ON A MODEL OF AN ELASTIC BODY FULLY IMMERSED IN A VISCOUS INCOMPRESSIBLE FLUID WITH SMALL DATA

We consider a model of an elastic body immersed between two layers of incompressible viscous fluid. The elastic displacement w is governed by the damped wave equation w(tt) + alpha w(t) + Delta w = 0 without any stabilization terms, where Delta > 0, and the fluid is modeled by the Navier-Stokes equations. We assume continuity of the displacement and the stresses across the moving interfaces and homogeneous Dirichlet boundary conditions on the outer fluid boundaries. We establish a priori estimates that provide the global -in -time well-posedness and exponential decay to a final state of the system for small initial data. We prove that the final state must be trivial, except for a possible small displacement of the elastic structure in the horizontal direction.