Biomedical Engineering Reference
In-Depth Information
For a given h D R
0
C ı, ı 2 B
Ǜ;K
we consider
X Df .t/ 2 V
div
I
2
.t/
LJ
LJ
S
w
2 H
0
.0;L/; .t/ D
2
.t/
LJ
LJ
S
w
g;
(1.108)
V
div
WD ff 2 V; div
h
f D 0a:e:on Dg; cf. (
1.101
):
For each test function 2 L
q
.0;T I X/; .T/ D 0,andforanyh 2 Y , such that
(
1.92
) holds, we construct solutions .
u
;/of the following problem defined on the
O
reference domain D, D @
t
Z
T
0
h@
t
.h
u
/; i
O
Z
u
/ h p div
h
Z
T
O
O
O
O
D
D
@
t
h@
y
. y
C
b
h
.
u
;
u
; /
C ..
u
; //
h
0
Z
T
Z
1
0
p
out
hj
x
D
L
1
j
x
D
L
p
in
hj
x
D
0
1
j
x
D
0
C
0
Z
T
Z
L
1
2
@
t
h
2
j
y
D
1
C
0
0
@
t
C @
x x
@
x x
C @
x
Z
t
0
.s; x/ds
@
x
Z
T
Z
L
C
0
0
Z
t
.s; x/ds
@
x x
R
0
:
C
0
By choosing ı 2 B
Ǜ;K
and applying the Korn's inequality we ensure that the viscous
bilinear form is coercive.
The following energy estimate holds for all 1 >q 2 uniformly in ı
q
L
q
.0;T
I
W
1;q
.D//
2
k
O
u
k
L
1
.0;T
I
L
2
.D//
Ck
O
u
k
(1.109)
2
2
2
L
1
.0;T
I
H
1
.0;L//
Ck@
t
k
L
1
.0;T
I
L
2
.0;L//
Ck@
t
k
L
2
.0;T
I
H
2
.0;L//
Ckk
c.T;p;K;Ǜ/
kp
@D
k
L
q
0
./;T;L
2
.@ǝ/
CkR
0
k
C
2
Œ0;L
:
2
Note that this energy estimate takes the same form as the one obtained
in the Newtonian case except that the space L
2
.0;T I H
1
.D// is replaced by
L
q
.0;T I W
1;q
.D//.
Now, let us define the following mapping,
F
W B
Ǜ;K
! Y I
F
.h/
F
.R
0
C ı/ D :
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