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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3_+}$$\end{document}. Such solutions are sometimes called Lemarié–Rieusset solutions in the whole space R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document}. 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(R+3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^3(\mathbb{R}^3_+)}$$\end{document} norm obtained by Barker and Seregin for solutions developing a singularity in finite time.