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We prove new existence and uniqueness results in full Sobolev spaces for thesteady-state Oseen problem in a smooth exterior domain of $\mathbb{R}^n$, $n\ge2$. These results are then employed, on the one hand, in the study of analogousproperties for the corresponding (linear) time-periodic case and, on the otherhand and more significantly, to prove analogous properties for their nonlinearcounterpart, at least for small data.

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