Biomedical Engineering Reference
In-Depth Information
Z
T
Z
1
0
p
out
hj
x
D
L
1
j
x
D
L
p
in
hj
x
D
0
1
j
x
D
0
C
0
Z
T
Z
L
1
2
b
u
2
/
2
j
y
D
1
C
u
2
@
t
h C .
b
0
0
C"
Z
T
0
h@
t
.h p/;i
(1.107)
Z
T
Z
"@
t
h@
y
. y p/ C "a
h
. p;/ C h div
h
O
u
C
0
D
Z
T
Z
L
"
2
C
@
t
h pj
y
D
1
0
0
@
t
C @
x x
@
x x
C @
x
Z
t
0
.s; x/ds
@
x
Z
T
Z
L
C
0
0
Z
t
.s; x/ds
@
x x
R
0
C .
b
u
2
/
C
0
for every . ;;/2 H
0
.0;T IV/ L
2
.0;T I H
1
.D// L
2
.0;T I H
0
.0;L//.
Theorem 1.4
(Existence of the Approximated Linearized Weak Solution).
Let
"; ; ı
be fixed. Assume (
1.75
)-(
1.77
), (
1.92
),
p
in
; p
out
2 L
q
0
.0;T I L
2
.0;1//:
Then there exists a weak solution of the
.;";ı/
-approximated problem transformed
to the fixed domain, in the sense of integral identity (
1.107
). Moreover,
8
<
L
q
0
.0;T IV
/
for
2<p<1;
L
q
0
.0;T IV
/ ˚ L
4=3
..0;T/ D/;
for
p D 2;
@
t
.h p/ 2 L
2
.0;T I H
1
.D//;
@
t
.h
u
/ 2
O
:
such that
Z
T
Z
T
Z
D
@
t
.h
u
/;
E
D
O
O
h
u
@
t
:
0
0
D
Solution of this approximated linearized problem has been obtained in the recent
work [
111
] by means of the standard energy method using the Galerkin approximate
solutions and showing the a priori estimates. To guarantee the coercivity of the
nonlinear viscous term we need to apply the generalized Korn inequality with
variable coefficients
R
D
j
D
c
.K;Ǜ/
R
D
jr
u
j
p
. This requires that at least
h 2 L
1
.0;T I W
1;
1
.0;L//,cf.[
111
] for more details. We would like to point
out that this is a crucial step forcing us to require also higher order friction term
in modeling the string equation.
Since the problem for a given ı,andfixed", is still nonlinear because of the
convection and nonlinear viscous terms, we have to obtain some compactness in
p
.
u
/j
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