Biomedical Engineering Reference
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We aim here at generalizing this condition in the framework of Sobolev spaces,
keeping in mind that all constants in Sobolev imbeddings depend on the geometry
of the domain. Our main result reads as follows:
Theorem 4.3.
Let
..
i
;!
i
/
i
D
1;:::;n
;
u
f
;p
f
/
be a classical solution to (FRBI) on
.0;T
/
and assume that contact occurs in
T
. Then,
(i)
there holds:
Z
T
0
kr
u
f
.t; /I L
d
C
1
.
F
.t//kdt DC1I
(4.68)
0
1
is a ball, there also holds:
Z
T
0
kr
B
(ii)
if
n D 1
and
2
u
f
.t; /I L
2
.
F
.t//kdt DC1:
(4.69)
We note that both criteria do not prevent from continuing a solution s.t.
k
u
f
I L
2
.
F
kr
u
f
I L
2
.
F
.t//k remain bounded with time. However, the
second implication prevents from requiring that r
.t//k
and
Q
F
/ after collision.
Proof of Item (i).
Let us consider ..
i
;!
i
/
i
D
1;:::;n
;
u
f
;p
f
/ a classical solution to
(FRBI) on some time-interval .0;T
/.Wedefine:
2
u
f
2 L
2
.
B
0
.t/ WD
R
d
n ;
8 t 2 Œ0;T
/;
and
dŒf
B
i
.t/g
i
D
0;:::;n
WD inf
n
dist.
2
o
;
B
i
.t/;
B
j
.t//;i ¤ j 2f0;:::;ng
8 t 2 Œ0;T
/:
By assumption, we have
lim inf
t
!
T
dŒf
B
i
.t/g
i
D
0;:::;n
D 0:
The cornerstone of the proof is the following proposition adapted from [
54
,
Theorem 3.1]:
Proposition 4.5.
Let
B
1
WD B.X
1
;ı/
and
B
2
WD B.X
2
;ı/
be two disjoint balls of
R
d
and denote
.X
2
X
1
/
jX
2
X
1
j
e
12
WD
;
12
WD dist.B
1
;B
2
/:
Let
2 W
1;p
.
R
d
/
such that
•
r D 0
on
R
d
,
•
.x/ WD
i
C !
i
.x X
i
/;
8 x 2 B
i
:
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