Biomedical Engineering Reference
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k. In what follows, we denote by C
i
, i 2
N
[f0g, generic positive constants
possessing this property. We commence to observe that, if k D 0, from well-known
results on the homogeneous Stokes problem and the property of the bilinear form b,
from (
3.109
) we readily show
v
0
Drp
0
D 0; k
u
0
k
2;2;
C
0
kf
0
k
2;
:
(3.119)
If k ¤ 0, proceeding exactly as in the proof of (
3.118
), from (
3.109
)
1
5
we conclude
2
2
2
{k
3
!
3
2
{k!kv
k
k
2
CkD.v
k
/k
k
u
k
k
2;
C{k!b.
u
k
;
u
k
/ D .f
k
;
u
k
/
; (3.120)
which, by Korn's, Poincaré's, and Schwarz's inequalities implies
1;2
C
1
kf
k
k
2;
k
u
k
k
2;
:
kv
k
k
Moreover, by (
3.109
)
4
and the trace theorem, we have
jkjk
u
k
k
1=2;2;
C
2
kv
k
k
1;2
(3.121)
that once combined with the previous inequality furnishes
2
jkj
k
u
k
k
1=2;2;
Cjkjkv
k
k
1;2
C
3
kf
k
k
2;
:
(3.122)
If we employ (
3.122
)into(
3.120
), with the help of (
3.106
), and recalling that jkj1
we show that
k
u
k
k
2;2;
C
4
kf
k
k
2;
:
(3.123)
Finally, combining (
3.123
) with classical estimates for solutions to the Stokes
system (
3.109
)
1
4
, we conclude
kv
k
k
2;2
Ckp
k
k
H
1
./=
R
C
5
jkjkf
k
k
2;
:
(3.124)
The existence result stated in the theorem is then a direct consequence of
(
3.119
), (
3.122
)-(
3.124
) and of Plancharel theorem. Finally, uniqueness is discussed
exactly as in Theorem
3.5
, and its proof will therefore be omitted.
t
3.6
Numerical Experiments
Objective of this section is to present some numerical experiments aimed at
investigating the “generic” absence of resonance in liquid-structure models of the
type considered in Sect.
3.5.3
(see Theorem
3.11
(a)). The idea is that if the absence
of resonance is only “generic,” namely, for T -periodic forces only a dense set of the
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