  This paper addresses the three-dimensional Navier-Stokes equations for an
incompressible fluid whose density is permitted to be inhomogeneous. We
establish a theorem of global existence and uniqueness of strong solutions for
initial data with small $\dot{H}^{\frac12}$-norm, which also satisfies a
natural compatibility condition. A key point of the theorem is that the initial
density need not be strictly positive.
