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SUMMARY:Dynamic Depletion of Vortex Stretching and Nonlinear\n       Stabi
 lity of 3D Incompressible Flows - Tom Hou\, Caltech
DTSTART:20070615T150000Z
DTEND:20070615T160000Z
UID:TALK6939@talks.cam.ac.uk
CONTACT:Nigel Peake
DESCRIPTION:Whether the 3D incompressible Euler or Navier-Stokes equations
 \ncan develop a finite time singularity from smooth initial data has been\
 nan outstanding open problem. Here we review some existing computational a
 nd theoretical work on possible finite blow-up of the 3D Euler equations.\
 nWe show that the geometric regularity of vortex filaments\, even in an ex
 tremely localized region\, can lead to dynamic depletion of vortex\nstretc
 hing\, thus avoid finite time blowup of the 3D Euler equations. Further\, 
 we perform large scale computations of the 3D Euler equations\nto re-exami
 ne the two slightly perturbed anti-parallel vortex tubes which is consider
 ed as one of the most attractive candidates for a\nfinite time blowup of t
 he 3D Euler equations. We found that there is tremendous dynamic depletion
  of vortex stretching and the maximum\nvorticity does not grow faster than
  double exponential in time. Finally\,\nwe present a new class of solution
 s for the 3D Euler and Navier-Stokes\nequations\, which exhibit very inter
 esting dynamic growth property. By\nexploiting the special nonlinear struc
 ture of the equations\, we prove nonlinear stability and the global regula
 rity of this class of solutions. 
LOCATION:MR2\, Centre for Mathematical Sciences
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