Biomedical Engineering Reference
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r
u
N
!r
u
in the sense of distributions. Therefore we have
Z
T
Z
Z
T
Z
Z
T
Z
G
D
.t/
r
u
N
lim
N
!1
D
.t/
r
u
:
0
max
0
0
Since this conclusion holds for all the test functions supported in .0;T/
.t/
,
from the uniqueness of the limit, we conclude
G
Dr
u
in .0;T/
.t/
.
Therefore, we have shown that
r
u
N
*r
u
weakly in L
2
..0;T/
max
/:
STEP 3.
We want to show that
R
0
R
F
G
N
D
R
0
R
F
r
u
W
q
for every
test function
q
D
q
ı A
,
q
2
X
F
. This will follow from STEP 2, the uniform
boundedness and convergence of the gradients r
W
q
t
N
u
N
provided by Lemma
2.2
,
and from the strong convergence of the test functions
q
N
!
q
provided by
Lemma
2.6
.Moreprecisely,wehavethatforevery
q
D
q
ı A
,
q
2
X
F
Z
T
Z
Z
T
Z
t
N
u
N
W
q
N
G
W
q
D
lim
N
!1
F
r
0
F
0
Z
T
Z
1
1 C
t
N
N
r
u
N
D
lim
N
!1
W
q
0
max
Z
T
Z
Z
T
Z
1
1 C
r
u
W
q
D
u
W
q
:
D
F
r
0
0
Here, we have used from (
2.110
)thatr
u
N
t
N
u
N
,andr
u
u
: This
Dr
Dr
completes proof.
t
Corollary 2.1.
For every
.
q
; ; / 2
X
we have
t
N
q
N
!r
q
; in L
2
..0;T/
F
/:
r
Proof.
Since
t
N
q
N
and
q
are the test functions for the velocity fields, the same
arguments as in Proposition
2.7
provide weak convergence of .r
t
N
q
N
/
N
2N
.
To prove strong convergence it is sufficient to prove the convergence of norms
kr
t
N
q
N
k
L
2
.
F
/
!kr
q
k
L
2
.
F
/
. This can be done, by using the uniform con-
vergence of .
t
N
/
N
2N
, in the following way:
Z
T
Z
Z
T
Z
1
1 C
t
N
jr
q
j
1
t
N
q
N
k
2
N
2
2
kr
L
2
.
F
/
D
!
1 C
jr
q
j
0
max
0
max
Z
T
Z
2
2
L
2
.
F
/
:
q
j
q
k
D
F
jr
Dkr
0
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