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Publication 19-CNA-002

On Local Singularities In Ideal Potential Flows With Free Surface

Jian-Guo Liu
Departments of Physics and Mathematics
Duke University
Durham, NC 27708
jliu@phy.duke.edu

Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@andrew.cmu.edu

Abstract: Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ideal free-surface flow with zero surface tension and gravity, we review existing works that describe “splash singularities”, singular hyperbolic solutions related to jet formation and “flip-through”, and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure. We illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation. Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.

Get the paper in its entirety as  19-CNA-002.pdf


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