  In this paper, we prove the global well-posedness of the classical solution
to the 2D Cauchy problem of the compressible Navier-Stokes equations with
arbitrarily large initial data when the shear viscosity $\mu$ is a positive
constant and the bulk viscosity $\l(\r)=\r^\b$ with $\b>\frac43$. Here the
initial density keeps a non-vacuum states $\bar\rho>0$ at far fields and our
results generalize the ones by Vaigant-Kazhikhov [41] for the periodic problem
and by Jiu-Wang-Xin [26] and Huang-Li [8] for the Cauchy problem with vacuum
states $\bar\rho=0$ at far fields. It shows that the solution will not develop
the vacuum states in any finite time provided the initial density is uniformly
away from vacuum. And the results also hold true when the initial data contains
vacuum states in a subset of $\mathbb{R}^2$ and the natural compatibility
conditions are satisfied. Some new weighted estimates are obtained to establish
the upper bound of the density.
