Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 15

parabolic-parabolic or a parabolic-ODE coupling that enables to consider strong
solutions as well as weak solutions.
Concerning three-dimensional elastic structure very few results are available.
One can refer to [ 76 , 87 , 152 ] in the steady state case and [ 34 , 35 , 114 ]forthe
full unsteady case. In the later works the authors consider the existence of strong
solutions for small enough data locally in time. Note that unrealistic compatibility
conditions are required for these existence results and that some drawbacks are
known to exist in the proof of [ 34 ] (they surely may be overcome but to the price of
tedious adaptations).
To be complete, let us mention that strong existence of the motion of elastic
bodies in a viscous compressible case was recently investigated in [ 18 ].
Concerning the fluid-beam or fluid-plate coupled system one considers in this
chapter, the 2D/1D steady state case has been considered in [ 86 ] for homogeneous
Dirichlet boundary conditions on the boundary f that is not the fluid-structure
interface. Existence of a unique regular enough solution is obtained for small enough
applied forces. To our knowledge this is the only work where both the transverse and
the longitudinal displacement are considered. In the unsteady framework the only
studied case, so far, is the case D 0. One may refer to [ 26 ] where a 3D/2D coupling
is studied and where the structure is a damped plate in flexion (i.e., D 0, >0)
and to [ 88 ] in the case of a plate in flexion (i.e., D 0, >0and LJ 2 D 0). The
previous results deal with the existence of weak solutions, i.e., in the energy spaces.
Note that in these problems the displacement of the structure is only in H 2 in space
which is not sufficient to imply a Lipschitz regularity of the fluid domain f .t/.
Nevertheless f .t/ is at least C 0 and taking the advantage of the only transverse
motion of the elastic interface one can prove that at least one weak solution exists.
These results also apply in the case of a 2D/1D coupled problem with D 0 and
D LJ 2 D 0 as we will see in Sect. 1.2.2 . More recently [ 129 ] have considered
the 2D/1D coupling with D 0, >0and LJ 2 0 (note that in this case the
fluid domain is Lipschitz) and involving also Neumann type boundary conditions
( 1.28 ), ( 1.29 ). The authors have given an alternative proof of existence of a weak
solutions based on ideas coming from numerical schemes introduced in [ 98 ]and
further developed in [ 56 , 57 , 62 , 110 , 121 ]. The proof is then based on a numerical
scheme where a splitting strategy is used for the structure part, emphasizing the
link between stable numerical schemes and strategies to prove existence of solution.
We would like to point out yet another theoretical results concerning the existence
of weak solution of the fluid-structure interaction problem based on numerical
techniques. In [ 111 ] the interaction of a non-Newtonian fluid with the viscoelastic
membrane was studied, for inflow/outflow boundary conditions ( 1.28 ), ( 1.29 )are
applied. The proof is based analogously as the numerical scheme described in [ 109 ]
on the artificial compressibility approach as well as on the global iterations with
respect to the moving domain, cf. also [ 110 ]. Such a link to numerics is also present
in [ 119 , 120 ] where the existence of strong solutions for 3D/2D, or 2D/1D coupled
problem involving a damped elastic structure is studied (see also [ 10 ] for 2D/1D
strong solutions). The proofs of [ 119 , 120 ] are based on a splitting strategy for the
Stokes system and on an implicit treatment of the added mass effect. Also we would