Whether the 3D incompressible Euler or Navier-Stokes equations
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 equations.
We show that the geometric regularity of vortex filaments, even in an
extremely localized region, can lead to dynamic depletion of vortex
stretching, thus avoid finite time blowup of the 3D Euler equations.
Further, we perform large scale computations of the 3D Euler equations
to re-examine the two slightly perturbed anti-parallel vortex tubes which
is considered as one of the most attractive candidates for a finite time
blowup of the 3D Euler equations. We found that there is tremendous dynamic
depletion of vortex stretching and the maximum vorticity does not grow faster
than double exponential in time. 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 nonlinear structure of the
equations, we prove nonlinear stability and the global regularity of this class of solutions.