Nonlinear stability of stationary solutions for curvature flow with triple junction

TL;DR

This paper establishes the nonlinear stability of stationary solutions in curvature flow networks with triple junctions, proving exponential stabilization and providing existence and energy estimates for solutions.

Contribution

It extends linear stability criteria to nonlinear stability and demonstrates exponential stabilization for curvature flow networks with triple junctions.

Findings

Linear stability criterion is sufficient for nonlinear stability.

Existence of classical smooth solutions is proven.

Exponential stabilization of the network is achieved.

Abstract

In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.