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

Abstract

We prove that the three-dimensional compressible Euler equations with surface tension along the moving free-boundary are well-posed; we then establish the limit as surface tension tends to zero. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [Supersonic Flow and Shock Waves, Appl. Math. Sci. 21, Springer-Verlag, New York, 1976] 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 requires the additional analysis of nonlinear wave equations generating sound waves. An existence theory is developed by a specially chosen parabolic regularization together with th...