Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 45

order to prove that the Galerkin approximation converges. Due to the additional
compressibility we are able to prove the following a priori estimates: the fluid veloc-
ity is bounded in L 1 .0;T I L 2 .D// \ L q .0;T I W 1;q .D// and its time derivative
is bounded in L q 0 .0;T I .W 1;q .D// 0 /, whereas the structure velocity is bounded in
L 1 .0;T I L 2 .0;L// \ L 2 ..0;T/I H 2 .0;L//, and its time derivative is bounded in
L 2 ..0;T/I L 2 .0;L//. Note that the latter estimate arises due to the -approximation
that implies that the right hand side of the structure equation can be represented in
terms of velocities and not of the fluid stress.
Now applying the Lions-Aubin lemma the compactness results and the corre-
sponding strong convergence in L q ..0;T/ D/ for u and in L 2 .0;T I H 1 .0;L//

for are obtained. The limiting process in the nonlinear viscous term is realized by
applying the Minty-trick and theory of monotone operators.
The weak solution from Theorem 1.4 depends on the parameters "; and on
the given geometry h. Keeping geometry fixed but passing to the limit with " !
0; !1we can obtain the weak solution of the original problem ( 1.73 )-( 1.89 )
defined on ı . Note that now when passing to the limit with and we cannot
apply the Lions-Aubin lemma anymore and to get the strong convergence for

O

u and
the equicontinuity in time is applied (see Lemma 1.4 ). Again the limiting process
in the nonlinear viscous term is realized by applying the Minty-trick and theory of
monotone operators.

O

Fixed Point Iterations

We have proved the existence of weak solution of the original problem in a domain
given by a known deformation function h D R 0 Cı . The aim of this subsection is to
prove that the mapping that associates with h (or equivalently to ı) has at least one
fixed point. To prove it we apply the Schauder fixed point theorem. The compactness
argument will be based on the equicontinuity in time, see Lemma 1.4 . Consequently
we obtain the final result: existence of weak solution for a fully coupled fluid-
structure interaction problem ( 1.73 )-( 1.89 ).
We denote the space Y D H 1 .0;T I L 2 .0;L//. Let us assume that ı belongs to
the ball B Ǜ;K defined by

n ı 2 Y Ikık Y C Ǜ;K ;0<Ǜ R 0 .x/ C ı.t; x/ Ǜ ;
LJ LJ LJ

B Ǜ;K D

@ x ı.t; x/ LJ LJ LJ K; .x;0/ D 0; 8x 2 Œ0;L; 8t 2 Œ0;T;
Z T
0 j@ t .t; x/j

K; 8x 2 Œ0;L o ;

2

where C Ǜ;K is a suitable constant large enough with respect to K;Ǜ and the data.
Note that here we ask the domain interface to be at least W 1; 1 in space so that
all the nonlinear geometric terms of the formulation coming from the change of
variables are well defined.