Global convergence for immortal elastic and ideal flows

Prove that for any immortal solution of the free elastic flow ∂tγ=−(kss+½k^3)ν or ideal flow ∂tγ=(kssss+k^2kss−½kk_s^2)ν with generic initial data, the evolving curve converges exponentially fast in the smooth topology, modulo similarity transformations, to a multiply-covered lemniscate or circle.

Background

Analogous exponential convergence results are known for curve diffusion flow (e.g., convergence to multiply-covered circles). Extending such global dynamical classifications to elastic and ideal flows would significantly advance the qualitative understanding of their long-time behavior. The authors propose a direct analogue, conjecturing exponential convergence under generic conditions.