  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 $L^q_{uloc,\sigma}(\R^d_+)$. We prove the
analyticity of the Stokes semigroup $e^{-t{\bf A}}$ in
$L^q_{uloc,\sigma}(\R^d_+)$ for $1<q\leq\infty$. This follows from the analysis
of the Stokes resolvent problem for data in $L^q_{uloc,\sigma}(\R^d_+)$,
$1<q\leq\infty$. 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.
