On the basin of attraction for the free boundary free elastic flow

TL;DR

The paper analyzes the free boundary free elastic flow, proving that planar curves between two parallel lines converge to a straight line if the initial scale-invariant energy is below a sharp threshold of $1.9615\pi$, by leveraging a critical interpolation inequality with constant $C_0\approx0.162278$ and a global a priori framework. The authors derive the flow’s evolution equations, establish local well-posedness via maximal $L^p$-regularity, and obtain uniform bounds that yield global existence under the basin condition. They also demonstrate that the conjectured $2\pi$ threshold cannot be attained and discuss critical initial data, including a near-critical curve whose dynamics illustrate the basin boundary. Numerical simulations corroborate the analytical picture, showing transitions between attraction to a horizontal line and growth, and revealing long-time tendencies toward a half-period of Bernoulli’s lemniscate as an attractor in rescaled variables. Together, these results clarify the stability landscape of free boundary elastic flows and illuminate potential asymptotic states beyond straight lines.

Abstract

The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting parallel lines perpendicularly. In this article we prove that the straight line has, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level $1.9615\, π$. We show that our method of proof cannot be pushed to the previously conjectured level $2π$, and in addition present numerical evidence that this conjecture may in fact be false.

Figures (12)

• Figure 1: A visualisation of the statement of Theorem \ref{['TMmain']}. Initial data $\gamma_0=\gamma(\cdot,0)$ satisfying \ref{['EQmtic']} are driven by the flow to the horizontal line $\gamma(\cdot,\infty) := \lim_{t\to\infty}\gamma(\cdot,t)$.
• Figure 2: Euler's rectangular elastica in $\mathcal{X}$ with one, two and three half-periods. With modulus $k=1/{\sqrt2}$, an arclength parametrisation $\gamma=(\gamma^1,\gamma^2):[0,L]\to\mathbb{R}^2$ is $\gamma^1(s)=\frac{1}{{b}}(2E(\operatorname{AM}({b} s,k),k)-{b} s)\,, \gamma^2(s)=-\frac{\sqrt2}{{b}}\,\operatorname{cn}({b} s,k),$ where ${b}=m \Gamma(\tfrac34)^2/\sqrt{\pi}$ depends on the number of half-periods $m$, and where each curve has the same length $L=\tfrac12(\Gamma(\tfrac14)/\Gamma(\tfrac34))^2\approx 4.37688.$ See Appendix \ref{['AA']} for the precise details. For the scale-invariant energy it holds that $\hat{\mathcal{E}}[\gamma]=2\pi m^2$. In particular, the left image has $\hat{\mathcal{E}}[\gamma]=2\pi$, and so more energy than is allowed by the hypothesis \ref{['EQmtic']} of Theorem \ref{['TMmain']}.
• Figure 3: The function $Q$, together with the constants $\frac{3}{2\pi^2}$, $\frac{1}{2\pi}$, and $0.162278$.
• Figure 4: The function $\tilde{Q}$, together with the constant $\frac{1}{2\pi}$.
• Figure 5: The initial curve $\gamma_c$, which is possibly not attracted by the straight lines.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Conjecture 3
• Lemma 4
• Lemma 5
• proof
• Remark 1: Trace space for maximal $L^p$-regularity
• Theorem 6
• proof
• Remark 2