Global existence and convergence to pressure waves in nonlinear fluid-structure interaction

We consider a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid, without damping on the elastic part. We prove local existence of strong solutions and global existence and uniqueness for small data. At the same time, depending on the geometric setting, non-trivial time-periodic solutions, called pressure waves, may persist. Our main result is the characterization of long-time behaviour of the elastic displacement: up to small rigid motions, either the system comes to rest or converges to a pressure wave.