Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits

TL;DR

This paper rigorously analyzes gradient flow equations on metric graphs with reservoirs, establishing existence, multiscale limits, and numerical validation for systems modeling mass exchange and storage.

Contribution

It introduces a formalism for coupled PDE-ODE systems on graphs, proves existence of solutions, and demonstrates convergence to reduced models via EDP framework.

Findings

Existence of solutions for coupled PDE-ODE systems on graphs.

Rigorous multiscale convergence to simplified gradient flows.

Numerical simulations confirming theoretical predictions.

Abstract

We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such systems of coupled ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs. Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling.