Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space

Global existence results for weak solutions of the so-called incompressible density-dependent Navier-Stokes equations are proven in the whole space $\mathbb{R}^N$ $(N \geq 2)$. In this note, the initial density is not required to be bounded from below by a positive constant and the viscosity may be a function of the density.