Global stability of traveling waves for an area preserving curvature flow with contact angle condition

TL;DR

This paper analyzes the long-term behavior of plane curves evolving under an area-preserving curvature flow with free endpoints on the x-axis, showing convergence to traveling waves under certain conditions.

Contribution

It establishes the global stability and convergence of convex embedded curves to traveling wave solutions in an area-preserving curvature flow with contact angle conditions.

Findings

Curves converge to traveling waves over time.

Convexity and boundedness ensure asymptotic stability.

Results apply to free-endpoint curvature flows with contact angles.

Abstract

We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that any moving curve converges to a traveling wave if the moving curve starts from an embedded convex curve and remains bounded in global time.