Uniqueness of self-similar solutions to the network flow in a given topological class

TL;DR

This paper establishes the uniqueness of expanding self-similar solutions and regular evolutions of network flows within fixed topological classes, using parabolic Allen-Cahn approximation methods.

Contribution

It proves the uniqueness of self-similar solutions and connected tree-like network evolutions in a fixed topological class, extending previous results with new approximation techniques.

Findings

Uniqueness of expanding self-similar solutions in a topological class

Uniqueness of regular evolutions of connected tree-like networks

Application of parabolic Allen-Cahn approximation to network flow

Abstract

In this paper we study the uniqueness of expanding self-similar solutions to the network flow in a fixed topological class. We prove the result via the parabolic Allen-Cahn approximation proved in \cite{triodginz}. Moreover, we prove that any regular evolution of connected tree-like network (with an initial condition that might be not regular) is unique in a given a topological class.