Well and ill-posedness for compressible Euler equations with vacuum

An interesting problem arising in gas and fluid dynamics is to understand the behavior of vacuum states, namely, the behavior of the system in the presence of vacuum. In this paper, we review and expand some local-in-time well-posedness results for one-dimensional compressible Euler equations for polytropic gases with a range of behaviors at the interface including the physical vacuum case. We also present some sort of ill-posedness results in some other cases. Finally, we present some conjectures about the change of behavior at the interface. This paper is a mixture of rigorous and formal calculations.