  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 $\mathbb
R^3_+$. Such solutions are sometimes called Lemari\'e-Rieusset solutions in the
whole space $\mathbb R^3$. 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
$L^3(\mathbb R^3_+)$ norm obtained by Barker and Seregin for solutions
developing a singularity in finite time.
