Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 19

formulation is preserved to avoid the added mass effect (only the geometrical part
being decoupled). In a second part, we explain the strategy used in [ 119 , 120 ], the
proof being based on a suitable decoupling of the fluid-structure problem where the
added mass effect is kept implicit.

1.2.2 Existence of Weak Solutions

In this subsection we will consider the case of the full nonlinear coupled problem
but with D 0. Consequently the fluid domain is a subgraph defined by:

2 ;x 2 .0;L/; 0 < y < R C .t;x/g:

f .t/ D .t/ Df.t;x;y/2 R

The displacement will satisfied a wave equation (thus D 0 in ( 1.11 )). In a
first part we will consider an additional viscous term (LJ 2 >0) then we study the
vanishing viscosity limit (LJ 2 ! 0) and we prove that there exists at least one
weak solution of the fluid-membrane coupled problem. Moreover to simplify we
will consider fluid boundary conditions that allow to obtain energy estimates and
consequently impose homogeneous Dirichlet boundary conditions ( 1.4 ). Note one
could also impose ( 1.28 ), ( 1.29 ).

The Damped Wave Equation: ˇ 2 >0

An energy estimate of the same type as ( 1.25 ) shows that one can look for an elastic
displacement in W 1; 1 .0;T I L 2 .0;L// \ H 1 .0;T I H 0 .0;L//. Nevertheless this
regularity implies that the fluid-structure interface is continuous for all time but not
Lipschitz. Consequently the set .t/ is an open set (till RC.t;x/ > 0) which not
Lipschitz. One of the first questions is then to properly define the functional spaces
to which the fluid velocity belongs to and to rigorously define the fluid trace velocity
on the moving interface.

Remark 1.4. In the case where one considers a beam equation (i.e., >0)the
energy estimates give that the elastic displacement belongs to L 1 .0;T I H 0 .0;L//.
Thus the mapping f is a C 1 diffeomorphism as long as R C >0.And
consequently .t/ is Lipschitz.
In our case, we introduce the open set T D[ t 2 .0;T/ .t/ ftg and we define

n v 2 L 2 . T /; rv 2 L 2 . T / o ;

L 2 .0;T I H 1 . .t/// D

. T / L 2 .0;T I H 1 . .t/// ; L 2 .0;T I H 0 . .t/// D D

n v 2 C 1 T ; divv D 0; v D 0 on .0;T/ f o ;

V f D