  We prove the unique existence of solutions of the 3D incompressible
Navier-Stokes equations in an exterior domain with small non-decaying boundary
data, for $t \in R$ or $t \in (0,\infty)$. In the latter case it is coupled
with small initial data in weak $L^{3}$. As a corollary, the unique existence
of time-periodic solutions is shown for the small periodic boundary data. We
next show that the spatial asymptotics of the periodic solution is given by the
same Landau solution at all times. Lastly we show that if the boundary datum is
time-periodic and the initial datum is asymptotically discretely self-similar,
then the solution is asymptotically the sum of a time-periodic vector field and
a forward discretely self-similar vector field as time goes to infinity. It in
particular shows the stability of periodic solutions in a local sense.
