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. We first review
some recent theoretical and computational studies of the 3D Euler
equations which show that there is a subtle dynamic depletion of
nonlinear vortex stretching due to local geometric regularity of vortex
filaments. Our study suggests that the convection term could have a
nonlinear stabilizing effect for certain flow geometry. We then 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 with a maximum effective
resolution of order $10^12$. 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. A 1D model is proposed to study the mechanism of the
finite time singularity. Recently we prove rigorously that this 1D model
develops finite time singularity.
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