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In-Depth Information
Rem
ark
B.2.1
In (
B.13b
),
u
k,m
+
1
is needed, as by (
B.11
)
u
k,
0
=
∈
V
for general
u
0
∈
K
◦
H
; the choice
U
k
(mk)
=
u
k,m
in (
B.13b
) would then imply
U
k
(t)
∈
V
u
0
/
for
0
<t<k
.
For every
k>
0,
U
k
is well-defined.
Theorem B.2.2
(i)
The mapping T
k
:{
u
0
,f
}→
U
k
(t) is bounded and Lipschitz-continuous
L
2
(J
;
V
∗
)
L
2
(J
from
H
×
→
;
V
)
,
uniformly in k
.
Fo r a n y
{
u
0
,f
}∈
H
×
L
2
(
0
,T
;
V
∗
)
,
the family
U
k
}
k>
0
is Cauchy in L
2
(
0
,T
) and its limit
u
∈
L
2
(J,
V
) is the unique weak solution of the PVI
(
B.9a
)-(
B.9e
),
which
satisfies
(
B.8
)
and
,
moreover
,
{
;
V
u
∈
C
0
(J,
H
).
(B.14)
(ii)
If
,
moreover
,
f
∈
S(
0
,T)
(
cf
.(
B.2
)),
then T
k
is bounded and Lipschitz-
continuous from
(
cf
.(
B.3
))
T
k
:
H
×
S(
0
,T)
→
I(
0
,T)
U
k
}
k>
0
is Cauchy in I(
0
,T)
.
(iii)
Finally
,
assuming f
and
{
=
g
+
h with
H
1
(J
;
V
∗
),
g
∈
BV (J
;
H
),
h
∈
(B.15a)
P
u
0
∈
K
,A
P
u
0
−
h(
0
)
∈
H
,
(B.15b)
U
k
}
k>
0
is uniformly bounded in I(
0
,T)
,
and u
{
=
also
lim
k
→
0
U
k
satisfies
u
∈
I(
0
,T) and the second line in
(
B.6
).
B.3 Proof of the Existence Result
We prove the existence result by establishing a priori estimates for time-semidiscrete
approximate solutions in (
B.12a
), (
B.12b
). We start by analyzing (
B.12a
), (
B.12b
)
for
m
=
0,
k>
0 and write
u
1
for
u
k,
1
and
f
0
in place of
f
k,
0
in (
B.11
). We
assume
∈
that
f
S(
0
,T)
,i.e.
f
L
1
(J
L
2
(J
;
V
∗
),
=
g
+
h, g
∈
;
H
), h
∈
and set
k
k
1
k
1
k
g
0
:=
g(τ)
d
τ, h
0
:=
h(τ )
d
τ.
(B.16)
0
0
Then, (
B.12a
), (
B.12b
) reads: find
u
1
∈
K
such that for all
v
∈
K
:
√
k
u
1
+
k
A
u
1
,u
1
−
v
V
∗
,
V
≤
(G
0
,u
1
−
v)
+
H
0
,u
1
−
v
V
∗
,
V
(B.17)
where
√
kh
0
∈
V
∗
.
G
0
:=
u
0
+
kg
0
∈
H
,H
0
:=
(B.18)
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