Institute for Advanced Study

**Abstract**: We study one-dimensional compressible Euler
equations arising in gas dynamics when the vacuum is allowed initially. It is
considered as a vacuum free boundary problem. The difficulty lies in the fact
that the system becomes degenerate along the vacuum boundary. Some local
existence result, where initial vacuum state is rather regular, is reviewed.
We present new a priori estimates for the physical vacuum boundary
in Lagrangian coordinates.