Biomedical Engineering Reference
In-Depth Information
Now, given
u
2
Y
h
, we denote
u
D
u
h
C with 2 C
1
.
R
d
C
n B
h
/ and split the
integral:
Z
D
Z
C 4
Z
D.
u
h
/ W D./ C
Z
2
2
2
:
C
n
B
h
jr
u
j
C
n
B
h
jr
u
j
C
n
B
h
jrj
R
d
R
d
R
d
R
d
C
n
B
h
u
u
n
2
D
.
R
d
By construction, we have D
lim
n
!1
n
, with
n
WD
C
n B
h
/,for
the topology associated with the norm kr I L
2
.
R
d
C
n B
h
/k. Hence, passing to the
limit in (
4.97
), we obtain:
2
Z
D.
u
h
/ W D./ D 0:
R
d
C
n
B
h
Finally, there holds:
Z
D
Z
C
Z
2
2
2
C
n
B
h
jr
u
h
j
C
n
B
h
jr
u
j
C
n
B
h
jrj
R
d
R
d
R
d
Z
2
:
C
n
B
h
jr
u
h
j
R
d
This ends the proof.
t
Relying on this minimizing property, one might prove that h 7! F.h/ is
locally lipschitz on .0; 1/. This would justify rigorously that the Cauchy problem
associated with (
4.91
) is locally well-posed. We do not go into the details and only
write the subsequent existence result:
For arbitrary
h
0
>0
and
h
0
2
R
, there exists a unique maximal solution
.T
;h/
to
(
4.91
)
with initial data:
h.0/ D h
0
:
h.0/ D h
0
;
Furthermore, we have the alternative:
•
either
T
D1
,
•
either
T
< 1
and
1
h.t/
C h.t/ C
h.t/
lim sup
t
!
T
DC1:
We are more interested here in analyzing the blow-up of the maximal solutions:
Proposition 4.10.
Assume that the gravity brings the sphere to the wall, i.e.,
mg
e
d
<0
. Given
h
0
>0
and
h
0
2
R
and
.T
;h/
the associated maximal solution,
there holds
T
D1
. In particular, there exists a function
h
min
W Œ0; 1/ ! .0; 1/
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