A dynamic approach to heterogeneous elastic wires

TL;DR

This paper studies the evolution of closed elastic wires with fixed length and winding number, analyzing a nonlocal flow that accounts for density and spontaneous curvature, proving well-posedness and convergence.

Contribution

It introduces a novel nonlocal quasilinear coupled parabolic system for elastic wires with fixed length and winding number, establishing well-posedness and long-term behavior.

Findings

Proved local well-posedness of the flow.

Established global existence of solutions.

Demonstrated full convergence of the flow.

Abstract

We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.