Biomedical Engineering Reference
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results (
4.44
)-(
4.47
) to pass to the limit in all terms of the weak formulation except
Z
T
Z
Z
T
Z
"
"
u
"
ǝ
u
"
"
u
"
ǝ
u
"
W D.
w
/ D
W D.
w
/:
0
0
O
We need further properties of the limit
u
"
!
u
to compute the asymptotics of this
term. As D.
w
/ 2 L
1
..0;T/
O
/ and
"
u
"
already converges weakly to
u
in
L
2
..0;T/
O
/ it would be sufficient here to obtain that
u
"
converges strongly to
u
in L
2
..0;T/ /, for instance. The rest of this section is devoted to the proof of a
variant of this statement.
Step 2. Asymptotics of the Nonlinear Terms.
We extend herein to the several
body case the method applied by Hoffmann and Starovoitov in [
43
].
Given an admissible test-function
w
2
K
Q
S
;we first construct a cylindrical
decomposition of the fluid-domain adapted to the test-function
w
.Ast 7!
M
Œ
t
i
is
t;"
continuous and the t 7!
M
i
are uniformly bounded in W
1;
1
..0;T/ / and
converge to t 7!
M
t
i
, there exists a non-decreasing sequence 0 D t
0
<t
1
< <
t
N
D T and associated open sets .
O
k
/
k
D
1;:::;N
such that the E
k
WD .t
k
;t
k
C
1
/
O
k
satisfy:
O
k
is the disjoint union of
f†
i
k
;i D 0;:::;ng
•
@
for a collection of .n C 1/
smooth †
i
k
, each one surrounding a rigid boundary (@ or
S
t
2
.t
k
;t
k
C
1
/
@
B
i
.t/ for
i
D 1;:::;n);
O
k
F
"
.t/ for all t 2
.
t
k
;t
k
C
1
/, for sufficiently small "I
•
•
.Œ0;T / n .
S
k
D
1
E
k
/ f.t;x/; s.t. D.
w
/.t;x/ D 0g.
Consequently, E
k
is a cylindrical domain for all k 2f1;:::;Ng and
w
is a.e. a rigid
velocity-field on any connected subset of the complement of
S
k
D
1
E
k
in .0;T/.
In particular, there holds:
Z
T
Z
ZZ
N
X
"
u
"
ǝ
u
"
u
"
ǝ
u
"
W D.
w
/ D
W D.
w
/:
0
E
k
k
D
1
Our problem reduces then to computing the limit of the N integrals on the right-
hand side of this last identity. We emphasize that N is fixed w.r.t.
w
independent of
" and that, in the integrals we want to compute now, we changed the space domain
into a
O
k
which has smooth boundaries. We might now apply all the classical results
on hydrodynamic spaces H.
O
k
/ and V.
O
k
/: existence of traces, duality, (compact)
embeddings:::see [
23
, Chap. III].
Let k 2f1;:::;Ng. We note that, as
w
is equal to a rigid velocity-field on the
smooth boundaries .†
i
k
/
i
D
1;:::;n
it is possible to adapt the construction in [
49
](see
Lemma 7.1 and more
gen
erally pp. 103-105) in order to obtain a divergence-free
W 2 C
1
.Œt
k
;t
k
C
1
O
k
/ such that W j
†
i
D
w
j
†
i
for all i D 1;:::;nand
LJ
LJ
LJ
LJ
W D.W/
LJ
LJ
LJ
LJ
k
u
"
ZZ
u
"
ǝ
u
"
I L
2
..t
k
;t
k
C
1
/I H
1
.//k
2
;
E
k
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