On Local Singularities in Ideal Potential Flows with Free Surface

Despite important advances in the mathematical analysis of the Eulerequations for water waves, especially over the last two decades, itis not yet known whether local singularities can develop from smoothdata in well-posed initial value problems. For ideal free-surfaceflow with zero surface tension and gravity, the authors reviewexisting works that describe ``splash singularities'', singularhyperbolic solutions related to jet formation and ``flip-through'',and a recent construction of a singular free surface by Zubarev andKarabut that however involves unbounded negative pressure. Theauthors illustrate some of these phenomena with numericalcomputations of 2D flow based upon a conformal mapping formulation.Numerical tests with a different kind of initial data suggest thepossibility that corner singularities may form in an unstable wayfrom specially prepared initial data.