  We consider a non-linear system modelling the dynamics of a linearly elastic
body immersed in an incompressible viscous fluid. We prove local existence of
strong solutions and global existence and uniqueness for small data. The main
result is the characterization of long-time behaviour of the elastic
displacement. We show convergence either to a rest state or rigid motion, or to
a time-periodic pressure wave that may occur only in specific geometric
settings.
