Biomedical Engineering Reference
In-Depth Information
Definition
4.5
will be easy to handle as it fits the construction scheme we detail
below. In particular, it will be obvious that the classical solutions we obtain satisfy
the regularity statements of this definition. A drawback is that the definition requires
the introduction of the change of variable X whose existence relies on the no-
contact assumption. In particular, it seems more adapted to introduce an “eulerian”
definition of classical solution in order to be able to consider contact in what follows.
This is the aim of the following definition we adapt from [
26
,
39
]:
Definition 4.6.
Given T>0, we call classical solution to (FRBI) on .0;T/ any
collection ..
1
;!
1
/;
u
f
;p
f
/ such that:
•
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
.
2
u
f
2 L
2
.
•
Q
F
/ with r
Q
F
/I
•
p
f
2 L
loc
.
Q
F
/ satisfies rp
f
2 L
2
.
Q
F
/I
•
..
1
;!
1
/;
u
f
;p
f
/ satisfies (FRBI) almost everywhere.
We note that, in this second definition, we consider
u
f
and p
f
as space/time
functions defined on the (open) set
Q
F
. In particular, the condition
u
f
2 H
1
.
Q
F
/
includes
time and space
derivatives. This has to be compared with the other
regularity statements which do only involve space derivatives.
Fortunately, both definitions of classical solution are consistent. In order to avoid
confusion, we provide the following proposition:
Proposition 4.4.
Definitions
4.5
and
4.6
are equivalent.
Proof.
Following the remark after Definition
4.5
, and in particular (
4.53
), we have
that any classical solution ..
1
;!
1
/;
u
f
;p
f
/ in the sense of Definition
4.5
is a
classical solution in the sense of Definition
4.6
.
Conversely, letting ..
1
;!
1
/;
u
f
;p
f
/ be a classical solution in the sense of
Definition
4.6
, similar computations to the ones entailing (
4.53
) yield that
U
f
2 H
1
.0;T I L
2
.
0
// \ L
2
.0;T I H
2
.
0
//; P
f
2 L
2
.0;T I H
1
.
0
//:
F
F
F
2 C.Œ0;TI H
1
.
0
//.To
The only point requiring more care is the proof that U
f
F
this end, we first note that
1
, there holds
•
on @
B
U
f
.t;y/ D
1
C !
1
.y G
1
/
?
2 H
1
.0;T I C
1
.@
0
1
//I
B
•
on @, U
f
vanishes.
Consequently, we might construct a velocity-field U
bdy
2 H
1
.0;T I H
2
.
0
// lifting
F
these boundary conditions and such that V WD U
f
U
bdy
satisfies also
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