Finite-Time Euler singularities: A Lagrangian perspective

TL;DR

This paper investigates whether finite-time singularities can occur in 3D incompressible inviscid flows, providing numerical evidence that high-symmetry vortex configurations do not lead to such singularities, using advanced Lagrangian and geometric analysis.

Contribution

The study applies Lagrangian and geometric non-blowup criteria to high-resolution simulations, offering evidence against finite-time singularities in high-symmetry vortex flows.

Findings

No finite-time singularity observed in high-symmetry vortex initial conditions.

Lagrangian tracer analysis supports non-blowup criteria.

Numerical data aligns with analytical non-blowup conditions.

Abstract

We address the question 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 being a promising candidate for a 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 by 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.