Biomedical Engineering Reference
In-Depth Information
Our aim is to apply the Schauder fixed point theorem and prove that the mapping
F
has at least one fixed point. This implies the existence of the weak solution to our
fully coupled problem. It is necessary to check the following properties:
•
F
.B
Ǜ;K
/ B
Ǜ;K
.R
0
C ı
.k/
//
k
is relatively compact in Y , for any sequence in .ı
.k/
/
k
2 B
Ǜ;K
(We show the equicontinuity in time, Lemma
1.4
, which implies, together with
the energy estimate, the strong convergence in Y .)
•
.
F
•
The continuity of the mapping
F
with respect to the strong topology in Y must
be proven.
Relative Compactness of the Fixed Point Mapping
F
We focus on the integral equicontinuity in time
and the relative compactness in Y:
Lemma
1.4
provides the equicontinuity result that holds independently on k.The
lemma is the analogue of the two Lemmas
1.2
and
1.3
and writes (with the same
notations as in the previous subsections, see (
1.54
))
Lemma 1.4.
For the weak solution
.
u
.k/
;@
t
.k/
/ D .
u
.k/
;
.k/
/
of the problem
associated with a given
ı
.k/
2 B
Ǜ;K
it holds
Z
T
Z
Z
T
Z
L
0
j@
t
.k/
.t C / @
t
.k/
.t/j
.k/
t
j
u
.k/
.t C /
u
.k/
.t/j
2
2
C
0
B
0
C.
1=q
C
1=2
/:
(1.110)
Here
.k
t
denotes the characteristic function of
h
.k/
.t/
. The constant
C D C.K;Ǜ/
does not depend on
k
.
From (
1.110
) we can get that
Z
T
Z
Z
T
Z
L
0
j@
t
.k/
.t C / @
t
.k/
.t/j
j
.k/
t
C
u
.k/
.t C /
.k/
u
.k/
.t/j
2
2
C
t
0
0
C.
1=q
C
1=2
/:
(1.111)
It implies that
.k
t
u
.k/
.t/, and thus
u
.k/
.t/ is relatively compact in L
2
..0;T/ B/.
Consequently, the Riesz-Fréchet-Kolmogorov compactness argument [
20
,The-
orem IV.26] (or see Lemma
1.1
) based on (
1.111
) implies the relative compactness
of @
t
.k/
;
u
.k/
in L
2
.0;T I L
2
.0;L//; L
2
.0;T I L
2
.B//; respectively. Additionally,
the standard interpolations give us the compactness of
u
.k/
in L
r
..0;T/ B/; 1
r<4and @
t
.k/
in L
s
..0;T/ .0;L//; 1 s<6.
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