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On singularity formation of a 3D model for incompressible Navier–Stokes equations

Hou, Thomas Y. and Shi, Zuoqiang and Wang, Shu (2012) On singularity formation of a 3D model for incompressible Navier–Stokes equations. Advances in Mathematics, 230 (2). pp. 607-641. ISSN 0001-8708. http://resolver.caltech.edu/CaltechAUTHORS:20120523-112912673

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Abstract

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.aim.2012.02.015DOIArticle
http://www.sciencedirect.com/science/article/pii/S0001870812000679PublisherArticle
http://arxiv.org/abs/0912.1316arXivDiscussion Paper
Additional Information:© 2012 Elsevier Inc. Received 3 February 2010; accepted 3 February 2012. Available online 24 March 2012. Communicated by Charles Fefferman. Dr. T. Hou would like to acknowledge NSF for their generous support through the Grants DMS-0713670 and DMS-0908546. The work of Drs. Z. Shi and S. Wang was supported in part by the NSF grant DMS-0713670. The research of Dr. S. Wang was also supported by China 973 Program (Grant No. 2011CB808002), the Grants NSFC 11071009 and PHR-IHLB 200906103. This work was done during Dr. Shu Wang’s visit to ACM at Caltech. He would like to thank Prof. T. Hou and Caltech for their hospitality during his visit. We would like to thank Profs. Joseph Keller, Congming Li, Xinwei Yu, and the referee for their comments which helped to improve the quality of this work.
Funders:
Funding AgencyGrant Number
NSFDMS-0713670
NSFDMS-0908546
China 973 program2011CB808002
National Natural Science Foundation of China11071009
National Natural Science Foundation of ChinaPHR-IHLB 200906103
Subject Keywords:Finite time singularities; Nonlinear nonlocal system; Incompressible Navier–Stokes equations
Record Number:CaltechAUTHORS:20120523-112912673
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120523-112912673
Official Citation:Thomas Y. Hou, Zuoqiang Shi, Shu Wang, On singularity formation of a 3D model for incompressible Navier–Stokes equations, Advances in Mathematics, Volume 230, Issue 2, 1 June 2012, Pages 607-641, ISSN 0001-8708, 10.1016/j.aim.2012.02.015. (http://www.sciencedirect.com/science/article/pii/S0001870812000679)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:31616
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:23 May 2012 20:55
Last Modified:23 Aug 2016 10:12

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