  Consider the motion of a viscous incompressible fluid in a 3D exterior domain
when a rigid body moves with prescribed time-dependent translational and
angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$
smoothing action near the initial time as well as generation of the evolution
operator was shown by Hansel and Rhandi (J. Reine Angew. Math. 2014) under
reasonable conditions. In this paper we develop the $L^q$-$L^r$ decay estimates
of the evolution operator and then apply them to the Navier-Stokes initial
value problem.
