Computational Mathematics and Scientific Computing Seminar

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Recent Progress on the Search of 3D Euler Singularities

Speaker: Thomas Hou, California Institute of Technology

Location: Warren Weaver Hall 1302

Date: Dec. 13, 2013, 10 a.m.

Synopsis:

Whether the 3D incompressible Euler equations can develop a singularity in fnite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. 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.