Free boundary problems of the Euler equation: hydrodynamical instabilities and energy estimates
Chongchun Zeng (Georgia Tech)

We consider the evolution of free surfaces of
incompressible and invicid fluids. Neglecting the gravity, we are interested in the cases of 1.) the motion of a droplet in the vacuum with or without surface tension and 2.) the motion of the interface between two fluids with surface tension. The evolution of these fluid boundaries and the velocity fields is determined by the free boundary problem of the Euler's equation. Each of these problems can be considered in a Lagrangian formulation on an infinite dimensional Riemannian manifold of volume preserving diffeomorphisms. In the absence of surface tension, the well-known Rayleigh-Taylor and Kelvin-Helmholtz instabilities appear naturally related to the signs of the curvatures of those infinite dimensional manifolds. The surface tension produces stronger conservative forces than the instabilities and thus regularizes the surface evolution. Finally, a scale of functionals as "energies" are defined and they bound high Sobolev norms of the velocity field as well as the mean curvature of the fluid boundary. Thus we establish the regularity of the solutions for a short time depending on the initial data.