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Recent Progress on the Search of 3D Euler Singularities
【2013.12.20 4:00-5:00pm,C311】

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 2013-12-02 

  Colloquia & Seminars 

  Speaker

 侯一钊教授,加州理工大学,美国艺术与科学院院士

  Title

     Recent Progress on the Search of 3D Euler Singularities             

  Time

  2013.12.20 4:00-5:00pm                            

  Venue

  C311 

  Abstract

Whether the 3D incompressible Euler equations can develop a singularity
in finite time from smooth initial data is one of the most challenging
problems in mathematical fluid dynamics. This is closely related to the
Clay Millennium Problem on 3D Navier-Stokes Equations. In this talk, we
will present strong numerical evidence that the 3D Euler equations develop finite time singularities. To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of order $10^12$ in each dimension near the point of the singularity, we are able to advance the solution up to $10^{-6}$ distance from the predicted singularity time while maintaining a pointwise relative error of $O(10^{-4})$ in vorticity. We have applied all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu to confirm the validity of the singularity. A careful local analysis also suggests that the blowing-up solution is highly anisotropic and is not of Leray type. However, the solution develops a self-similar structure near the point of the singularity in the radial and axial directions as the singularity time is approached. This is a joint work with Dr. Guo Luo from Caltech.

  Affiliation

 侯一钊,美国加州理工大学讲座教授,美国艺术与科学院院士。侯一钊教授在科学计算领域的多个方面获得杰出成果,曾获得冯康科学计算奖、美国工业与应用数学学会Wilkinson数值分析与科学计算奖、美国计算力学会计算与应用科学奖、晨兴应用数学金奖、国际数学家大会45分钟邀请报告等荣誉和奖项。

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