Potentially Singular Solutions of the 3D Incompressible Euler Equations

Abstract

Whether the 3D 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 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 $(3\times 10^{12}{)}^2$ near the point of the singularity, we are able to advance the solution up to $\ddot{I}{„}_2=0.003505$ 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 $\ddot{I}‰$ and observing a $(3\times 10^8)$-fold increase in the maximum vorticity $\Vert \ddot{I}‰{\Vert }_{\infty }$. 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.

version 1: 81 pages, 86 figures; version 2: 57 pages, 44 figures, removed part of the technical details and streamlined the presentation