Finite-time Euler singularities: A Lagrangian perspective

Abstract

We address the question of whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows as promising candidates for finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high-resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made in the analytical non-blowup criteria introduced by Deng et al. [Commun. PDE 31 (2006)] connecting vortex line geometry (curvature, spreading) to velocity increase, to rule out singular behavior. © 2012 Elsevier Ltd. All rights reserved.