Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits
Authors
- Heinze, Georg
- Pietschmann, Jan-Frederik
- Schlichting, André
ORCID: 0000-0003-4140-491X
2020 Mathematics Subject Classification
- 35A15 35A35 35R02 60B10 65M08
Keywords
- PDEs on graphs and networks, dissipative evolution equations, variational methods applied to PDEs, gradient systems, evolutionary Gamma-convergence, EDP convergence, finite volume methods for initial value and initial-boundary value problems, involving PDE
DOI
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.
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