Lagrangian and geometric analysis of finite-time Euler singularities

TL;DR

This paper introduces a high-resolution numerical method combining geometric analysis and non-blowup criteria to investigate potential finite-time singularities in 3D Euler flows, providing evidence against singularity formation in specific symmetric cases.

Contribution

The paper develops a novel numerical approach that integrates Lagrangian vortex line geometry with analytical criteria to reliably detect or rule out finite-time singularities in Euler flows.

Findings

No finite-time singularity observed in tested symmetric initial conditions.

High-resolution simulations up to 8192^3 mesh points conducted.

Method effectively distinguishes between singular and near-singular flow evolution.

Abstract

We present a numerical method of analyzing possibly singular incompressible 3D Euler flows using massively parallel high-resolution adaptively refined numerical simulations up to 8192^3 mesh points. Geometrical properties of Lagrangian vortex line segments are used in combination with analytical non-blowup criteria by Deng et al [Commun. PDE 31 (2006)] to reliably distinguish between singular and near-singular flow evolution. We then apply the presented technique to a class of high-symmetry initial conditions and present numerical evidence against the formation of a finite-time singularity in this case.