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Dynamics of Elastic Wires: Preserving Area without Nonlocality

Leonie Langer

TL;DR

This paper introduces an area-preserving, sixth-order $H^{-1}$-gradient flow for the planar elastic energy $\mathcal{E}(\gamma)$ with penalization $\mathcal{E}_\lambda(\gamma)=\mathcal{E}(\gamma)+\lambda\mathcal{L}(\gamma)$. It constructs time-dependent Hilbert spaces $H_\gamma$ and derives the gradient flow in the $H_\gamma^{-1}$ metric, yielding the evolution $\partial_t^\perp\gamma=-\nabla_{s_\gamma}^2(\nabla_{s_\gamma}^2\kappa_\gamma+\tfrac12|\kappa_\gamma|^2\kappa_\gamma-\lambda\kappa_\gamma)$, which preserves enclosed area and decreases the penalized energy. The authors prove global existence for all $\lambda\ge0$; when $\lambda>0$, the flow admits smooth subconvergence to stationary area-constrained points, and a constrained Łojasiewicz–Simon inequality is developed to obtain full convergence. They characterize stationary solutions as constrained critical points, highlighting both elastica and non-elastica shapes, and show that removing the length penalty ($\lambda=0$) can prevent convergence, with possible infinite-time blow-up. This provides a local, nonlocal-free, area-preserving alternative to classical elastic flows and yields a robust framework for studying long-time behavior of area-constrained elasticae.

Abstract

We derive an $H^{-1}$-gradient flow of the elastic energy which preserves the enclosed area of evolving planar curves. For this new sixth-order evolution equation, we prove a global existence result. Additionally, by penalizing the length, we show convergence to an area constrained critical point of the elastic energy.

Dynamics of Elastic Wires: Preserving Area without Nonlocality

TL;DR

This paper introduces an area-preserving, sixth-order -gradient flow for the planar elastic energy with penalization . It constructs time-dependent Hilbert spaces and derives the gradient flow in the metric, yielding the evolution , which preserves enclosed area and decreases the penalized energy. The authors prove global existence for all ; when , the flow admits smooth subconvergence to stationary area-constrained points, and a constrained Łojasiewicz–Simon inequality is developed to obtain full convergence. They characterize stationary solutions as constrained critical points, highlighting both elastica and non-elastica shapes, and show that removing the length penalty () can prevent convergence, with possible infinite-time blow-up. This provides a local, nonlocal-free, area-preserving alternative to classical elastic flows and yields a robust framework for studying long-time behavior of area-constrained elasticae.

Abstract

We derive an -gradient flow of the elastic energy which preserves the enclosed area of evolving planar curves. For this new sixth-order evolution equation, we prove a global existence result. Additionally, by penalizing the length, we show convergence to an area constrained critical point of the elastic energy.
Paper Structure (20 sections, 28 theorems, 190 equations, 1 figure)

This paper contains 20 sections, 28 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Derivation of the flow equation
  3. An appropriate Hilbert space setting
  4. The area preserving elastic flow as a gradient flow
  5. Basic properties of the flow
  6. Short time existence
  7. Global existence and subconvergence
  8. Geometrical lemmata and interpolation inequalities
  9. Bounds on the length
  10. Proof of global existence and subconvergence
  11. Convergence
  12. Stationary solutions as constrained critical points
  13. A constrained Łojasiewicz--Simon gradient inequality
  14. Proof of full convergence
  15. Stationary solutions
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $\lambda\geq0$, there exists a geometrically unique global smooth solution $\gamma\colon[0,\infty)\times{\mathbb{S}^1}\to\mathbb R^2$ of eq:ivp. If $\lambda>0$, there exists a family of smooth diffeomorphisms $\Phi(t)\colon{\mathbb{S}^1}\to{\mathbb{S}^1}$, $t\in(0,\infty)$, such that $\gamma

Figures (1)

  • Figure 1: Exemplary initial data for $\omega=0$ and $\mathcal{A}\neq0$ in \ref{['fig:statsolomega0Aneq0']} and for $\omega=1$ and $\mathcal{A}=0$ in \ref{['fig:statsolomega1A0']}.

Theorems & Definitions (57)

  • Theorem 1.1: Global existence and convergence
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 47 more