, (99)
u = −
1
ϑ
a.e. in Ω, η = −
1
u
Γ
a.e. on Γ, (100)
∂
∂t
χ
χ
Γ
+ A
χ
χ
Γ
−
s
′
0,b
(
χ
)
s
′
0,Γ
(
χ
Γ
)
=
λ
′
b
(
χ
)u
λ
′
Γ
(
χ
Γ
)u
Γ
in H, (101)
and the initial conditions
(ϑ, η)|
t=0
= (ϑ
0
, η
0
), (
χ
,
χ
Γ
)|
t=0
= (
χ
0
,
χ
0
Γ
) a.e. in Ω and on Γ. (102)
Sometimes, for brevity, we shall indicate a solution just as a pair (ϑ,
χ
) rather than as a quadruplet
(ϑ, η, u,
χ
).
13
Remark 3.3. It is worth giving s ome explanation on the boundary behavior of ϑ. Since u = −1/ϑ ∈
L
2
(0, T ; V) by (96), it turns out that u has a trace u
Γ
on Γ, which b elongs to V
Γ
for almost every value
of the time variable thank s to the definition of V. On the other hand, we cannot simply write η = ϑ
Γ
since the trace of ϑ needs not exist. We have, instead, to intend η as (minus) the reciproca l of the trace
of u, as specified by the second (100). When we consider smoother solutions (for instance, in the a priori
estimates, or in the case when ϑ
0
is more summable, cf. (1 11) below), this problem does no t occur since
the higher regularity of ϑ permits to give sense to its trace. Regarding the phase variable, the situation
is simpler; indeed, by (97),
χ
is always smooth enough to have a trace
χ
Γ
.
3.2 Existence and uniqueness results
In this part we state our main results regarding well posedness of Problem (P) and regularization proper-
ties of weak solutions. In what follows we will denote by c a positive constant, which may vary from line
to line (or even in the same formula), de pending only on the data of the problem. Specific dependences
will be indicated when needed. Moreover, we will denote by Q a nonnegative-valued, continuous and
monotone increasing function of its arguments. Our main result can be stated as follows:
Theorem 3.4. Under assumptions (80)-(83), (86)-(89), and (92), there exists a unique weak solutio n to
Problem (P). More over, for all t > 0 ther e holds the energy identity
E[(ϑ(t), η(t)), (
χ
(t),
χ
Γ
(t))] +
Z
t
0
k∇u(s)k
2
H
+ k
χ
t
(s)k
2
H
ds = E
0
+
Z
t
0
(H(s), u(s))
H
ds. (103)
If, in addition to (89), the fo rcing function satisfies
H = (h
b
, h
Γ
) ∈ H
1
(0, T ; V
′
) ∩ L
2
(0, T, L
3+ǫ
(Ω) × L
3+ǫ
(Γ)), (104)
then we also have, for any τ ∈ (0, T ), the following regularization properties:
kuk
L
∞
(τ,T ;V )
+ kuk
L
∞
((τ,T )×Ω)
≤ Q(E
0
, τ
−1
), (105)
k
χ
k
L
∞
(τ,T ;H
2
(Ω))
+ kf(
χ
)k
L
∞
(τ,T ;H)
≤ Q (E
0
, τ
−1
), (106)
ku
Γ
k
L
∞
(τ,T ;V
Γ
)
+ ku
Γ
k
L
∞
((τ,T )×Γ)
≤ Q (E
0
, τ
−1
), (107)
k
χ
Γ
k
L
∞
(τ,T ;H
2
(Γ))
+ kf
Γ
(
χ
Γ
)k
L
∞
(τ,T ;H
Γ
)
≤ Q(E
0
, τ
−1
). (108)
The proof of the existence part is base d on an approximation-a priori bounds-passage to the limit pro-
cedure. Additional estimates lead to (105)-(108). In particular the L
∞
-bounds in (105) and in (107)
are obtained by adapting a Moser iter ation scheme with regularization devised in [40]. Note that the
existence part generalizes to the dynamic boundary c onditions case the result of Ito and Kenmochi [24].
In the case of singular potentials, we can also prove that, at least for strictly positive times,
χ
is
uniformly separated from the singularities of the potentials. In order to avoid unnecessary technical
complications, we shall state this property under the additional assumption that
dom f = dom f
Γ
= (−1, 1), |f
Γ
(r)| ≥ κ
s
|f(r)| − C
s
∀ r ∈ (−1, 1) (109)
and for some κ
s
∈ (0, 1], C
s
≥ 0. Namely, we assume the potentials to be normalized so that the pure
states are repre sented by the values ±1, b oth on the bulk and o n the boundary. Moreover, in view of
(83), we require |f
Γ
| to be larger than (a positive constant times) |f|, at least in proximity of ±1.
14
Corollary 3.5. Let the assumptions of Theorem 3 .4 hold, together with (109). Then, for any τ > 0,
there exists ω ∈ (0, 1), depending on τ but independent of T , such that
|
χ
(t, x)| ≤ 1 − ω almost everywhere in (τ, T ) × Ω and in (τ, T ) × Γ. (110)
Next, we discuss the asymptotic regularization of the temperature field. Actually, we have already noted
(cf. (105), (107)) that condition (10 4) on the forcing function H is sufficient in order for ϑ to become
uniformly separated from zero for strictly positive times. The following result (which generaliz es [40,
Thm. 2.7], where Neumann conditions are considered) states that ϑ is bounded from above, at least
for strictly positive times, provided that (104) holds and the initial temperature enjoys some additional
summability property.
Proposition 3.6. Let the ass umptions of Theorem 3.4 hold. Let als o assume that
(ϑ
0
, η
0
) ∈ L
3+ǫ
(Ω) × L
3+ǫ
(Γ), for some ǫ > 0, (111)
either (109) holds, or
χ
0
∈ L
∞
(
Ω, dm). (112)
Then, any weak solution to Problem (P) satisfies, for any τ ∈ (0, T ),
kϑk
L
∞
((τ,T )×Ω)
+ kηk
L
∞
((τ,T )×Γ)
≤ Q
E
0
, τ
−1
, kϑ
0
k
L
3+ǫ
(Γ)
, kη
0
k
L
3+ǫ
(Γ)
, k
χ
0
k
L
∞
(
Ω,dm)
. (113)
We point out that, in the case of singular potentials, the uniform boundedness of
χ
0
required by (113) is
a direct consequence of (88).
As a byproduct of Prop. 3.6, we can also prove additional regularity of the time derivative of ϑ:
Corollary 3.7. Under the assumptions of Proposition 3.6, we have
ϑ ∈ H
1
(τ, T ; H) for any τ > 0. (114)
As a consequence, (99) can b e decoupled and interpreted in the strong form (66) as a relation in H.
4 Proofs of the main results
4.1 Proof of Theorem 3.4: a priori estimates
As a first step, we detail the main estimates constituting the cor e of the existence proof. In order
to simplify the exposition, we limit ours e lves to perform formal a prior i bounds on the solutions of
Problem (P). In the next section we will see that these bo unds imply weak sequential stability. It is clear
that, in a formal proof, the estimates should be performed in the framework of a proper approximation
scheme (e.g., a Faedo-Galerkin approximation or a time discretization), possibly combined with some
regular ization of the data. However, this kind of procedure has been already described in full detail in
several papers related to similar models (see , e.g., [17] a nd [20]) and, actually, the ar guments given in
these papers could be easily adapted to our cas e.
That said, we start by presenting the estimates. In all wha t follows, we shall assume to have sufficient
regular ity to justify all the computations. In particular, we ask ϑ to be smooth enough to have a trace,
so that η = ϑ
Γ
(and correspondingly 1/η = 1/ϑ
Γ
).
15
Energy estimate. Test (99) by
1 − 1/ϑ
1 − 1/η
and (101) by
χ
t
χ
Γ,t
. Noting that two terms cancel out
and using that H has zero mean value, we obtain
d
dt
E[ϑ,
χ
] +
Z
Ω
|∇(−1/ϑ)|
2
dx +
Z
Γ
|∇
Γ
(−1/η)|
2
dS +
Z
Ω
|
χ
t
|
2
dx +
Z
Γ
|
χ
Γ,t
|
2
dS = −(H, 1/ϑ)
H
. (115)
Using that m(H) = 0 (cf. (79)), we ca n write
(H, 1/ϑ)
H
=
H, 1/ϑ − m
Ω
1/ϑ
H
=
Z
Ω
h
b
− m
Ω
1
ϑ
+
1
ϑ
dx +
Z
Γ
h
Γ
− m
Ω
1
ϑ
+
1
η
dS.
Now, the integral over Ω is easily estimated, using the Poincar´e-Wirtinger inequality, as
Z
Ω
h
b
1
ϑ
− m
Ω
1
ϑ
dx ≤ c
ε
kh
b
k
2
+ εk∇(1/ϑ)k
2
∀ ε > 0.
On the other hand, we treat the integral over Γ in this way:
Z
Γ
h
Γ
1
η
− m
Ω
1
ϑ
dS ≤ kh
Γ
k
Γ
k1/η − m
Ω
(1/ϑ)k
Γ
≤ ckh
Γ
k
Γ
k1/ϑ − m
Ω
(1/ϑ)k
V
≤ ckh
Γ
k
Γ
k1/ϑ − m
Ω
(1/ϑ)k + k∇(1/ϑ)k
≤ ckh
Γ
k
Γ
k∇(1/ϑ)k ≤ c
ε
kh
Γ
k
2
Γ
+ εk∇(1/ϑ)k
2
∀ ε > 0, (116)
where in the second and in the fourth inequalities we have used, respective ly, the trace theorem and the
Poincar´e-Wirtinger inequality. Hence, taking ε small enough and integrating (115), we obtain
E(t) +
1
2
Z
t
0
k∇(1/ϑ)k
2
+ k∇
Γ
(1/η)k
2
Γ
+ 2k
χ
t
k
2
H
ds ≤ E
0
+ ckHk
2
L
2
(0,T ;H)
. (117)
To control the full V - norm o f 1/ϑ, we have to provide a bound of its mean value (actually, only the
gradient is estimated in (117)). To this aim, we use Lemma 3.1 with v = −1/ϑ, obtaining
k1/ϑk
L
1
(Ω)
≤ |Ω|e
c
1
R
Ω
(log ϑ)
−
+
c
2
|Ω|
k∇(1/ϑ)k
L
1
(Ω)
, (118)
and the first term on the right-hand side is uniformly bounded thanks to (117) and to the expressio n (90)
of the energy functional. Moreover, by the trace theor e m, we get
k1/ηk
V
Γ
≤ c
k1/ϑk
V
+ k∇
Γ
(1/η)k
Γ
. (119)
Hence, collecting the above co mputations, (115) gives the a priori bound
E[ϑ,
χ
](t) +
Z
T
0
k(1/ϑ)(s)k
2
V
+ k
χ
t
(s)k
2
H
ds ≤ Q(E
0
, T, kHk
2
L
2
(0,T ;H)
) ∀ t ≤ T. (120)
Estimate of the nonlinear terms. Estimate (120) g ives that u = −1/ϑ ∈ L
2
(0, T ; V) and
χ
∈
L
∞
(0, T ; V), implying, thanks to (85), that the right-hand side o f the phase equation (101) belongs
16
to L
2
(0, T ; H). Now, let us test (101) (both on the bulk and on the boundary ) by f(
χ
). Using the
monotonicity of f and assumption (86) we then get
d
dt
Z
Ω
F (
χ
) dm + kf(
χ
)k
2
+
Z
Γ
f(
χ
Γ
)
f
Γ
(
χ
Γ
) − δ
χ
Γ
+
λ
′
Γ
(
χ
Γ
)
ϑ
Γ
dS
≤
Z
Ω
f(
χ
)
δ
χ
−
λ
′
b
(
χ
)
ϑ
dx ≤
1
2
kf(
χ
)k
2
+ ck
χ
k
2
+ c
1 + k
χ
k
2
L
4
(Ω)
kuk
2
L
4
(Ω)
. (121)
The key point is represented by the control of the last term on the left-hand side, and here the compati-
bility condition (83 ) comes into play. Indeed, using (83), we readily ar rive at
Z
Γ
f(
χ
Γ
)
f
Γ
(
χ
Γ
) − δ
χ
Γ
+
λ
′
Γ
(
χ
Γ
)
ϑ
Γ
≥
c
s
2
kf(
χ
Γ
)k
2
Γ
− c − ck
χ
Γ
k
2
Γ
− c
1 + k
χ
Γ
k
2
L
4
(Γ)
ku
Γ
k
2
L
4
(Γ)
. (122)
Hence, integrating (121) in time and using (122) and (120), we infer
kf(
χ
)k
L
2
(0,T ;H)
+ kf(
χ
Γ
)k
L
2
(0,T ;H
Γ
)
≤ Q(E
0
, T ). (123)
Here and below, we allow Q to depend additionally on H. Moreover, we recall that kF (
χ
0
)k
L
1
(
Ω,dm)
≤
Q(E
0
) thanks to (88) and (83).
Next, (120), (123), and a comparison of terms in (61) (i.e., the bulk component of (99) in the strong
formulation – recall that we ass ume the solutions to be s mooth at this level), give
k∆
χ
k
L
2
(0,T ;H)
≤ Q (E
0
, T ). (124)
Consequently, using standard trace and elliptic regularity theorems (cf., e.g. [5, Theorem 2.7.7 and
Theorem 3.1.5]), it is not difficult to ar rive at
k∂
ν
χ
k
L
2
(0,T ;H
Γ
)
≤ Q(E
0
, T ). (125)
This allows to test the boundary equation (62) by f
Γ
(
χ
Γ
). Proceeding as above (but witho ut us ing (61))
and controlling the term ∂
ν
χ
directly by means of (125), we finally arrive at
k∆
Γ
χ
Γ
k
L
2
(0,T ;H
Γ
)
+ kf
Γ
(
χ
Γ
)k
L
2
(0,T ;H
Γ
)
≤ Q(E
0
, T ). (126)
Now, let us come to the temperature equation. By (120) and assumption (86), we have
k∂
t
λ
b
(
χ
)k
L
2
(0,T ;L
3/2
(Ω))
≤ kλ
′
b
(
χ
)k
L
∞
(0,T ;L
6
(Ω))
k
χ
t
k
L
2
(0,T ;H)
≤ Q (E
0
, T ), (127)
and a similar relation on Γ. Actua lly, when −s
0,b
and −s
0,Γ
are singular potentials, we automatically
have a uniform L
∞
-bound on
χ
. Hence, we obtain more precisely
k∂
t
λ
b
(
χ
)k
L
2
(0,T ;H)
≤ kλ
′
b
(
χ
)k
L
∞
(Q)
k
χ
t
k
L
2
(0,T ;H)
≤ Q (E
0
, T ). (128)
By a comparis on of ter ms in the (coupled) weak formulation (99), we then get in both c ases
kϑ
t
k
L
2
(0,T ;V
′
)
≤ Q (E
0
, T ). (129)
17
Finally, to get the H-regularity of ϑ, we test (99) by ϑ. Then, using the first of (87), (8 9) and (128),
noting that
Z
Ω
λ
′
b
(
χ
)
χ
t
ϑ
≤ c
1 + k
χ
k
L
∞
(Ω)
k
χ
t
kkϑk ≤ ck
χ
t
k
2
+ c
1 + k
χ
k
2
H
2
(Ω)
kϑk
2
, (130)
thanks also to (86), observing that a similar relation holds on Γ, and applying Gronwall’s lemma, it is
not difficult to arrive at
kϑk
L
∞
(0,T ;H)
≤ Q(E
0
, T