Dynamics of vacuum states for the compressible Navier-Stokes equations

Abstract

The dynamics of vacuum states for 1D compressible Navier-Stokes equations are considered. For any global entropy weak solution, we show that the flow density is continuous on both space and time, and is positive everywhere for all the time, if no vacuum state exists initially (i.e., non-formation of vacuum states happens). Furthermore, we prove that there is a global weak solution which contains one piece of discontinuous finite vacuum for some time period, meanwhile the vacuum is shown to be compressed at an algebraic rate and then vanishes within finite time.