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On an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equations

Author:Shu Wang    Publicsh date:2016-05-13    Clicks:561
报告时间:2016年5月13日(星期五)上午10:30—11:30 报告地点:科技楼南楼702 北京工业大学应用数理学院院长王术教授: 题目:On an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equati ......
报告时间:2016年5月13日(星期五)上午10:30—11:30
报告地点:科技楼南楼702
北京工业大学应用数理学院院长王术教授:
题目:On an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equations 

摘要:We study the singularity formation and global regularity of an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equations. This 3D model is derived from the axisymmetric Navier-Stokes equations with swirl using a set of new variables. The model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. We prove rigorously that the 3D model develops finite time singularities for a large class of initial data with finite energy and appropriate boundary conditions. Moreover, we also prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. The related problems are surveyed and some recent results will also be reviewed. 

References:

1.Hou, Thomas Y.; Li, Congming; Shi, Zuoqiang; Wang, Shu(王术); Yu, Xinwei. On singularity formation of a nonlinear nonlocal system. Arch. Ration. Mech. Anal. 199 (2011), no. 1, 117–144.

2. Hou, Thomas Y.; Shi, Zuoqiang; Wang, Shu(王术). On singularity formation of a 3D model for incompressible Navier-Stokes equations. Adv. Math. 230 (2012), no. 2,607–641.

3. Hou, Thomas Y., Lei, Z., Luo, G., Wang, Shu(王术), Zou, C. On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations.   Arch Rational Mech. Anal., 212(2014), 683-706.
4. Wang, Shu(王术). On a new 3D model for incompressible Euler and Navier-Stokes equations. Acta Mathematica Scientia.30B(6)(2010): 2089-2102.