Local Energy Weak Solutions for the Navier–Stokes Equations in the Half-Space

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.