  In this paper, we shall prove the global existence of weak solutions to 3D
inhomogeneous incompressible Navier-Stokes system $({\rm INS})$ with initial
density in the bounded function space and having a positive lower bound and
with initial velocity being sufficiently small in the critical Besov space,
$\dot B^{\f12}_{2,1}.$ This result corresponds to the Fujita-Kato solutions of
the classical Navier-Stokes system. The same idea can be used to prove the
global existence of weak solutions in the \emph{critical functional framework}
to $({\rm INS})$ with one component of the initial velocity being large and can
also be applied to provide a lower bound for the lifespan of smooth enough
solutions of $({\rm INS}).$
