
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data has been one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this longstanding 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 noflow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6thorder Galerkin and 6thorder finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (nonblowup) criteria, including BealeKatoMajda, ConstantinFeffermanMajda, and DengHouYu, to confirm the validity of the singularity. A careful local analysis also suggests that the blowingup solution develops a selfsimilar structure near the point of the singularity, as the singularity time is approached.

