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

This paper is devoted to the study of the Stokes and Navier-Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely $L_{uloc,\sigma }^q({\mathbb{R}}_{+}^d)$. We prove the analyticity of the Stokes semigroup $e^{-t\mathbf{A}}$ in $L_{uloc,\sigma }^q({\mathbb{R}}_{+}^d)$ for $1<q\le \mathrm{\infty }$. This follows from the analysis of the Stokes resolvent problem for data in $L_{uloc,\sigma }^q({\mathbb{R}}_{+}^d)$, $1<q\le \mathrm{\infty }$. We then prove bilinear estimates for the Oseen kernel, which enables to prove the existence of mild solutions. The three main original aspects of our contribution are: (i) the proof of Liouville theorems for the resolvent problem and the time dependent Stokes system under weak integrability conditions, (ii) the proof of pressure estimates in the half-space and (iii) the proof of a concentration result for blow-up solutions of the Navier-Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.