Unbounded growth of the energy density associated to the Schr\"odinger map and the binormal flow

TL;DR

This paper investigates the binormal flow, linked to vortex filament dynamics and the Schrödinger map, revealing unbounded energy density growth related to high-frequency oscillations, which may influence singularity formation in Euler equations.

Contribution

It demonstrates the unbounded growth of energy density in the binormal flow, connecting geometric curve evolution to high-frequency oscillations and potential singularities in fluid dynamics.

Findings

Energy density can grow unboundedly over time.

High-frequency oscillations of tangent vectors increase without limit.

Results relate to potential singularity development in Euler equations.

Abstract

We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schr\"odinger map with values on the 2-D sphere, and to the 1-D cubic Schr\"odinger equation. Although these equations are completely integrable we show the existence of an unbounded growth of the energy density. The density is given by the amplitude of the high frequencies of the derivative of the tangent vectors of the curves, thus giving information of the oscillation at small scales. In the setting of vortex filaments the variation of the tangent vectors is related to the derivative of the direction of the vorticity, that according to the Constantin-Fefferman-Majda criterion plays a relevant role in the possible development of singularities for Euler equations.