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A
:=
A
+
=
where
λI
satisfies (
B.4b
) with
λ
0. Then, (
B.45
) for the new quantities
implies
e
−
λt
w(t)
I(
0
,T )
C
sup
0
<θ<T
e
−
2
λτ
r(τ)
d
τ
2
.
θ
1
e
−
λt
s(t)
≤
w(
0
)
H
+
S(
0
,T )
+
(B.47)
0
We apply Lemma
B.3.4
to the time-discrete PVI (
B.12a
), (
B.12b
). We assume
(
B.4b
) with
λ
=
0 and choose
1
k
1
k
p
m
=
p(τ)
d
τ, q
m
=
q(τ)
d
τ
(B.48)
J
k,m
J
k,m
where
s
=
p
+
q
satisfies (
B.39
). Then, by Lemma
B.3.4
,
1
2
α
−
P
M
≤
p
L
1
(
0
,Mk
;
H
)
,Q
M
≤
w
0
H
+
q
L
2
(
0
,Mk
;
V
∗
)
.
Then (
B.32
) implies
max
{
X
M
,Y
M
}≤
C
{
w
0
H
+
s
S(
0
,Mk)
}
.
(B.49)
Remark B.3.8
In (
B.49
), we assumed (
B.4b
) with
λ
=
0. If
λ>
0, we replace (
B.48
)
by
J
k,m
e
−
λτ
p(τ)
d
τ
J
k,m
e
−
λτ
d
τ
J
k,m
e
−
λτ
q(τ)
d
τ
J
k,m
e
−
λτ
d
τ
p
m
=
,
m
=
.
(B.50)
λk)
m
w
m
, and analogously
We define for
k
sufficiently small
w
m
=
(
1
−
p
m
,
q
m
.
Then the same reasoning as in Remark
B.3.7
gives for
k
sufficiently small
max
C
S(
0
,Mk)
,
{
X
M
, Y
M
}≤
e
−
λτ
s(τ)
w
0
H
+
(B.51)
λk)
M
λk
M
/(
1
if we use 1
/(
1
−
≤
exp
(
−
−
λk))
.
We now apply (
B.49
), (
B.51
) to obtain a priori estimates for the approximate
solution
U
k
(t)
obtained from (
B.12a
), (
B.12b
), (
B.
13a
), (
B.13b
). To this end, we
identify the sequence
{
w
m
}
with the step function
w
k
(t)
such that
w
k
(t)
J
k,m
=
w
m
,m
=
1
,...,M.
(B.52)
Lemma B.3.9
Assume
(
B.4b
)
with λ
=
0.
Then there exists C>
0
depending only
on α such that
w
k
(t
+
k)
I
(
0
,T )
≤
C(
w
0
H
+
s(t)
S(
0
,T )
)
(B.53)
and
√
k
w
k
(t
+
k)
I
(
0
,T )
≤
C(
w
1
H
+
w
1
V
+
s(t)
S(
0
,T
+
k)
).
(B.54)
Proof
Since
T
Mk
,(
B.49
) and (
B.52
)give(
B.53
). To show (
B.54
), we may as-
sume
T>k
. We apply (
B.49
) to the shifted sequence
=
m
{
w
m
+
1
}
1
.
=
We are now in position to give a priori estimates for the sequence of solutions to
(
B.12a
), (
B.12b
).
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