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arXiv · 2504.10446

Evolution equations on co-evolving graphs: long-time behaviour and the graph continuity equation

José Antonio Carrillo·Antonio Esposito·László Mikolás
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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.

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José Antonio Carrillo, Antonio Esposito, László Mikolás. 2025-04-14. Evolution equations on co-evolving graphs: long-time behaviour and the graph continuity equation. https://arxiv.org/abs/2504.10446

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