Title: A priori estimates for the free-boundary Euler equations with surface tension in three dimensions,
Abstract: We derive a priori estimates for the incompressible free-boundary Euler
equations with surface tension in three spatial dimensions. Working in
Lagrangian coordinates, we provide a priori estimates for the local existence
when the initial velocity, which is rotational, belongs to $H^3$ and the trace
of initial velocity on the free boundary to $H^{3.5}$, thus lowering the
requirement on the regularity of initial data in the Lagrangian setting. Our
methods are direct and involve three key elements: estimates for the pressure,
the boundary regularity provided by the mean curvature, and the Cauchy
invariance.