Biomedical Engineering Reference
In-Depth Information
2.6.4
Weak Solutions
Notation and Function Spaces
Notation.
To define weak solutions of the moving-boundary problem (
2.91
)-
(
2.102
) and of the moving-boundary problem (
2.111
)-(
2.118
)definedonafixed
domain, the following notation will be useful:
•
a
S
will denote the bilinear form associated with the elastic energy of the thick
structure:
Z
2
D
.
d
/ W
D
. / C .r
d
/ .r /
:
a
S
.
d
; / D
(2.122)
S
Here “W” denotes the scalar product defined in (
2.7
).
•
b will denote the following trilinear form corresponding to the (symmetrized)
nonlinear advection term in the Navier-Stokes equations:
Z
Z
1
2
1
2
b.t;
u
;
v
;
w
/ D
.
u
r/
v
w
.
u
r/
w
v
:
(2.123)
F
.t/
F
.t/
•
The linear functional which associates the inlet and outlet dynamic pressure
boundary data with a test function
v
will be denoted by:
D P
in
.t/
Z
v
z
P
out
.t/
Z
hF.t/;
v
i
in=out
v
z
:
in
out
Function Spaces.
For the fluid velocity we would like to work with the classical
function space associated with weak solutions of the Navier-Stokes equations. This,
however, requires some additional consideration. Namely, since our thin structure
is governed by the linear wave equation, lacking the bending rigidity terms, weak
solutions cannot be expected to be Lipschitz-continuous. Indeed, from the energy
inequality (
2.103
) we only have 2 H
1
.0;1/, and from Sobolev embedding we get
that 2 C
0;1=2
.0;1/, which means that
F
.t/ is not necessarily a Lipshitz domain.
However,
F
.t/ is locally a sub-graph of a Hölder continuous function. In that case
one can define “Lagrangian” trace
.t/
W C
1
.
F
.t// ! C./;
.t/
W v 7! v.t;
z
;rC .t;
z
//:
(2.124)
Furthermore, it was shown in [
32
,
79
,
118
] that the trace operator
.t/
can be
extended by continuity to a linear operator from H
1
.
F
.t// to H
s
./, 0 s<
4
.
For a precise statement of the results about “Lagrangian” trace, see Theorem
2.2
.
Now, the velocity solution space can be defined in the following way:
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