Biomedical Engineering Reference
In-Depth Information
0
•
the shapes of the bodies
i
are open connected subsets of
having smooth
boundaries and such that
(
4.19
)
holds true,
B
the densities
i
2 L
1
.
0
•
B
i
/
satisfy
(
4.1
)
.
Given u
0
2 H./
and
T>0
, there exists a weak solution
..'
i
;
i
/
i
D
1;:::;n
;
u
/
to
(FRBI)
on
.0;T/
.
The proof is a combination of [
20
, Lemma 2.2] and of the construction of weak
solutions adapted to non-smooth body shapes that we presented in Sect.
4.3.1
.We
recall the main ingredients.
First Step: Extension of Weak Solutions up to Contact.
In Theorem
4.1
, existence
of weak solutions is proven locally in time. In order to extend existence of weak
solutions up to contact between rigid bodies, we remark that the time of existence is
only limited by the L
2
-norm of the initial data and the initial distance between rigid
bodies. Existence of weak solutions up to contact then yields from a concatenation
principle for weak solutions to (FRBI) that we adapt from [
20
, Lemma 2.2 and
Sect. 4].
Let ..'
i
;
i
/
i
D
1;:::;n
;
u
/ be a weak solution with initial data ..
B
i
;
i
/
i
D
1;:::;n
;
u
0
/
M
t
i
the isometries such that
defined on .0;T
0
/. Let us denote
t
i
.
0
M
B
i
/ D
B
i
.t/;
8 t 2 .0;T
0
/:
As
u
2 L
1
.0;T I L
2
.// and the eulerian velocity of
M
t
i
coincides with
u
,wehave
that
M
t
i
2 W
1;
1
.0;T/ C.Œ0;T
0
/. Hence, we might define
B
0
i
D
M
T
0
i
.
0
i
/;
B
8 i Df1;:::;ng:
Assuming that we remain far from contact, these new shapes are open subsets of
having smooth boundaries and satisfying (
4.19
). Similarly, when t ! T
0
the
densities
i
.t; / converge almost everywhere (and thus in all L
p
-spaces for finite
p)to
Q
i
.x/ D
i
.Œ
T
0
i
1
x/
1
B
M
:
0
i
Obviously, these new initial densities satisfy (
4.1
). Finally, for all
w
2
D
./ s.t.
D.
w
/ D 0 in the neighborhood of
.T/, we might introduce
w
as a test-function in
the weak formulation (
4.32
), for arbitrary 2 C
c
.0;T
0
/, with support sufficiently
close to T
0
. This yields that
S
Z
.t;/
u
.t;x/
w
.t;x/ 2 H
1
.0;T
0
/ C.Œ0;T
0
/:
f.t/D
Because of energy estimate (
4.34
), there holds:
jf.t/j
p
E
c
.0/k
p
.t;/
w
I L
2
./k:
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