  Numerical simulations of the incompressible Euler equations are performed
using the Taylor-Green vortex initial conditions and resolutions up to
$4096^3$. The results are analyzed in terms of the classical analyticity strip
method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of
the time-decay of the width of the analyticity strip $\delta(t)$ is observed at
the highest resolution for $3.7<t<3.85$ while preliminary 3D visualizations
show the collision of vortex sheets. The BKM criterium on the power-law growth
of supremum of the vorticity, applied on the same time-interval, is not
inconsistent with the occurrence of a singularity around $t \simeq 4$.
  These new findings lead us to investigate how fast the analyticity strip
width needs to decrease to zero in order to sustain a finite-time singularity
consistent with the BKM theorem. A new simple bound of the supremum norm of
vorticity in terms of the energy spectrum is introduced and used to combine the
BKM theorem with the analyticity-strip method. It is shown that a finite-time
blowup can exist only if $\delta(t)$ vanishes sufficiently fast at the
singularity time. In particular, if a power law is assumed for $\delta(t)$ then
its exponent must be greater than some critical value, thus providing a new
test that is applied to our $4096^3$ Taylor-Green numerical simulation.
  Our main conclusion is that the numerical results are not inconsistent with a
singularity but that higher-resolution studies are needed to extend the
time-interval on which a well-resolved power-law behavior of $\delta(t)$ takes
place, and check whether the new regime is genuine and not simply a crossover
to a faster exponential decay.
