On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid

In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently in Lindblad and Luo (Commun Pure Appl Math 71:1273-1333, 2018) and Luo (Ann PDE 4(2):1-71, 2018), the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surface tension, nor estimates previously derived for a liquid with surface tension and fixed sound speed, are applicable here. In order to obtain our result, we devise a suitable sound-speed-weighted energy that takes into account the coupling of the fluid motion with the boundary geometry. Estimates are closed by exploiting the full non-linear structure of the Euler equations and invoking several geometric properties of the boundary in order to produce some remarkable cancellations. We stress that we do not assume the fluid to be irrotational.