Biomedical Engineering Reference
In-Depth Information
Remark 1.8.
The mapping
f
could also be the flow associated with the fluid
velocity. In this case, no convection terms appears in (
1.64
). This change of variables
has, for instance, been done in [
89
]or[
18
] in the context of fluid-solid interaction.
The compatibility conditions that the data have to satisfy are now
8
<
min
Œ0;L
.R C
0
/>0;
div
u
0
D 0 in
.0/;
u
0
D 0 on
0
; periodic in x
u
0
.t;x;RC
0
.x// D .0;
1
.x//
T
on .0;L/;
Z
L
(1.68)
:
1
D 0:
0
The result presented in [
120
]for D 0 and written in terms of .v;q;/can be
summarized as follows:
H
#
.
.0// H
#
.0;L/
Theorem 1.2.
Let
us
consider
.
u
0
;
0
;
1
/
2
H
#
.0;L/
satisfying
time
T
>0
such
(
1.68
)
,
then
there
exists
a
that
(
1.64
)
,
(
1.11
)
,
(
1.7
)
,
(
1.3
)
,
(
1.12
)
has
got
a
unique
solution
.v;q;/
that
belongs to
L
2
.0;T
I H
#
.
O
f
// \ H
1
.0;T
I L
#
.
O
f
//
L
2
.0;T
I H
#
.
O
f
//
H
1
.0;T
I H
#
.0;L// \ H
2
.0;T
I L
#
.0;L//
, where the subscript
#
stands for
the
x
-periodicity.
Remark 1.9.
For the initial conditions we will assume, for the sake of simplicity,
that
0
D 0. Consequently
O
f
D
.0/. Remark that the results in [
119
,
120
] seem
to be only correct in this case (or for a
0
more regular than H
#
.0;L/)andthat
some, surely feasible, adaptations should to be done when
0
¤ 0. Nevertheless to
suppose that
0
D 0 is a quite reasonable assumption that is not so restrictive.
The steps of the proof are the following:
•
Study the linear coupled problem, with, in particular, nonhomogeneous diver-
gence condition;
•
Study the nonlinear terms and prove that they are small for small times to obtain
that there exists a unique fixed point thanks to Picard theorem for small enough
time or for small enough data.
The first point consists then in studying the following nonhomogenous coupled
problem:
f
@
t
v v Crq D f; in
O
f
;
div v D div g; in
O
f
;
v D .0;@
t
/
T
; on †;
v D 0; on
0
;
s
e@
tt
Ǜ
2
@
xx
C @
x
LJ
2
@
xx
@
t
D..v;q/n/
2
C h; on .0;L/;
(1.69)
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