Compressible flows with a density-dependent viscosity coefficient

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.