Toward the finite-time blowup of the 3D axisymmetric Euler equations: A numerical investigation

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 × 10¹²)² near the point of the singularity, we are able to advance the solution up to τ₂ = 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 × 10⁸)-fold increase in the maximum vorticity ||ω||∞. 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.