Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈ℝ2 where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the interface to the dynamic system of elasticity. The characteristic feature of this coupled model is that the resolvent is not compact and the energy function characterizing balance of the total energy is weakly degenerated. These combined with the lack of mechanical dissipation and intrinsic nonlinearity of the dynamics render the problem of asymptotic stability rather delicate. Indeed, the only source of dissipation is the viscosity effect propagated from the fluid via interface. It will be shown that under suitable geometric conditions imposed on the geometry of the interface, finite energy function associated with weak solutions converges to zero when the time t converges to infinity. The required geometric conditions result from the presence of the pressure acting upon the solid.