  In this paper, we establish the global existence of small solutions to the
inhomogeneous Navier-Stokes system in the half-space. The initial density only
has to be bounded and close enough to a positive constant, and the initial
velocity belongs to some critical Besov space. With a little bit more
regularity for the initial velocity, those solutions are proved to be unique.
In the last section of the paper, our results are partially extended to the
bounded domain case.
