On the vortex filament conjecture for Euler flows

TL;DR

This paper investigates the evolution of vortex filaments in ideal fluids, demonstrating that under certain conditions, they follow binormal curvature flow, supported by new mathematical estimates and stability analysis.

Contribution

It provides a rigorous proof that vortex filaments evolve according to binormal curvature flow, combining novel estimates with stability techniques.

Findings

Vortex filaments follow binormal curvature flow to leading order.

New estimates on Hamiltonian-Poisson structures are developed.

Stability estimates support the evolution analysis.

Abstract

In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $\mathbb{R}^3$, we prove that the curve evolves to leading order by binormal curvature flow. Our approach combines new estimates on the distance of the corresponding Hamiltonian-Possion structures with stability estimates recently developed in Ref. 15.