  We first prove decay estimates and spacetime integral bounds for Stokes flows
in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the
spaces of uniformly locally $r$-integrable functions. Using these estimates, we
construct mild solutions of the Navier-Stokes equations in the amalgam spaces
satisfying the corresponding spacetime integral bounds. Time-global solutions
are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide
new bounds for the strong solutions classically constructed by Kato and the
more recent solutions in uniformly local spaces constructed by Maekawa and
Terasawa. As an application we obtain a result on the stability of suitability
for weak solutions to the perturbed Navier-Stokes equation where the drift
velocity solves the Navier-Stokes equations and has small data in a local $L^3$
class. Extending an earlier result, we also construct global-in-time local
energy weak solutions in $E^2_q$, $1\le q <2$.
