Speaker: Tom Hou
Title: The Interplay between Local Geometric Properties and the Global Regularity of the 3D Incompressible Euler Equations
Abstract: Whether the 3D incompressible Euler equation can develop a finite time
singularity from smooth initial data has been an outstanding open problem.
Here we review some existing computational and theoretical work on possible
finite blow-up of the 3D Euler equation. We show that there is a sharp
relationship between the geometric properties of the vortex filament and the
maximum vortex stretching. By exploring this local geometric property of the
vorticity field, we have obtained a global existence of the 3D incompressible
Euler equations provided that the unit vorticity vector and the velocity field
have certain mild regularity property in a very localized region containing
the maximum vorticity. Further, we perform large scale computations of the
3D Euler equations to re-examine the alleged finite-time blowup of the two antiparallel vortex tubes, which has been considered as one of the most promising candidates for a finite-time blowup of the 3D Euler equations. Our numerical studies indicate that the maximum vorticity does not grow faster
than double exponential in time. The velocity field and the enstrophy remain bounded throughout the
regularity of vortex lines seems to be responsible for the dynamic depletion
of vortex stretching.
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