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 R³+. Such solutions are sometimes called Lemarié–Rieusset solutions in the whole space R³. 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 L³(R³+) norm obtained by Barker and Seregin for solutions developing a singularity in finite time.