Dynamic stability classification for curve diffusion, elastic, and ideal flows

Classify the dynamically stable closed solutions of the three planar curvature flows—curve diffusion flow ∂tγ=−kssν, free elastic flow ∂tγ=−(kss+½k^3)ν, and ideal flow ∂tγ=(kssss+k^2kss−½kk_s^2)ν—by proving that the only dynamically stable solutions are: (a) circles for the curve diffusion flow; (b) multiply-covered Bernoulli lemniscates and multiply-covered circles for the elastic flow; and (c) multiply-covered ideal lemniscates and multiply-covered circles for the ideal flow. Here dynamic stability means there exists ε>0 such that any curve η with ||γ−η||_{C^∞}<ε generates a solution trajectory that converges, modulo similarity transformations, to γ.

Background

The paper constructs infinite families of new homothetic solutions (jellyfish expanders for elastic flow, epicyclic shrinkers for curve diffusion flow, and epicyclic expanders for ideal flow). Numerical evidence suggests a rich dynamical landscape and raises questions about which solutions can be dynamically stable under their respective flows.

Existing results indicate stability of circles in certain settings and suggest non-stability of newly constructed expanders. The authors formalize this into a comprehensive classification conjecture specifying exactly which multiply-covered canonical shapes (circles and lemniscates) are dynamically stable for each flow.