Local energy weak solutions for the Navier-Stokes equations in the half-space

Abstract

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 Lemarié-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.

59 pages

References in corpus (1)

• Estimates for the Navier-Stokes equations in the half-space for non localized data

Cited by in corpus (1)

• Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions