Weak formulation 73
4. The explicit problem
For v ∈ C [0, T ]; V ∩ C
1
[0, T ]; H and t ∈]0, T ], we introduce the auxiliary norm kvk
2 t
:= kvk
2 C
[0,t ];V
+ kv
t
k
2 C
[0,t ];H
. 20
and let s = s
1
+ s
2
∈ L
1
0, T ; H + W
1,1
0, T ; V
′
, 21
with s
2
0 = s
20
. 22
We recover useful properties for the following reduction of Problem 1, which is expressed in terms of the single variable function u, for the sake of convenience.
P
ROBLEM
2. Find u ∈ C [0, T ]; V ∩ C
1
[0, T ]; H fulfilling 12, 14-15, and hu
t t
t , vi + hwt , vi + hLut , vi = hs
1
t + s
2
t , vi 23
for a.a. t ∈]0, T [ and any v ∈ V , for some w ∈ L
∞
0, T ; H . Indeed, we can state
T
HEOREM
3. Assume that Hypotheses a and b, 21-22 and 16 are satisfied. Then Problem 2 has a unique solution. Moreover, letting { ˆ
u , ˆ
v , ˆ
s
20
, ˆ s
1
, ˆ s
2
}, { ˜u , ˜
v , ˜
s
20
, ˜ s
1
, ˜ s
2
} be two sets of data and letting ˆ
u, ˜ u represent the related solutions, the estimate
k ˆu − ˜uk
2 t
≤C
1
k ˆu − ˜u
k
2
+ | ˆv − ˜v
|
2
+ kˆs
20
− ˜s
20
k
2 ∗
+ kˆs
1
− ˜s
1
k
2 L
1
0,t ;H
+ kˆs
2
− ˜s
2 t
k
2 L
1
0,t ;V
′
24 holds for any t ∈]0, T ], where the constant C
1
depends just on ℓ, c, C, and T . Proof. We start from the existence result given in [4], where stronger assumptions are required
on data. Therefore we regularize s, u , v
and choose three families {s
ε
}, {u
ε
}, {v
ε
} such that s
ε
∈ W
1,1
0, T ; H , s
ε
→ s in
L
1
0, T ; H ∩ W
1,1
0, T ; V
′
, u
ε
∈ V, Lu
ε
∈ H, u
ε
→ u in
V , v
ε
∈ V, v
ε
→ v in
H as ε ↓ 0. For instance, u
ε
could be taken as the solution of the elliptic problem see [3, Ap- pendix]
u
ε
+ εLu
ε
= u .
for ε sufficiently small cf. 3. Thanks to Lemma 1 and [4, Lemma 3.3, p. 88], Problem 2 with s, u
, v replaced by s
ε
, v
ε
, u
ε
has a unique solution u
ε
, w
ε
satisfying u
ε
∈ W
1,∞
0, T ; V ∩ W
2,∞
0, T ; H , w
ε
∈ L
∞
0, T ; V
′
. Actually, in our case w
ε
belongs to L
∞
0, T ; H owing to 6. Now, we can use [4, estimate 3.5, p. 87] as contracting estimate. Indeed, since 14 and
74 S. Durando
5 yield w, u
t
≥ 0 a.e. in ]0, T [, the procedure followed in [4, Lemma 3.2, p. 87] enables us
to infer k ˆu
ε
− ˜u
ε
k
2 t
≤C
1
k ˆu
ε
− ˜u
ε
k
2
+ | ˆv
ε
− ˜v
ε
|
2
+ kˆs
ε 20
− ˜s
ε 20
k
2 ∗
+ kˆs
ε 1
− ˜s
ε 1
k
2 L
1
0,t ;H
+ kˆs
ε 2
− ˜s
ε 2
t
k
2 L
1
0,t ;V
′
25 with obvious notation, and note that 0 solves Problem 2 with null data
ku
ε
k
2 t
≤ C
1
ku
ε
k
2
+ |v
ε
|
2
+ ks
ε 20
k
2 ∗
+ ks
ε 1
k
2 L
1
0,t ;H
+ ks
ε 2
t
k
2 L
1
0,t ;V
′
26 for all t ∈]0, T ] and for some constant C
1
independent of ε. Consequently, also kw
ε
k
L
∞
0,T ;H
is uniformly bounded. Due to well-known compactness results, we can find subsequences converging weakly star
∗
⇀ . Let u, w, and ε
n
↓ 0 fulfill u
n ∗
⇀ u
in L
∞
0, T ; V ∩ W
1,∞
0, T ; H , w
n ∗
⇀ w in
L
∞
0, T ; H , where u
n
and w
n
stand for u
ε
n
and w
ε
n
, respectively. Now, one can show that the pair u, w solves the equations of Problem 2. In fact, it turns out that
u
n
→ u strongly in C
[0, T ]; V ∩ C
1
[0, T ]; H , 27
w ∈ Bu
t
a.e. in ]0, T [ . 28
The proof of 27 consists in a direct check of the Cauchy condition in C [0, T ]; V ∩C
1
[0, T ]; H for u
n
, by applying again [4, estimate 3.5, p. 87]. Further, accounting for the strong convergence of {u
n t
} in L
2
0, T ; H and arguing as in the proof of 19, thanks to [1, Lemma 1.3, p. 42] one verifies the second condition 28.
At this point, we can first take the limit in 25 on some subsequence ε
n
↓ 0 and recover 24. Then, the uniqueness of u and w follows from 24 and a comparison in 23.
5. Existence, uniqueness and continuous dependence
Kategori