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
.
We prove the analyticity of the Stokes semigroup
in
for
.
This follows from the analysis of the Stokes resolvent problem for data in
,
.
We then prove bilinear estimates for the Oseen kernel, which enables us to
prove the existence of mild solutions. The three main original aspects of our
contribution are: the proof of Liouville theorems for the resolvent problem and the
time-dependent Stokes system under weak integrability conditions, the proof of
pressure estimates in the half-space, and 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.