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Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model

Hou, Thomas Y. and Shi, Zuoqiang (2012) Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. Discrete and Continuous Dynamical Systems, 32 (5). pp. 1449-1463. ISSN 1078-0947. https://resolver.caltech.edu/CaltechAUTHORS:20120326-132813054

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Abstract

By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li [12]. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of [12], we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.3934/dcds.2012.32.1449 DOIArticle
http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=6874PublisherArticle
http://arxiv.org/abs/1011.5514arXivDiscussion Paper
Additional Information:© 2011 American Institute of Mathematical Sciences. Received March 2011; revised May 2011. The research was in part supported by the National Science Foundation through the grant DMS-0908546.
Funders:
Funding AgencyGrant Number
NSFDMS-0908546
Subject Keywords:3D Euler equations; SQG equation; finite time blow-up; growth rate of maximum vorticity; geometric properties
Issue or Number:5
Classification Code:2000 Mathematics Subject Classication. Primary: 76B03; Secondary: 35L60, 35M10
Record Number:CaltechAUTHORS:20120326-132813054
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20120326-132813054
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29846
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:17 Apr 2012 22:10
Last Modified:03 Oct 2019 03:45

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