  Whether the 3D incompressible Navier-Stokes equations can develop a finite
time singularity from smooth initial data is one of the most challenging
problems in nonlinear PDEs. In this paper, we present some new numerical
evidence that the incompressible axisymmetric Navier-Stokes equations with
smooth initial data of finite energy seem to develop potentially singular
behavior at the origin. This potentially singular behavior is induced by a
potential finite time singularity of the 3D Euler equations that we reported in
the companion paper (arXiv:2107.05870). We present numerical evidence that the
3D Navier--Stokes equations develop nearly self-similar singular scaling
properties with maximum vorticity increased by a factor of 10^7. We have
applied several blow-up criteria to study the potentially singular behavior of
the Navier--Stokes equations. The Beale-Kato-Majda blow-up criterion and the
blow-up criteria based on the growth of enstrophy and negative pressure seem to
imply that the Navier--Stokes equations using our initial data develop a
potential finite time singularity. We have also examined the
Ladyzhenskaya-Prodi-Serrin regularity criteria. Our numerical results for the
cases of (p,q) = (4,8), (6,4), (9,3) and (p,q)=(\infty,2) provide strong
evidence for the potentially singular behavior of the Navier--Stokes equations.
Our numerical study shows that while the global L^3 norm of the velocity grows
very slowly, the localized version of the L^3 norm of the velocity experiences
rapid dynamic growth relative to the localized L^3 norm of the initial
velocity. This provides further evidence for the potentially singular behavior
of the NavieStokes equations.
