Keywords: Asymptotics, Exterior flows, Navier-Stokes equations, self-similar Abstract: 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 all t ∈ R or t > 0. In the case t > 0 it is coupled with a small initial data in weak L3. If the boundary data is time-periodic, the spatial asymptotics of the time-entire solution is given by a Landau solution which is the same for all time. If the boundary data is time-periodic and the initial data is asymptotically discretely self-similar, 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. This is a joint work with Kyungkuen Kang and Hideyuki Miura.