The transition problem between time-independent motions of a body in a viscous liquid

A body $\mathcal{B}$ moves in an unbounded Navier-Stokes liquid by time-independent translatory motion. Suppose that at time $t=0$, $\mathcal{B}$ smoothly changes its motion to an arbitrary rigid motion, reached at time $t=1$. We then show that the associated Navier-Stokes problem has a unique solution connecting the two steady-states generated by the motion of $\mathcal{B}$, provided all the involved velocities of $\mathcal{B}$ are sufficiently small.