[Paper Review] A sixth order flow of plane curves with boundary conditions

James McCoy, Glen Wheeler+1 more|arXiv (Cornell University)|Oct 26, 2017

Geometric Analysis and Curvature Flows4 references3 citations

TL;DR

This paper studies a sixth-order curvature flow for plane curves with Neumann boundary conditions between parallel lines, proving that small-energy initial curves converge exponentially in $C^∞$ topology to straight line segments. The flow evolves curves via the $L^2$-gradient of the energy $\int k_s^2\,ds$, and under small initial energy, curvature and all its derivatives decay exponentially to zero, ensuring global existence and smooth convergence.

ABSTRACT

We show that small energy curves under a particular sixth order curvature flow with generalised Neumann boundary conditions between parallel lines converge exponentially in the smooth topology in infinite time to straight lines.

Motivation & Objective

• To analyze the long-time behavior of a sixth-order geometric flow for plane curves with Neumann-type boundary conditions.
• To establish global existence and smooth convergence of solutions under small initial energy.
• To characterize the asymptotic limit of such curves as straight line segments.
• To prove exponential decay of curvature and its derivatives in $L^2$ and pointwise norms.

Proposed method

• Formulate the $L^2$-gradient flow of the energy $E[\gamma] = \frac{1}{2}\int k_s^2\,ds$ for curves with boundary conditions.
• Impose Neumann boundary conditions: $k_s(\pm1,t) = k_{sss}(\pm1,t) = 0$ to ensure natural boundary terms vanish.
• Derive evolution equations for curvature and its derivatives using integration by parts and geometric identities.
• Use interpolation inequalities and $L^2$ bounds to control curvature derivatives uniformly in time.
• Establish exponential decay of $\|k_{ss}\|_{L^2}^2$ under a small energy assumption.
• Apply Hale-Raugel’s convergence theorem and stability analysis to conclude unique convergence to a horizontal line segment.

Experimental results

Research questions

• RQ1 Under what conditions does a sixth-order curvature flow with boundary conditions converge to a straight line?
• RQ2 How does the small energy assumption affect the long-time behavior of the flow?
• RQ3 What role do Neumann boundary conditions on curvature derivatives play in ensuring global existence and smoothness?
• RQ4 Can exponential decay of curvature and its derivatives be established for this sixth-order flow?
• RQ5 Is the limiting curve unique under the given flow and boundary conditions?

Key findings

• Solutions exist globally in time for initial curves with small energy and zero winding number.
• The $L^2$-norm of curvature and all its derivatives remain uniformly bounded for all time.
• The $L^2$-norm of $k_{ss}$ decays exponentially under the small energy condition.
• All curvature derivatives decay exponentially in both $L^2$ and pointwise $C^0$ norms.
• The curve converges exponentially in $C^\infty$ topology to a unique horizontal line segment.
• The limiting line segment lies within a bounded region, with its distance from the initial curve estimated a-priori.