  We are concerned with the Cauchy problem of the two-dimensional (2D)
nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field
density. It is proved that if the initial density decays not too slow at
infinity, the 2D Cauchy problem of the density-dependent Navier-Stokes
equations on the whole space $\mathbb{R}^2$ admits a unique global strong
solution. Note that the initial data can be arbitrarily large and the initial
density can contain vacuum states and even have compact support. Furthermore,
we also obtain the large time decay rates of the spatial gradients of the
velocity and the pressure which are the same as those of the homogeneous case.
