Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 149

Theorem 2.3. Sequences .v N / N 2N and . u N / N 2N are relatively compact in

L 2 .0;T I L 2 .0;1// and L 2 .0;T I L 2 . F // , respectively.
Proof. We use Theorem 2.1 with q D 2,andX D L 2 . We verify that both
assumptions (i) and (ii) hold.
Assumption (i): To show that the sequences .v N / N 2N and . u N / N 2N are rel-
atively compact in L 2 .0;1/ and L 2 . F /, respectively, we use Lemma 2.3 and
the compactness of the embeddings H s . F /! L 2 . F / and H s=2 .0;1/ ,!
L 2 .0;1/, respectively, for 0<s<1=2. Namely, from Lemma 2.3 we know
that sequences . u N / N 2N and .v N / N 2N are uniformly bounded in L 2 .0;T I H s . F //
and L 2 .0;T I H s=2 .0;1//, respectively, for 0<s<1=2. The compactness of the
embeddings H s . F /,! L 2 . F / and H s=2 .0;1/ ,! L 2 .0;1/ verify Assumption
(i)ofTheorem 2.1 .
Assumption (ii): We prove that the “integral equicontinuity,” stated in assumption
(ii) of Theorem 2.1 , holds for the sequence .v N / N 2N . Analogous reasoning can be
used for . u N / N 2N . Thus, we want to show that for each ">0, there exists a ı>0
such that

2

k h v N v N k

L 2 .! I L 2 .0;1// <";

8jhj <ı;independently of N 2 N ; (2.159)

where ! is an arbitrary compact subset of . Indeed, we will show that for each
">0, the following choice of ı:

ı WD minfdist.!;@/=2;"=.2C/g

provides the desired estimate, where C is the constant from Lemma 2.1 (indepen-
dent of N).
Let h be an arbitrary real number whose absolute value is less than ı.Wewantto
show that ( 2.159 ) holds for all t D T=N. This will be shown in two steps. First,
we will show that ( 2.159 ) holds for the case when t h (Case 1), and then for the
case when t < h (Case 2).
A short remark is in order: For a given ı>0, we will have t < ı for infinitely
many N, and both cases will apply. For a finite number of functions .v N /, we will,
however, have that t ı. For those functions ( 2.159 ) needs to be proved for all
t such that jhj <ı t, which falls into Case 1 below. Thus, Cases 1 and 2
cover all the possibilities.

Case 1: t h . We calculate the shift by h to obtain (see Fig. 2.11 ,left):

Z jt
jt h kv N v j C 1

N 1

X

2
L 2 .! I L 2 .0;1//

2
L 2 .0;1/

k h v N v N k k

N

j D 1

N 1

X
j D 1 kv N v j C 1

2

D h k

L 2 .0;1/ hC<"=2<":

N