122 E. Bonetti
fulfilling θ
0 = 2 ,
34 χ
1
0, χ
2
0 = χ
1
, χ
2
, 35
and satisfying, almost everywhere in 0, T , c
− θ α
′′
θ χ
2
div u∂
t
θ + k
Aθ + k ∗ Aθ = F + H + L∂
t
χ
1
+θ α
′
θ − αθ div u∂
t
χ
2
+ θ α
′
θ χ
2
∂
t
div u in
V
′
36 ζ ∂
t
χ
1
χ
2
+ ∂ I
K
χ
1
, χ
2
∋ −lθ − θ
∗
−αθ div u
in H
2
37
Hu + Bαθ χ
2
= S + G
in W
′
. 38
In particular, the boundedness result in 33 for div u follows from the next lemma,
which can be proved by use of the Lax-Milgram theorem and exploiting standard esti- mates and regularity results on elliptic equations cf. [9].
L
EMMA
1. Let θ , χ
2
belong to L
2
Q such that |χ
2
| ≤ c
K
a.e. in Q. Then, under
assumptions 25, 26, 28, and 29, there exists a unique solution u ∈ L
∞
0, T ; W
solving the resulting equation 38. Moreover, the following bound holds 39
kdiv uk
L
∞
Q
≤ c
1
, for a constant c
1
depending only on , C, kαk
L
∞
R
and the convex K. In particular, the previous lemma allows us to specify hypothesis 28 cf. Remark
1. Indeed, in order to get positivity of the coefficient of the temperature time derivative in the energy balance 36, by virtue of 20, 28, 29, and 39, it is now clear that it
is sufficient to ask for a constant c
α
sufficiently small in the sense that there holds cf. [14, 15]
40 c
− θ α
′′
θ χ
2
div u ≥ c
2
:= c − θ
c
c
α
c
K
c
1
0. Let us in addition note that the specific heat turns out bounded
|c − θ α
′′
θ χ
2
div u| ≤ c
+ θ
c
c
α
c
K
c
1
. Finally, we have also to introduce a technical assumption, we need to exploit basic a
priori estimates on the solutions of the problem see, i.e., [8] for similar proceeding. Thus, we require that there holds
41 θ
c
θ
c
+ 1c
α
c
K 2
≤ c
2
λ + 2µ3.
Let us note that both 40 and 41 are in accordance with experiments see [20].
Some results on the well-posedness 123
3. Proof of Theorem 1
Existence result stated by the Theorem 1 can be proved by applying a semi-implicit time discretization scheme combined with an a priori estimate - passage to the limit
procedure. For the sake of synthesis we only outline the proof but omit the details for which we refer to [4]. We first introduce the time step of our backward finite differences
scheme τ := T N , N being a fixed positive integer. Hence, the time discrete scheme for the problem 31-38 relies on the approximation of 36-38 by
c − 2
i−1
α
′′
2
i−1
X
i−1 2
div U
i−1
2
i
− 2
i−1
τ + k
A2
i
+ k ∗ I
τ
Aθ
τ i
= L X
i 1
− X
i−1 1
τ + 2
i−1
α
′
2
i−1
− α2
i−1
div U
i−1
X
i 2
− X
i−1 2
τ +2
i−1
α
′
2
i−1
X
i−1 2
div U
i
− div U
i−1
τ + F
i
+ H
i
in V
′
42 ζ
X
i 1
−X
i−1 1
τ X
i 2
−X
i−1 2
τ
+ ∂ I
K
X
i 1
, X
i 2
∋ −l2
i
− θ
∗
−α2
i
div U
i−1
in H
2
43 HU
i
+ Bα2
i
X
i 2
= G
i
+ S
i
in W
′
, 44
where I
τ
in 42 denotes the one step backward translation operator i.e. I
τ
at = at − τ and θ
τ
the piecewise constant function related to the vector of solutions 2
i
by 45
θ
τ
t = 2
i
, if t ∈ i − 1τ, i τ ],
for i = 1, ..., N . Note that the term k ∗ I
τ
Aθ
τ i
turns out to be explicit in the scheme see [1]. Finally, F
i
, H
i
, G
i
, and S
i
stand for suitable time independent functions
discretizing the data F, H , G, and S i.e. F
i
= τ
−1
R
iτ i−1τ
Fs ds. Thus, if we let 2
= θ , X
i0
= χ
i
for i = 1, 2, and U the corresponding unique solution of 44
written for i = 0 cf. Lemma 1, by use of a fixed point theorem we are able to prove existence of a discrete solution for 42-44 for any i ≥ 1, at least for τ sufficiently
small. Henceforth, we perform suitable estimates on the discrete solutions independent of the parameter τ , in order to pass to the limit as τ ց 0 by use of weak and weak
star compactness arguments or by direct Cauchy proof. To this aim, let us introduce the following notation: given a N +1-vector of time independent functions a
, ..., a
N
we term by a
τ
the related piecewise constant function a
τ
cf. 45 and by e
a
τ
the piecewise linear in time interpolation function, namely
46 e
a
τ
t = a
i
+ a
i
− a
i−1
τ t − i τ ,
t ∈ [i − 1τ, τ ].
124 E. Bonetti
Thus, if we use the above notation, it is straightforward to rewrite the discrete system 42-44, as follows
c − I
τ
θ
τ
α
′′
θ
τ
χ
2τ
divu
τ
∂
t
e θ
τ
+ k Aθ
τ
+ k ∗ I
τ
Aθ
τ τ
= H
τ
+ F
τ
+ L∂
t
e χ
1τ
+ I
τ
θ
τ
α
′
θ
τ
− αθ
τ
divu
τ
∂
t
e χ
2τ
+I
τ
θ
τ
α
′
θ
τ
χ
2τ
∂
t
div e
u
τ
47 ζ
∂
t
e χ
1τ
∂
t
e χ
2τ
+ ∂ I
K
χ
1τ
, χ
2τ
∋ −lθ
τ
− θ
∗
−αθ
τ
I
τ
divu
τ
48
Hu
τ
+ Bαθ
τ
χ
2τ
=
Gτ + Sτ 49
with 50
e θ
τ
0 = θ ,
e χ
iτ
0 = χ
i
i = 1, 2. Hence, by exploiting suitable a priori estimates on the system 42-44, we can prove
that there exists σ 0 such that for τ ≤ σ the following bounds hold independently of τ
e θ
τ H
1
0,T ;H ∩L
∞
0,T ;V
+ kθ
τ
k
L
∞
0,T ;V
≤ c 51
2
X
i=1
k e
χ
iτ
k
H
1
0,T ;H ∩L
∞
Q
+ k e
χ
iτ
k
L
∞
Q
≤ c 52
k e
u
τ
k
H
1
0,T ;W
+ ku
τ
k
L
∞
0,T ;W
+ kdiv u
τ
k
L
∞
Q
≤ c. 53
The reader can refer to [8] and [10] for a detailed presentation of an estimating pro- cedure as that we have used to prove 51-53 and to [1] for a possible argument to
handle the convolution product k ∗ I
τ
Aθ
τ τ
. Thus, by use of compactness argu- ments from 51-53, and 45, 46, we can deduce up to subsequences the following
convergence results, as τ ց 0 e
θ
τ ∗
⇀θ in H
1
0, T ; H ∩ L
∞
0, T ; V , e
θ
τ
→ θ in C [0, T ]; H
54 θ
τ ∗
⇀θ in L
∞
0, T ; V , θ
τ
→ θ in L
∞
0, T ; H 55
e χ
j τ ∗
⇀χ
j
in H
1
0, T ; H ∩ L
∞
Q, χ
j τ ∗
⇀χ
j
in L
∞
Q, j = 1, 2
56 e
u
τ
⇀ u in H
1
0, T ; W, u
