Vortex filament conjecture for general Euler flows

Establish the existence of solutions to the three-dimensional incompressible Euler equations whose vorticity remains concentrated along a curve evolving by the binormal flow for long times in the general case.

Background

The paper describes the binormal flow as a toy model for the evolution of vortex filaments in incompressible fluids governed by the Euler equations. A central unresolved issue is whether this geometric evolution can be realized by actual Euler solutions whose vorticity remains concentrated on a one-dimensional curve over long time intervals.

The authors identify this question as the vortex filament conjecture and note that existing results establish only special cases, including axially symmetric configurations, helical and leap-frogging solutions, and related results for Navier–Stokes or superfluid models. The general case therefore remains unresolved.

References

One can ask if it is true that it is possible to prove the existence of solutions of the Euler equations whose vorticity remains concentrated along a curve (filament) under the evolution of the binormal flow for long time (see ). This problem is known as vortex filament conjecture and it is still open in the general case.

— The binormal flow: a toy model for turbulence  (2610.13457 - Danesi et al., 9 Oct 2026) in Section 1, Introduction

It is an open question if this result can be extended to more singular solutions to allow for the existence of corners in the initial vortex filament.

— The binormal flow: a toy model for turbulence  (2610.13457 - Danesi et al., 9 Oct 2026) in Section 1, Introduction, concluding paragraph