Stability of regular shrinkers in the network flow

TL;DR

This paper investigates the stability of self-similarly shrinking solutions, known as regular shrinkers, in network flow, revealing that most are unstable under perturbations, especially those with multiple enclosed regions.

Contribution

It provides a stability analysis of regular shrinkers, identifying which configurations are stable or unstable, and shows that most with multiple enclosed regions can be perturbed away.

Findings

Regular shrinkers with two or more enclosed regions are unstable.

Among single-enclosed-region shrinkers, 4-ray star, 5-ray star, fish, and rocket are unstable.

Most regular shrinkers are unstable under perturbations.

Abstract

The singularities of network flow are modeled by self-similarly shrinking solutions called regular shrinkers. In this paper, we study the stability of regular shrinkers. We show that all regular shrinkers with two or more enclosed regions can be perturbed away. Among the regular shrinkers with one enclosed region, 4-ray star, 5-ray star, fish, and rocket are unstable.