  We prove new existence and uniqueness results in full Sobolev spaces for the
steady-state Oseen problem in a smooth exterior domain of $\mathbb{R}^n$, $n\ge
2$. These results are then employed, on the one hand, in the study of analogous
properties for the corresponding (linear) time-periodic case and, on the other
hand and more significantly, to prove analogous properties for their nonlinear
counterpart, at least for small data.
