Whether the three-dimensional incompressible Euler equations can develop a singularity
in finite time from smooth initial data is one of the most challenging problems in mathematical
fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question
from a numerical point of view by presenting a class of potentially singular solutions to the Euler
equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition
along the axial direction and a no-flow boundary condition on the solid wall. The equations
are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on
specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions.
With a maximum effective resolution of over $(3 × 10^{12})^2$ near the point of the singularity, we are
able to advance the solution up to $\tau_2 = 0.003505$ and predict a singularity time of $t_s ≈ 0.0035056$,
while achieving a pointwise relative error of $O(10^{−4})$ in the vorticity vector ω and observing a
$(3 × 10^8)$-fold increase in the maximum vorticity $\|ω\|_{\infty}$. The numerical data are checked against
all major blowup/non-blowup criteria, including Beale–Kato–Majda, Constantin–Fefferman–Majda,
and Deng–Hou–Yu, to confirm the validity of the singularity. A local analysis near the point of the
singularity also suggests the existence of a self-similar blowup in the meridian plane.