Attainability of Time-Periodic flow of a Viscous Liquid Past an Oscillating Body

A body $\mathcal{B}$ is started from rest by translational motion in an otherwise quiescent Navier-Stokes liquid filling the whole space. We show, for small data, that if after some time $\mathcal{B}$ reaches a spinless oscillatory motion of period $\mathcal{T}$, the liquid will eventually execute also a time periodic motion with same period $\mathcal{T}$. This problem is a suitable generalization of the famous Finn’s starting problem for steady-states, to the case of time-periodic motions