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

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• 作者：Thomas Y. Hou^a ; ^{hou@acm.caltech.edu} ; Zuoqiang Shi^b ; ^{zqshi@math.tsinghua.edu.cn} ; Shu Wang^c ; ^{wangshu@bjut.edu.cn}
• 关键词：Finite time singularities ; Nonlinear nonlocal system ; Incompressible Navier&ndash ; Stokes equations
• 刊名：Advances in Mathematics
• 出版年：2012
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：1 June, 2012
• 卷：230
• 期：2
• 页码：607-641
• 文件大小：252 K

摘要

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in 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.