  In this paper, we analyse the long-time behavior of solutions to a coupled
system describing the motion of a rigid disk in a 2D viscous incompressible
fluid. Following previous approaches, we look at the problem in the system of
coordinates associated with the center of mass of the disk. Doing so, we
introduce a further nonlinearity to the classical Navier Stokes equations. In
comparison with the classical nonlinearities, this new term lacks time and
space integrability, thus complicating strongly the analysis of the long-time
behavior of solutions. We provide herein two refined tools : a refined analysis
of the Gagliardo-Nirenberg inequalities and a thorough description of
fractional powers of the so-called fluid-structure operator. On the basis of
these two tools we extend previous decay estimates to arbitrary initial data
and show local stability of the Lamb-Oseen vortex.
