# Evolution Equations on Co-evolving Graphs: Long-Time Behaviour and the Graph-Continuity Equation

## TL;DR

This paper studies how evolution equations behave on changing graphs and connects them to continuity equations to understand long-term behavior.

## Contribution

The paper establishes a new link between evolution equations on co-evolving graphs and nonlinear continuity equations.

## Key findings

- Weak solutions of the graph-continuity equation correspond to the push-forward of initial data through flow maps.
- Upwinding dynamics on graphs with monotonic velocity lead to long-time convergence to uniform mass distribution.

## Abstract

We focus on evolution equations on co-evolving infinite graphs and establish a rigorous link with a class of nonlinear continuity equations, whose vector fields depend on the graphs considered. More precisely, weak solutions of the so-called graph-continuity equation are shown to be the push-forward of their initial datum through the flow map solving the associated characteristics’ equation, which depends on the co-evolving graph considered. This connection can be used to prove contractions in a suitable distance, although the flow on the graphs requires a too limiting assumption on the overall flux. Therefore, we consider upwinding dynamics on graphs with pointwise and monotonic velocity and prove long-time convergence of the solutions towards the uniform mass distribution.