We prove that the 3-D compressible Euler equations with surface tension along the
moving free-boundary are well-posed. Specifically, we consider isentropic dynamics and
consider an equation of state, modeling a liquid, given by Courant and Friedrichs as
$p(\rho) = \alpha \rho^ \gamma - \beta$ for consants $\gamma >1$ and $ \alpha, \beta
> 0$. The analysis is made difficult by two competing nonlinearities associated with the
potential energy: compression in the bulk, and surface area dynamics on the free-boundary.
Unlike the analysis of the incompressible Euler equations, wherein boundary regularity
controls regularity in the interior, the compressible Euler equation require the additional
analysis of nonlinear wave equations generating sound waves. An existence theory is
developed by a specially chosen parabolic regularization together with the vanishing
viscosity method. The artificial parabolic term is chosen so as to be asymptotically
consistent with the Euler equations in the limit of zero viscosity. Having solutions for
the positive surface tension problem, we proceed to obtain a priori estimates which are
independent of the surface tension parameter. This requires choosing initial data which
satisfy the Taylor sign condition. By passing to the limit of zero surface tension, we
prove the well-posedness of the compressible Euler system without surface on the
free-boundary, and without derivative loss.