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The interplay between analysis and computation in the study of 3D Euler singularities.

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Tom Hou (Caltech)

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 question is
closely related to the Clay Millennium Problem on 3D Navier-Stokes
Equations. A potential singularity in the 3D Euler equations is
significant because it may be responsible for the onset of energy
cascade in turbulent flows. We first review some recent theoretical
and computational studies of the 3D Euler equations. 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. The singularity is a ring like singularity that occurs
at a stagnation point in the symmetry plane located at the boundary of
the cylinder. A careful local analysis also suggests that the
blowing-up solution is highly anisotropic and is not of Leray type. A
1D model is proposed to study the mechanism of the finite time
singularity. We have recently proved rigorously that the 1D model
develops finite time singularity. Finally, we present some recent
progress in developing an integrated analysis and computation strategy
to analyze the finite time singularity of the original 3D Euler
equations.