Biomedical Engineering Reference
In-Depth Information
More precisely, Lemma 3.3 from [
118
] states the following:
Lemma 2.4
([
118
]).
Let
2 C
0;Ǜ
,
0<Ǜ<1
, and let u
2 H
1
.
/
. Define
u
.r;
z
/ D
u
.
z
;.1C .
z
//r/; .
z
; r/ 2
F
:
(2.158)
Then u
2 W.0;1I s/
for
0<s<Ǜ
.
Thus, Lemma
2.4
implies that
u
N
.t;:/ 2 W.0;1I s/ for 0<s<1=2.Now,using
the fact W.0;1I s/,! H
s
.
F
/ we get
2
H
s
.
F
/
Ck
u
N
.t;:/k
2
H
1
.
.t
t/
/
;a:a:t2 .0;T/; 0 < s < 1=2:
k
u
N
.t;:/k
By integrating the above inequality w.r.t. t we get the first statement of Lemma
2.3
.
To prove the second statement of Lemma
2.3
we use Theorem 3.1 of [
118
], which
states that the notion of trace for the functions of the form (
2.157
)forwhich
u
N
2
H
1
and
N
2 C
0;1=2
, can be defined in the sense of H
s=2
, 0<s<1=2.For
completeness, we state Theorem 3.2 of [
118
] here.
Theorem 2.2
([
118
]).
Let
Ǜ<1
and let
be such that
2 C
0;Ǜ
.0;1/; .
z
/
min
> 1;
z
2 Œ0;1; .0/ D .1/ D 1:
Then, the trace operator
W C
1
.
/ ! C./
2 C
1
.
/
its “Lagrangian trace” u
.
z
;1C
that associates with each function u
.
z
// 2 C./
, defined via
(
2.158
)
for
r D 1
,
W
u
7!
u
.
z
;1C .
z
//;
can be extended by continuity to a linear operator from
H
1
.
/
to
H
s
./
for
0
s<Ǜ=2
.
By recalling that v
N
D .
u
N
/
j
, this proves the second statement of Lemma
2.3
.
t
Notice that the difficulty associated with bounding the gradient of
u
N
is
somewhat artificial, since the gradient of the fluid velocity
u
N
defined on the
physical domain is, in fact, uniformly bounded (by Proposition
2.6
). Namely, the
difficulty is imposed by the fact that we decided to work with the problem defined
on a fixed domain
F
, and not on the family of moving domains. This decision,
however, simplifies other parts of the main existence proof. The “expense” that we
had to pay for this decision is embedded in the proof of Lemma
2.3
.
We are now ready to use Theorem
2.1
to prove compactness of the sequences
.v
N
/
N
2N
and .
u
N
/
N
2N
.
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