The Total Variation Flow in Metric Graphs

TL;DR

This paper investigates the Total Variation Flow on metric graphs, establishing foundational properties, existence, uniqueness, and explicit solutions, thereby advancing the mathematical understanding of flow dynamics in network-like structures.

Contribution

It introduces a rigorous framework for Total Variation Flow in metric graphs, including definitions, properties, and explicit solutions, which were previously unexplored.

Findings

Solutions reach the initial data mean in finite time

Existence and uniqueness of solutions are proven

Explicit solutions are derived

Abstract

Our aim is to study the Total Variation Flow in Metric Graphs. First, we define the functions of bounded variation in Metric Graphs and their total variation, we also give an integration by parts formula. We prove existence and uniqueness of solutions and that the solutions reach the mean of the initial data in finite time. Moreover, we obtain explicit solutions.