Blow-up or no blow-up? The interplay between theory and numerics

Thomas Y. Hou, Caltech

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data has been an outstanding open problem. Recent studies indicate that the local geometric regularity of vortex filaments can lead to dynamic depletion of vortex stretching. Guided by the local non-blow-up theory, we have performed large scale computations of the 3D Euler equations on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. The local geometric regularity of vortex filaments and the anisotropic solution structure play an important role in depleting the nonlearity dynamically and thus preventing a finite time blow-up.

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