  In this paper, we study the global well-posedness of classical solution to 2D
Cauchy problem of the compressible Navier-Stokes equations with large initial
data and vacuum. It is proved that if the shear viscosity $\mu$ is a positive
constant and the bulk viscosity $\lambda$ is the power function of the density,
that is, $\lambda(\rho)=\rho^{\beta}$ with $\beta>3$, then the 2D Cauchy
problem of the compressible Navier-Stokes equations on the whole space
$\mathbf{R}^2$ admit a unique global classical solution $(\rho,u)$ which may
contain vacuums in an open set of $\mathbf{R}^2$. Note that the initial data
can be arbitrarily large to contain vacuum states. Various weighted estimates
of the density and velocity are obtained in this paper and these self-contained
estimates reflect the fact that the weighted density and weighted velocity
propagate along with the flow.
