## Abstract

We prove that the Stokes operator with Robin boundary conditions
is the generator of a bounded holomorphic semigroup on
L^\infty_\sigma({\mathbb R}^n_+), which is even strongly continuous
on the space \BUC_\sigma({\mathbb R}^n_+). Contrary
to that result it is also proved that there exists no Stokes
semigroup on L^1_\sigma({\mathbb R}^n_+), except if we assume the
special case of Neumann boundary conditions. Nevertheless, we also
obtain gradient estimates for the solution of the Stokes equations
in L^1_\sigma({\mathbb R}^n_+) for all types of Robin boundary
conditions.