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The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space R 3 + . Such solutions are sometimes called Lemarié-Rieusset solutions in the whole space R 3 . The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz-Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical L 3 (R 3 + ) norm obtained by Barker and Seregin for solutions developing a singularity in finite time.

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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 ∈ R or t ∈ (0, ∞). 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 timeperiodic 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.