Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 24

the regularization parameter. It is here a key step in the proofs of existence of
weak solutions. One may try to apply the Aubin's lemma to prove the desired
compactness. Nevertheless it is not straightforward to apply this lemma in the
case of divergence free functions defined on moving time-dependent domains. We
can refer to [ 74 , 145 ] for incompressible Navier-Stokes in moving domains or to
[ 39 , 40 , 146 ] for existence weak solutions of fluid solid coupled problem.
One option to obtain compactness is to study quantities like: k u .t Ch/ u .t/k L tx
and k@ t .t C h/ @ t .t/k L tx . Indeed, we are going to use the following lemma
that characterizes the compact sets of L p .0;T I X/ where X is a Banach space
(see [ 150 ]).
Lemma 1.1. Let X be a Banach space and F,! L q .0;T I X/ with 1 q<1 .
Then F is a relatively compact set of L q .0;T I X/ if and only if
i) n R t 2

t 1 f.t/dt;f 2 F o is relatively compact in X , 80<t 1 <t 2 <T

ii) kf.tC h/ f.t/k L q .0;T I X/ ! 0 as h goes to zero, uniformly with respect to f

in F .

We will now apply Lemma 1.1 to the sequence F D . u ;@ t /, indexed by the
regularization parameter, q D 2 and X D L 2 .B/ L 2 .0;L/. Note that here we
have introduced a set B that contains all the fluid domains .t/ for any t 2 .0;T/
and we have extended the fluid velocity by defining

( u in .t/
.0;@ t / T in B n .t/:

u D (1.54)

Remark 1.5. This extension relies strongly on the fact that there is only t he
transverse motion of the elastic structure. Note moreover that, for this reason, u
is divergence free and that if @ t is in H 1 in space (which is the case since LJ 2 >0)
this extension is also in H 1 .B/ in space.

The first point i) of Lemma 1.1 is clearly satisfied thanks to energy estimates and
we have to verify the second point. Given any h>0, we denote g .t; / D g.th; /
and g C .t; / D g.t C h; /. The assertion ii/ is a consequence of the following
lemma:

Lemma 1.2. Let T>0 such that min Œ0;T Œ0;L .1C/ Ǜ>0 . Then for all h>0
small enough, we have
Z T

Z Z T Z L

C p h; j u . u / j

2

.@ t @ t / 2 C (1.55)

0 B 0 0

and

Z T Z

C p h;

B j u . u / j

2

(1.56)

0