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

Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit