Biomedical Engineering Reference
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restriction of the L
2
.;dy/ scalar product to K
0
Œ
0
B
1
;reads:
Z
Z
.U;V/ D
U V C
1
U V;
0
1
F
0
B
Z
U V C m
1
V
U
C
J
1
!
V
!
U
;
D
0
F
with the notations
U
j
B
1
.x/ D
U
C !
U
.x G
1
/
?
; resp. V
j
B
1
.x/ D
V
C !
V
.x G
1
/
?
:
In [
58
], the authors prove that the operator .D.A/;A/ is self-adjoint and positive
yielding a contraction semi-group t 7! S.t/on K
0
Œ
1
;. We note that, as classical
in incompressible fluid problems, the formalism presented here gets rid of the
pressure P. This pressure is reintroduced afterwards as follows. In the case F
NS
,
F
LM
,andF
AM
vanish, let U.t;/ WD S.t/U
0
. For sufficiently smooth data U
0
,we
have U.t;/ 2 D.A/ for all t>0,sothatweset
B
0
(
1
.t/ C !
1
.t/.y G
1
/
?
; on
0
B
1
;
U.t;x/ D
0
:
U
f
.t;x/;
on
F
By construction, U
f
is divergence-free and satisfies boundary conditions:
U
f
.t;y/ D
1
C !
1
.y G
1
/
?
;
0
8 y 2 @
B
1
; U
f
.t;y/ D 0 on @:
Then, we have
@
t
U D AU on .0;T/:
(4.62)
0
/ yields that:
Multiplying this equation by the trivial extension of any W 2
D
.
F
Z
.@
t
U
f
U
f
/ W D 0:
F
0
Applying DeRham theory, we construct a pressure P
f
such that ..
1
;!
1
/;U
f
;P
f
/
satisfies (
4.54
). Multiplying then (
4.62
) with any W 2 C
c
./ \ KŒ
1
;yields:
B
Z
m
1
W
P
1
C
J
1
!
W
!
1
C
@
t
U W
0
F
Z
Z
Z
2.y G
1
/
?
D.U/nd!
W
D
U W
2D.U/nd
W
0
1
0
1
F
0
@
B
@
B
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