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Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations

Hou, Thomas Y. and Li, Ruo (2007) Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 18 (4). pp. 637-642. ISSN 1078-0947. doi:10.3934/dcds.2007.18.637. https://resolver.caltech.edu/CaltechAUTHORS:20160322-091323469

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Abstract

We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, t, approaches a possible singularity time, T. The solution outside the inner core region is assumed to be regular, but it does not satisfy self-similar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as t → T in L^p for some p ϵ (3,∞), we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.3934/dcds.2007.18.637DOIArticle
http://www.aimsciences.org/journals/displayArticles.jsp?paperID=2618PublisherArticle
http://arxiv.org/abs/math/0603126arXivDiscussion Paper
Alternate Title:Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations
Additional Information:© 2007 American Institute of Mathematical Sciences. Received for publication March 2007. Available Online: May 2007. We would like to thank Profs. Congming Li and Dongho Chae for their useful comments and suggestions. The first author is supported by NSF under the NSF FRG grant DMS-0353838 and ITR Grant ACI-0204932. The second author was partially supported by the National Basic Research Program of China under the grant 2005CB321701.
Funders:
Funding AgencyGrant Number
NSFDMS-0353838
NSFACI-0204932
National Basic Research Program of China2005CB321701
Subject Keywords:Navier-Stokes equations, Euler equations, locally self-similar, blow-up
Issue or Number:4
Classification Code:2000 Mathematics Subject Classification. Primary: 76D03, 76D05; Secondary: 76B03
DOI:10.3934/dcds.2007.18.637
Record Number:CaltechAUTHORS:20160322-091323469
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160322-091323469
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:65576
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:30 Mar 2016 23:51
Last Modified:10 Nov 2021 23:47

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