Biomedical Engineering Reference
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and, on the other hand, that the distance between
B
1
.t/ and @ remains bounded
from below by a strictly positive constant, it is possible to construct a vector-field
ƒ 2 H
1
..0;T/I C
c
.// satisfying the following properties:
•
divƒ D 0 on .0;T/ ,
•
ƒ.t;x/ D
1
C !
1
.x G
1
/
?
in a neighborhood of
B
1
.t/,forallt 2 Œ0;T/,
•
Supp.ƒ.t; // for all t 2 Œ0;T.
Classical results on differential systems entail that the associated flow .t;x/ 7!
X.t;x/ solution to
@
t
X.t;y/ D ƒ.t;X.t;y//;
8 .t;y/ 2 .0;T/ ;
X.t;y/ D y;
8 y 2 ;
is then well defined globally and satisfies:
•
X 2 C
1
.Œ0;TI Diff
1
.// (where Diff
1
./ stands for the set of smooth
diffeomorphisms of ),
•
X.t;
0
1
/ D
B
1
.t/,forallt 2 .0;T/,
•
X.t;/ is an isometric mapping on
B
0
1
for all t 2 Œ0;T,
•
X.t;y/ D y for all .t;y/ 2 .0;T/ @.
B
Hence, we might define the associated change of unknown:
A.t;y/ D a.t;X.t;y//;
8 .t;y/ 2 Q
F
:
(4.52)
We keep the convention that capital letters are associated with the change of
unknown computed in this construction. Next, a classical solution is defined as
follows:
Definition 4.5.
Given T>0, we call classical solution to (FRBI) on .0;T/ any
collection ..
1
;!
1
/;
u
f
;p
f
/ satisfying
•
1
2 H
1
.0;T/ and !
1
2 H
1
.0;T/I
•
there exists ı>0such that the associated body motion t 7!
B
1
.t/ satisfies
dist.
B
1
.t/;@/ > ı;
8 t 2 .0;T/I
•
U
f
2 H
1
.0;T I L
2
.
0
// \ C.Œ0;TI H
1
.
0
// \ L
2
.0;T I H
2
.
0
//I
F
F
F
•
P
f
2 L
2
.0;T I H
1
.
0
//I
•
..
1
;!
1
/;
u
f
;p
f
/ satisfies (FRBI) almost everywhere.
We note that, as long as no contact occurs, straightforward computations entail
that the regularity statements in the above definition yield:
F
@
t
u
f
2 L
2
.
2
u
f
/ 2 L
2
.
rp
f
2 L
2
.
Q
F
/; .
u
f
; r
u
f
; r
Q
F
/;
Q
F
/: (4.53)
Consequently, the last statement in this definition makes sense.
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