Biomedical Engineering Reference
In-Depth Information
Continuity of the Mapping
F
We can show by limiting process for k !1in the weak formulation satisfied by
.
u
.k/
;
.k/
/ associated with h
.k/
that
.h
.k/
/ !
F
F
.h/
u
.k/
;
.k/
/ converges
whenever h
.k/
converges to h in B
Ǜ;K
. As already shown above .
O
! in H
1
.0;T I L
2
.0;L//
as k !1. Due to the boundedness of from a priori estimate (
1.109
)and
the imbeddings in one dimension we have even stronger result—the uniform
convergence of @
x
.k/
in C.Œ0;T Œ0;L/. Indeed,
u
;/ in Y , i.e., we have
.k/
strongly to some .
O
.k/
2 L
1
.0;T I H
2
.0;L// \ W
1;
1
.0;T I L
2
.0;L//
(1.112)
,! C
0;1
LJ
.0;T I H
2LJ
.0;L//
for 0<LJ<1. From the continuous imbedding of H
2LJ
.0;L/ into H
2LJ
.0;L/
and the Arzelá-Ascoli Lemma we conclude that a subsequence of
.k/
converges
strongly in C.Œ0;TI H
s
.0;L//; 0 < s < 2.Sincefors>3=2we also have
continuous imbedding H
s
.0;L/ ,! C
1
Œ0;L, we can conclude, that
.k/
!
strongly in C.0;TI C
1
Œ0;L/.
Let us summarize available convergences
u
.k/
*
u
weakly in L
q
.0;T I W
1;q
.D//;
O
u
.k/
O
strongly in L
r
..0;T/ B/; 1 r<4;
!
N
u
u
.k/
u
strongly in L
r
..0;T/ D/; 1 r<4;
.k/
* weakly in H
1
.0;T I H
2
.0;L//;
O
!
O
(1.113)
.k/
*
weakly* in L
1
.0;T I L
2
.0;L//;
.k/
! uniformly in C.0;TI C
1
Œ0;L/;
@
t
.k/
! @
t
strongly in L
s
..0;T/ .0;L//; 1 s<6:
Limiting Process
Now we let k !1in the weak formulation satisfied by .
u
.k/
;
.k/
/. The previous
convergences allow to pass to the limit. As already mentioned the main difficulty at
this step is to deal with test functions that depend on the solution. Nevertheless as
mentioned in Sect.
1.2.2
, one can choose test functions that do not depend on k.Let
us mention two important limits for k !1:
Z
T
b
h
.k/
.
u
; /
u
.k/
;
u
.k/
; / b
h
.
O
O
u
;
O
O
! 0;
0
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