Whether the 3D incompressible Euler equation can develop a finite time
singualrity from smooth initial data has been an outstanding open problem.
It has been believed that a finite singularity of the 3D Euler equation
could be the onset of turbulence. Here we review some existing computational
and theoretical work on possible finite blow-up of the 3D Euler equation.
Further, we show that there is a sharp relationship between the geometric
properties of the vortex filament and the maximum vortex stretching. By
exploring this geometric property of the vorticity field, we have obtained
a global existence of the 3D incompressible Euler equation provided that
the normalized unit vorticity vector has certain mild regularity property
in a very localized region containing the maximum vorticity. Our assumption
on the local geometric regularity of the vorticity field is consistent
with recent numerical experiments. Further, we discuss how viscosity may
help preventing singularity formation for the 3D Navier-Stokes equation,
and present a new result on the global existence of the viscous Boussinesq
equation.