  We prove well-posedness for the 3-D compressible Euler equations with moving
physical vacuum boundary, with an equation of state given by the so-called
gamma gas-law for gamma > 1. The physical vacuum singularity requires the sound
speed c to go to zero as the square-root of the distance to the moving
boundary, and thus creates a degenerate and characteristic hyperbolic
free-boundary system wherein the density vanishes on the free-boundary, the
uniform Kreiss--Lopatinskii condition is violated, and manifest derivative loss
ensues. Nevertheless, we are able to establish the existence of unique
solutions to this system on a short time-interval, which are smooth (in Sobolev
spaces) all the way to the moving boundary, and our estimates have no
derivative loss with respect to initial data. Our proof is founded on an
approximation of the Euler equations by a degenerate parabolic regularization
obtained from a specific choice of a degenerate artificial viscosity term,
chosen to preserve as much of the geometric structure of the Euler equations as
possible. We first construct solutions to this degenerate parabolic
regularization using a new higher-order Hardy-type inequality; we then
establish estimates for solutions to this degenerate parabolic system which are
independent of the artificial viscosity parameter. Solutions to the
compressible Euler equations are found in the limit as the artificial viscosity
tends to zero. Our regular solutions can be viewed as degenerate viscosity
solutions. Out methodology can be applied to many other systems of degenerate
and characteristic hyperbolic systems of conservation laws.
