Existence of an ideal lemniscate for the ideal flow

Establish the existence of a self-similarly expanding figure-8 (lemniscate-type) closed solution to the ideal flow ∂tγ=(kssss+k^2kss−½kk_s^2)ν; that is, construct a homothetic expander for the ideal flow with lemniscate topology (the “ideal lemniscate”) and prove it rigorously.

Background

The proposed dynamic stability classification (above) presupposes the existence of an ideal lemniscate, analogous to the Bernoulli lemniscate for elastic flow and the figure-8 shrinker for curve diffusion. While the paper constructs epicyclic expanders for the ideal flow, a lemniscate-type expander is not established and remains a prerequisite for the broader classification program.

References

This conjecture has as a precondition the discovery of an ideal lemniscate, a self-similarly expanding figure-8 type solution to the ideal flow, which is yet to be established rigorously.

— Jellyfish exist  (2601.21227 - Andrews et al., 29 Jan 2026) in Open questions and outlook, Section 1 (Introduction)