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

A body $\mathscr B$ moves in an unbounded Navier-Stokes liquid by time-independent translatory motion. Suppose that at time $t=0$, $\mathscr B$ 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 $\mathscr B$, provided all the involved velocities of $\mathscr B$ are sufficiently small.