Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

• Published in: Multiscale modeling & simulation Vol. 12; no. 4; pp. 1722 - 1776
• Main Authors: Luo, Guo, Hou, Thomas Y.
• Format: Journal Article
• Language: English
• Published: 01.01.2014
• Subjects: Axisymmetric; Euler equations; Mathematical analysis; Mathematical models; Self-similarity; Singularities; Three dimensional; Vorticity
• ISSN: 1540-3459; 1540-3467
• DOI: 10.1137/140966411

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 $(3 \times 1012})2}$ 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 $t_{s} \approx 0.0035056$, while achieving a pointwise relative error of $O(10-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 108})$-fold increase in the maximum vorticity $\lVert\omega\rVert_{\infty}$. 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.