Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 159

Now because of the uniform convergence ( 2.169 ) of the sequence . t N / N 2N ,
and the uniform boundedness of .k@ r u N k L 2 . F / / N 2N , which is consequence of
Proposition 2.6 , we can take N 1 N 0 such that
Z T

Z

2 < "

.t/ \ t N .t/ j u N

N u N j

2 ;N N 1 :

0

This inequality, together with ( 2.171 ) proofs that N u N ! u strongly in

L 2 ..0;T/ max /.
STEP 2. We will now show that N r u N *r u weakly in L 2 ..0;T/ max /:

First notice that from

r u N

t N u N

Dr

t N u N / N 2N in L 2 ..0;T/ F /, established
in Proposition 2.6 , we get that the sequence . N

and from uniform boundedness of .r

r u N / N 2N converges weakly in
L 2 ..0;T/ max /. Let us denote the weak limit of . N

G . Therefore,

r u N / N 2N by

Z T Z Z T Z

G D

N r u N ; 2 C c ..0;T/ max /: lim

N !1

0 max 0 max

We want to show that G D r u .
For this purpose, we first consider the set . max n .t// and show that

G

D 0

there, and then the set .t/ and show that G Dr u there.
Let be a test function such that supp .0;T/

max n .t/ .Usingthe
uniform convergence of the sequence t N , obtained in ( 2.169 ), there exists an N
such that N . x / D 0, N N , x 2 supp. Therefore, we have
Z T Z Z T Z

G D

N r u N lim

N !1

D 0:

0 max 0 max

max n .t/ .
Now, let us take a test function such that supp .0;T/ .t/. Again using
the same argument as before, as well as the uniform convergence of the sequence
t N , obtained in ( 2.169 ), we conclude that there exists an N such that N . x / D
1, N N , x 2 supp . Therefore, we have
Z T

G D 0 on .0;T/

Thus,

Z Z T Z Z T Z

G D

N r u N

.t/ r u N

lim
N !1

D lim

N !1

:

0 max 0 max 0

From the strong convergence N u N ! u obtained in STEP 1, we have that on the

set supp , u N ! u in the sense of distributions, and so, on the same set supp ,