Finite time singularities for the free boundary incompressible Euler equations
type: Item
instance of: scholarly article
title: Finite time singularities for the free boundary incompressible Euler equations
zbMATH Open document ID: 1291.35199
author: Angel Castro
author: Francisco Gancedo
author: Javier Gómez-Serrano
author: Diego Córdoba
author: Charles Fefferman
published in: Annals of Mathematics. Second Series
publication date: 2013-10-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.2170
review text: The 2D incompressible free boundary Euler problem is used to study the evolution of a fluid region and a vacuum region separated by a smooth (initial) interface. The fluid is irrotational and the surface tension is neglected. The main point is to describe the evolution of the vacuum-fluid interface \(WF\). Even if at the initial moment \(WF\) is smooth (but not necessary a graph), after a finite time a singularity appears -- the curve touches itself. Numerical results, based on the Beale-Hou-Lovengrub method, are used to motivate some conjectures. Then the authors explain what results can be proved. Interval arithmetic is used to produce a rigorous computer-assisted proof for the existence of an exact solution, close to the approximate numerical solution, which ends into a particular singularity. A local existence for analytic initial data in a transformed (``tilda'') domain is proved, by using the abstract form of the Cauchy-Kowalewski theorem.
Mathematics Subject Classification ID: 35Q31
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76D27
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 6220729
zbMATH Keywords: incompressible Euler equations
zbMATH Keywords: potential theory
zbMATH Keywords: water waves problem
reviewed by: Gelu Paşa
MaRDI profile type: Publication
OpenAlex ID: W2964346256
Wikidata QID: Q101086040
arXiv ID: 1112.2170
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DOI: 10.4007/ANNALS.2013.178.3.6
description: scientific article; zbMATH DE number 6220729