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In this paper, we consider the compressible NavierStokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support. The viscosity coefficient is assumed to be a power function of the density so that the Cauchy problem is well-posed. New global existence results are established when the density function connects to the vacuum states continuously. For this, some new <i>a priori</i> estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum. We will also give a non-global existence theorem of regular solutions when the initial data are of compact support in Eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum.

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