Recent progress on dynamic stability and global
regularity of 3D incompressible Euler and Navier-Stokes
equations
April 20, 2011
3:00pm SEC 202
Abstract
Whether the 3D incompressible Euler and Navier-Stokes equations can develop
a finite time singularity from smooth initial data with finite energy has
been one of the most long standing open questions. We review some recent
theoretical and computational studies which show that there is a subtle
dynamic depletion of nonlinear vortex stretching due to local geometric
regularity of vortex filaments. The local geometric regularity of vortex
filaments can lead to tremendous cancellation of nonlinear vortex
stretching, thus preventing a finite time singularity. Our studies also
reveal a surprising stabilizing effect of convection for the 3D
incompressible Euler and Navier-Stokes equations. Finally, we present a new
class of solutions for the 3D Euler and Navier-Stokes equations, which
exhibit very interesting dynamic growth property. By exploiting the special
structure of the solution and the cancellation between the convection term
and the vortex stretching term, we prove nonlinear stability and the global
regularity of this class of solutions.
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Last modified: April 11 2016 - 18:14:43