Regularity estimates of a fluid-free surface evolution

In this work the evolution of a fluid droplet in vacuum is considered. This means that the surface tension and the fluid forces are in equilibrium at the free boundary. The fluid is governed by the incompressible quasi-steady Stokes equation. We present higher order energy estimates for this setting in the planar case. In particular bounds of the curvature and its tangential derivative combined with the second and third spacial derivatives of the fluid velocity as respective dissipation. These estimates are shown to hold until the point of a topological degeneracy. They provide quantitative bounds, that depend on specific properties of the initial geometry only. The work contrasts previous approaches, which are based on the use of local coordinates and instead performs all estimates in an Eulerian setting. Indeed, the estimates provided here are geometrically intrinsic and collapse only once these intrinsic qualities break.