  We are concerned with the existence and uniqueness issue for the
inhomogeneous incompressible Navier-Stokes equations supplemented with H^1
initial velocity and only bounded nonnegative density. In contrast with all the
previous works on that topics, we do not require regularity or positive lower
bound for the initial density, or compatibility conditions for the initial
velocity, and still obtain unique solutions. Those solutions are global in the
two-dimensional case for general data, and in the three-dimensional case if the
velocity satisfies a suitable scaling invariant smallness condition. As a
straightforward application, we provide a complete answer to Lions' question in
[25], page 34, concerning the evolution of a drop of incompressible viscous
fluid in the vacuum.
