Biomedical Engineering Reference
In-Depth Information
In this second case also, ..'
i
;
i
/
i
D
1;:::;n
;
u
"
/
">0
, is a sequence of weak solutions
to (FRBI) associated with the container .
"
/
">0
and initial conditions given by
..
i
;
";
i
/
i
D
1;:::;n
;
u
0;"
/. One then shows that a weak cluster point of this sequence
of weak solutions to (FRBI) in the "-geometry yields a weak solution to (FRBI) in
the limit geometry.
For simplicity, we detail the computations in the second application only. We
point out the arguments that need to be adapted when dealing with a sequence of
penalized solutions. First, we extend all fields outside
"
by 0. This yields sequences
we still denote '
i
,
i
and
u
"
. As initial velocity-fields are bounded in H./ and
body densities are also uniformly bounded, it yields that, up to extract a subsequence
we do not relabel for simplicity, there holds:
";0
B
'
i
! '
i
in L
1
..0;T/ /
w
;
(4.42)
i
!
i
in L
1
..0;T/ /
w
;
(4.43)
u
"
!
u
in L
1
.0;T I L
2
.//
w
and in L
2
.0;T I H
0
.//
w
.
(4.44)
Our aim is to prove that the collection ..'
i
;
i
/
i
D
1;:::;n
;
u
/ is a weak solution to
(FRBI) for initial datum ..
i
;
i
/
i
D
1;:::;n
;
u
0
/ on .0;T/ . The main ingredients
B
of the proof are
passage to the limit in the transport equation satisfied by .'
i
;
i
/, and construc-
tion of the isometries
•
t
i
associated with the body motions,
M
•
passage to the limit in the nonlinear term for any test-function
w
2
K
Œ
Q
S
;:
Z
T
Z
"
u
"
ǝ
u
"
W D.
w
/:
0
We consider these two steps separately. For the second step, we detail a method
due to [
43
] well-adapted to this fluid/body problem. In this section, we consider
solutions prior to contact. We recall that energy estimate (
4.34
) is sufficient
to guarantee that the sequence ..'
i
;
i
/
i
D
1;:::;n
;
u
"
/
">0
remains far from contact
uniformly on some time interval .0;T/ independent of ">0.
Step 1. Construction of Isometries.
As we noticed already, the results of [
18
]
apply to the sequence of divergence-free velocity-fields .
u
"
/
">0
. On the one hand,
this sequence converges to
u
in L
2
.0;T I H
0
.//
w
, on the other hand the sequence
of initial data .'
";
i
;
";
i
/ converge a.e. to
1
i
and
i
1
i
, respectively. Consequently,
0
0
B
B
there holds
'
i
! '
i
; in C.Œ0;TI L
q
.// for all finite q;
(4.45)
i
!
i
; in C.Œ0;TI L
q
.// for all finite q;
(4.46)
and '
i
and
i
satisfy (
4.28
) with '
0
given by their respective initial data, for all
2 C
c
.Œ0;T//. This yields in particular that '
i
.t; / is the indicator function of
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