New results for the Oseen problem with applications to the Navier-Stokes equations in exterior domains

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.