Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits
For researchers in gradient flows and network dynamics, this provides a rigorous framework for coupled ODE-PDE systems on metric graphs with reservoirs, though the contribution is incremental within the established EDP convergence framework.
The paper derives and analyzes gradient flows on metric graphs with reservoirs, establishing existence of solutions and rigorously proving convergence to reduced models under scaling limits via EDP convergence with embeddings. Numerical experiments confirm the theoretical results.
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.