# On the energy of critical solutions of the binormal flow

## TL;DR

This paper investigates critical solutions of the binormal flow, establishing a conserved energy that exhibits a jump discontinuity at singularity formation, linking fluid dynamics, ferromagnetism, and Schrödinger equations.

## Contribution

It introduces a natural energy for critical solutions of the binormal flow and proves its conservation except at singularities, providing new insights into vortex filament dynamics.

## Key findings

- Existence of a conserved energy for critical solutions.
- Energy exhibits a jump discontinuity at singularity formation.
- Links between fluid mechanics, ferromagnetism, and Schrödinger equation.

## Abstract

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schr\"odinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity.