Relaxation of the flow of triods by Curve Shortening Flow via the vector-valued parabolic Allen-Cahn equation

TL;DR

This paper establishes a connection between vector-valued parabolic Allen-Cahn equations and the evolution of triod interfaces under curve shortening flow, providing a new analytical approach to interface dynamics.

Contribution

It introduces solutions to a vector-valued Allen-Cahn equation that approximate triod interfaces evolving via curve shortening flow as epsilon approaches zero.

Findings

Solutions approximate triod interfaces under flow as epsilon decreases

Provides a new analytical framework linking PDEs and geometric flows

Advances understanding of interface evolution in complex networks

Abstract

In this paper we find solutions $u_\epsilon$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $\epsilon \to 0$ develops as interface a given triod evolving under curve shortening flow.