Remarks on the breakdown of smooth solutions for the 3-D Euler equations
type: Item
instance of: scholarly article
title: Remarks on the breakdown of smooth solutions for the 3-D Euler equations
zbMATH Open document ID: 0573.76029
DOI: 10.1007/BF01212349
author: Tosio Kato
author: J. Thomas Beale
author: Andrew J. Majda
published in: Communications in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The authors consider the Euler equations for the motion of an incompressible, inviscid fluid in the whole space. They show that, if a solution is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches. Therefore, it is not possible for other kinds of singularities to form before the vorticity becomes unbounded. On the other hand, if the vorticity remains bounded, a smooth solution persists. In other words, the maximum norm of the vorticity controls the breakdown of smooth solutions for the 3-D Euler equations. The same result, with the vorticity substituted by the deformation tensor, has been recently proved by \textit{G. Ponce} in Commun. Math. Phys. 98, 349-353 (1985; Zbl 0589.76040).
Mathematics Subject Classification ID: 76Bxx
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 3915851
zbMATH Keywords: Euler equations
zbMATH Keywords: incompressible, inviscid fluid in the whole space
zbMATH Keywords: maximum vorticity
zbMATH Keywords: critical time
zbMATH Keywords: singularities
zbMATH Keywords: breakdown of smooth solutions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01212349
OpenAlex ID: W1996952201
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description: scientific article; zbMATH DE number 3915851