Biomedical Engineering Reference
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where
.
u
n
;h
n
/ 2 ŒH
2
./\H
0
./ŒH
2
./\H./
. Moreover, there exists
a constant
C D C.n;!;;;;;
0
;;H
0
/>0
such that
k
u
n
k
2;2
Ckh
n
k
2;2
C kf
n
k
2
;
for all
0 jnjN:
(3.86)
Finally, if
.
w
;b/
is another
T
-periodic solution to
(
3.80
)
corresponding to the
same data with
w
2 W
2;2
.0;T I L
2
./ \ W
1;2
.0;T I H
1
.// \ L
2
.0;T I H
2
.//;
b 2 W
1;2
.0;T I L
2
.// \ L
2
.0;T I H
2
.//
then necessarily
.
u
;h/ D .
w
;b/
.
(b)
Suppose the Lamé coefficients satisfy ((
3.3
) and)
C D 0
. Then, there exists
at least one resonant period for (
3.80
)-(
3.81
).
Proof.
In view of what we already proved in this section and of Theorem
3.5
(a), we
have to show only part (b). To this end, without loss we may take H
0
D H
0
e
3
,so
that, setting
u
D .
u
3
;
u
0
/,Eq.(
3.80
) with f 0 becomes
9
=
.
u
3
/
tt
u
3
D 0
u
0
tt
u
0
D
1
0
curlh H
0
in .0; 1/;
(3.87)
0
curl curlh D curl.
u
0
t
H
0
/
divh D 0
1
h
t
;
Recalling (
3.82
), from (
3.87
) it is clear that the space
H
0
WD
ǚ
.
u
;
u
t
;h/ 2
H
W
u
D .
u
3
;0/;
u
t
D ..
u
3
/
t
;0/; h D 0g
is a subspace of
.t/. Moreover,
from (
3.87
)
1
and the fact that
u
3
vanishes on @ at all times, by a simple procedure
we infer
H
left invariant by the action of the semigroup
U
2
2
2
2
k
u
3
.t/k
2
C kr
u
3
.t/k
2
D k
u
3
.0/k
C kr
u
3
.0/k
2
; for all t 0;
and the desired property follows from Theorem
3.7
.
t
3.5.3
A Liquid-Structure Interaction Problem Showing
Generic Absence of Resonance
In the present and following sections we will investigate the occurrence of resonance
within a certain class of linear models of liquid-structure interactions. The motion
of the liquid is assumed to be governed by the Stokes (linearized) equations, while
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