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On Local Singularities in Ideal Potential Flows with Free Surface
Authors:Liu  Jian-Guo  Pego  Robert L
Institution:Department of Physics and Department of Mathematics,Duke University, Durham, NC 27708, USA. and Department of Mathematical Sciences and Center for Nonlinear Analysis,Carnegie Mellon University, Pittsburgh, Pennsylvania, PA 12513, USA.
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, the authors 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. The authors 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.
Keywords:Incompressible flow  Water wave equations  Splash singularity  Flip-through  Dirichlet hyperbolas  Conformal mapping
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