On the local existence of the free-surface Euler equation with surface tension

Here σ > 0 is the surface tension, while H represents twice the mean curvature of the boundary ∂ (t). Problems related to local or global existence of solutions of free surface evolution under the Euler flow, with or without surface tension, have attracted considerable attention in the last decades. For both cases different approaches have been developed; however, the search is still in progress for the lowest regularity spaces where the existence or uniqueness of solutions hold. For the history of both problems, cf. [2,17,32] and references therein. While in the zero surface tension case the problem is known to be unstable, and thus the RayleighTaylor stability condition has to be imposed, this is not necessary when the surface tension is nonzero since the surface tension provides a stabilizing effect close to the boundary.