Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

We consider a three–dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with an elastic plate located on one part of the fluid boundary. We do not neglect the deformation of the fluid domain which consequently depends on the displacement of the structure. The purpose of this work is to study the solutions of this unsteady fluid–structure interaction problem, as the coefficient modeling the viscoelasticity (resp. the rotatory inertia) of the plate tends to zero. As a consequence, we obtain the existence of at least one weak solution for the limit problem (Navier–Stokes equation coupled with a plate in flexion) as long as the structure does not touch the bottom of the fluid cavity.