CIMMS Lunchtime Series: Well-posedness of the free-surface Euler equations with or without surface tension

Steve Shkoller, Department of Mathematics, University of California, Davis

The incompressible Euler equations with a free surface describe the motion of an ideal, inviscid fluid whose boundary or interface moves with the the fluid. We prove well-posedness for this PDE for short-time and any initial data in sufficiently smooth Sobolev spaces, with or without surface tension. The method requires spacetime a priori estimates, as well as the construction of a new class of approximate solutions for the Euler equations.