A model for singularity formation in three-dimensional Euler and Navier-Stokes flows

TL;DR

This paper introduces a formal model for singularity formation in 3D Euler and Navier-Stokes flows, using an approximation of local two-dimensionality to analyze vortex structures and predict finite-time singularities.

Contribution

It presents a novel approximate model that captures singularity formation in 3D fluid flows, linking local vortex dynamics to finite-time blow-up in Euler and Navier-Stokes equations.

Findings

Model exhibits finite-time Euler singularity at an isolated point.

Navier-Stokes problem shows finite-time singularity at high Reynolds number.

Singularities are compatible with BKM and CF conditions.

Abstract

We present a formal, approximate model for singularity formation in classical fluid dynamics in three dimensions. The construction utilizes an approximation of local two-dimensionality to study an anti-parallel hairpin vortex structure with a cross-section equivalent to the 2D Chaplygin-Lamb dipole vortex. The model exhibits a finite time Euler singularity at an isolated point, with only finite local stretching of vortex lines. The model also suggests an associated Navier-Stokes problem, which exhibits a finite-time point singularity, provided that a Reynolds number is sufficiently large. The singularities are compatible with both the BKM [1] and CF[2] conditions. The vorticity support is infinite in volume but the singularity forms as a result of local processes requiring only finite energy input.