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 local geometric properties of vortex filaments 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 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 can prove the global
regularity of this class of solutions.