  We prove the global existence of weak solutions for the 2-D compressible
Navier-Stokes equations with a density-dependent viscosity coefficient
($\lambda=\lambda(\rho)$). Initial data and solutions are small in energy-norm
with nonnegative densities having arbitrarily large sup-norm. Then, we show
that if there is a vacuum domain at the initial time, then the vacuum domain
will retain for all time, and vanishes as time goes to infinity. At last, we
show that the condition of $\mu=$constant will induce a singularity of the
system at vacuum. Thus, the viscosity coefficient $\mu$ plays a key role in the
Navier-Stokes equations.
