Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 28

T k C 1 T k k =2 and goes to zero, which is a contradiction. This achieves the
proof of existence of a weak solution as long as no contact occurs between the
elastic structure and the bottom of the fluid cavity.

Undamped Wave Equation: ˇ 2 ! 0

We explain in this subsection how one can pass to the limit in the coupled problem as
the structure viscosity LJ 2 goes to zero. As we will see, the fluid dissipation enables
to control the space of high frequencies of the structure velocity without any added
viscosity on the wave equation.
The elastic displacement in the case LJ 2 D 0 has only a hyperbolic regularity
and belongs to L 1 .0;T I H 0 .0;L// \ W 1; 1 .0;T I L 2 .0;L//. We easily verify that
these regularities are sufficient to define all the functional spaces and to give a sense
to the trace of the fluid velocity at the interface: u .t;x;RC.t;x// on .0;L/.They
are also sufficient to prove that the existence time of the weak solutions obtained
for LJ 2 >0is bounded from below independently of LJ 2 . Indeed is uniformly
continuous in space and time independently of LJ 2 . Consequently this ensures that
there exists a time T>0such that R C stays away from zero.
Yet the uniform energy estimates are not sufficient to obtain the compactness in
L 2 .0;T I L 2 .0;L// of the structure velocity. As noted at Remark 1.6 , in the previous
subsection this compactness relied on the fact that @ t 2 L 2 .0;T I H 1 .0;L// thanks
to the parabolic regularization of the elastic equation. Here one cannot hope to
obtain ( 1.55 )or( 1.56 ) and in particular we cannot hope to have the convergence

rate p h. But in order to obtain a uniform decay, as h goes to zero, it is sufficient
to have some space regularity of @ t . The idea is then to take advantage of the
kinematic condition .0;@ t .t;x// T D u .t;x;RC .t;x// on .0;L/, and of the fact

that the fluid is viscous.
Indeed, if had Lipschitzian regularity, since u 2 L 2 .0;T I H 1 . .t//, one
would deduce that @ t 2 L 2 .0;T I H 1=2 .0;L// as the trace of u and the space
of high frequencies of the structure velocity would then be controlled. Here these
regularities are not satisfied by any weak solution (unless >0). But thanks to
Sobolev injections 2 C 0 .Œ0;TŒ0;L/\L 1 .0;T I H 0 .0;L// and this regularity
of the fluid-structure interface enables to obtain that @ t 2 L 2 .0;T I H s .0;L// for
any s < 1=4.
The question is now: is this dissipation sufficient to derive the same kind of result
enunciate at Lemma 1.2 ? The answer is yes and can be summarized by the following
lemma:

Lemma 1.3. Let T>0 such that min Œ0;T .0;L/ .1 C / Ǜ>0 . We have 8">
0; 9h 0 >0; s. t. 8LJ 2 >0;8h h 0
Z T

Z Z T Z L

j u . u / j

2

.@ t @ t / 2 C " (1.60)

0 B 0 0