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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. doi:10.3934/dcds.2012.32.1449.

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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
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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.
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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
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29846
Deposited By: Tony Diaz
Deposited On:17 Apr 2012 22:10
Last Modified:09 Nov 2021 19:31

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