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Migrating elastic flows

Tomoya Kemmochi, Tatsuya Miura

TL;DR

This work addresses migration of elastic flows for open planar curves under a natural boundary condition, comparing length-preserving and length-penalized formulations. Using a variational framework with $\partial_t\gamma = -2\nabla_s^2\kappa-|\kappa|^2\kappa+\lambda\kappa$ and bending energy $B[\gamma]=\int|\kappa|^2 ds$, the authors prove long-time existence and convergence to elastica-like stationary states under pinned endpoints, and establish migration from the upper to the lower half-plane for small endpoint separation $\ell$ relative to length $L$. They construct well-prepared initial data and provide extensive numerical simulations showing loop-sliding and migration across both flow types, with a strong dependence on the nonlocal parameter $\lambda$ in the length-penalized case. The results illuminate how boundary constraints and nonlocal terms govern migration in higher-order geometric flows, offering the first analytic insights into elastic-flow migration and guiding future studies on related open-curve dynamics.

Abstract

Huisken's problem asks whether there is an elastic flow of closed planar curves that is initially contained in the upper half-plane but `migrates' to the lower half-plane at a positive time. Here we consider variants of Huisken's problem for open curves under the natural boundary condition, and construct various migrating elastic flows both analytically and numerically.

Migrating elastic flows

TL;DR

This work addresses migration of elastic flows for open planar curves under a natural boundary condition, comparing length-preserving and length-penalized formulations. Using a variational framework with and bending energy , the authors prove long-time existence and convergence to elastica-like stationary states under pinned endpoints, and establish migration from the upper to the lower half-plane for small endpoint separation relative to length . They construct well-prepared initial data and provide extensive numerical simulations showing loop-sliding and migration across both flow types, with a strong dependence on the nonlocal parameter in the length-penalized case. The results illuminate how boundary constraints and nonlocal terms govern migration in higher-order geometric flows, offering the first analytic insights into elastic-flow migration and guiding future studies on related open-curve dynamics.

Abstract

Huisken's problem asks whether there is an elastic flow of closed planar curves that is initially contained in the upper half-plane but `migrates' to the lower half-plane at a positive time. Here we consider variants of Huisken's problem for open curves under the natural boundary condition, and construct various migrating elastic flows both analytically and numerically.
Paper Structure (9 sections, 10 theorems, 43 equations, 10 figures)

This paper contains 9 sections, 10 theorems, 43 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Existence of migrating elastic flows
  3. Long-time existence and convergence
  4. Stationary solutions
  5. Flows converging to a unique arc
  6. Choice of well-prepared initial data
  7. Numerical examples of migrating elastic flows
  8. Length-preserving case
  9. Length-penalized case

Key Result

Theorem 1.1

There exists $c\in(0,1]$ with the following property: Let $L>0$ and $p_0,p_1\in \Lambda_0\subset \mathbf{R}^2$ such that $0<|p_0-p_1|< cL$. Then there exists a smooth solution $\gamma:\bar{I}\times[0,\infty)\to\mathbf{R}^2$ to the length-preserving elastic flow eq:flow of length $L$ under the natura

Figures (10)

  • Figure 1: Critical points; the upper arc and the upper loop.
  • Figure 2: Numerical result of Example \ref{['ex:constrained_asymmetric_long']}
  • Figure 3: Numerical result of Example \ref{['ex:constrained_asymmetric_short']}
  • Figure 4: Numerical result of Example \ref{['ex:constrained_symmetric']}
  • Figure 5: Numerical result of Example \ref{['ex:unconstrained_asymmetric_long']}
  • ...and 5 more figures

Theorems & Definitions (28)

  • Theorem 1.1: Migrating elastic flow
  • Theorem 2.1: DLP2014
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Basic properties of critical points
  • proof
  • Lemma 2.5
  • Proposition 2.6: Convergence to the lower arc
  • Lemma 2.7
  • proof
  • ...and 18 more