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

Georg Heinze, Jan-Frederik Pietschmann, André Schlichting

TL;DR

The paper develops a variational gradient-flow framework for diffusion on finite metric graphs with mass reservoirs at vertices, encoding both edge diffusion and edge-vertex exchange via a coupled energy-dissipation structure. It proves existence by a microscopic discretization and then establishes several multiscale limits (Kirchhoff, fast-edge diffusion, and combinatorial graph) through EDP convergence with embeddings, providing limit gradient flows on reduced graphs. Numerical simulations on a triangle validate the theory, demonstrate short-time behavior for non-well-prepared data, and illustrate the convergence of entropies and curve distances under rescaling. Overall, the work connects a detailed edge-vertex gradient system to simpler gradient flows on reduced graph structures, offering a rigorous pathway between continuous metric-graph dynamics and discrete graph limits with potential extensions to nonlocal interactions.

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.

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

TL;DR

The paper develops a variational gradient-flow framework for diffusion on finite metric graphs with mass reservoirs at vertices, encoding both edge diffusion and edge-vertex exchange via a coupled energy-dissipation structure. It proves existence by a microscopic discretization and then establishes several multiscale limits (Kirchhoff, fast-edge diffusion, and combinatorial graph) through EDP convergence with embeddings, providing limit gradient flows on reduced graphs. Numerical simulations on a triangle validate the theory, demonstrate short-time behavior for non-well-prepared data, and illustrate the convergence of entropies and curve distances under rescaling. Overall, the work connects a detailed edge-vertex gradient system to simpler gradient flows on reduced graph structures, offering a rigorous pathway between continuous metric-graph dynamics and discrete graph limits with potential extensions to nonlocal interactions.

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.

Paper Structure

This paper contains 24 sections, 30 theorems, 202 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Abstract gradient systems and EDP convergence with embedding
  3. Gradient flow interpretation and energy dissipation principle
  4. Existence via microscopic derivation
  5. Multiscale limits
  6. Numerical simulations
  7. Related work
  8. Notation
  9. Gradient systems and EDP convergence with embeddings
  10. Gradient structure over metric graphs with reservoirs
  11. Continuity equation structure
  12. Dissipation potentials
  13. Energy dissipation principle
  14. Microscopic derivation from pure jump process
  15. Microscopic model
  16. ...and 9 more sections

Key Result

Corollary 2.6

Let $({\mathsf M}^\varepsilon,{ 0.70\hbox{$\m@th\bigtriangledown$} }^\varepsilon, {\mathcal{E}}^\varepsilon,{\mathcal{R}}^\varepsilon) \xrightarrow{{\mathsf{EDP}}} ({\mathsf M}^0,{ 0.70\hbox{$\m@th\bigtriangledown$} }^0, {\mathcal{E}}^0,{\mathcal{R}}^0)$. Assume $({\mat Then, there exists an EDP solution $(\rho^0,j^0)\in{\mathsf{CE}}^0$ in the sense of Definition def:

Figures (8)

  • Figure 1.1: Example of a complete three-state metric graph ${\mathsf M}_3\coloneqq({\mathsf V}_3,{\mathsf E}_3,{\mathsf L}_3)$ defined in terms of the nodes ${\mathsf V}_3\coloneqq\{*\}{{\mathsf v}_1,{\mathsf v}_2,{\mathsf v}_3}$, edges ${\mathsf E}_3\coloneqq\{*\}{e_1,e_2,e_3}$ with $e_1\coloneqq{\mathsf v}_1{\mathsf v}_2$, $e_2\coloneqq{\mathsf v}_2{\mathsf v}_3$, $e_3\coloneqq{\mathsf v}_3{\mathsf v}_1$, edge lengths $\ell_3:{\mathsf E}_3\to (0,\infty)$ and metric edges ${\mathsf L}_3$ as in \ref{['e:def:metric_edges']}.
  • Figure 1.2: Left: The extended graph $(\hat{{\mathsf V}},\hat{{\mathsf E}})$ constructed from the complete three-state metric graph illustrated in Figure \ref{['fig:sketch_graph']} with the rescaled rates from \ref{['eq:scaling:TerminalLimit']}, that is a high rate of leaving the contracted metric edges $e\in {\mathsf E}$. Right: The limit for $\varepsilon\to 0$, which leads to reduced dynamics on a the three-state combinatorial graph $({\mathsf V}_3,{\mathsf E}_3)$ with harmonic averaged rates given in \ref{['eq:intro:TerminalLimit']}.
  • Figure 4.1: Sketch of the discretization of the edges into new vertices. Blue: Intervals $I^e_{n,k}$.
  • Figure 6.1: Sketch of the triangular graph used in all numerical experiments.
  • Figure 6.2: Evolution of the measures $\upgamma^\varepsilon = u^\varepsilon\upomega^\varepsilon$ for the prelimit system with rescaling $\omega^\varepsilon_{\mathsf v} \coloneqq \varepsilon \omega_{\mathsf v}$ and ${\mathscr k}^{\varepsilon,e}_v = {\mathscr k}^{e}_v / \varepsilon$ with different values of $\varepsilon$ as well as for the discrete Kirchhoff system \ref{['eq:discrete_u_interior_kirchhoff']}--\ref{['eq:discrete_vertex_kirchhoff']}.
  • ...and 3 more figures

Theorems & Definitions (79)

  • Definition 2.1: Continuity equation
  • Definition 2.2: Gradient system in continuity equation format
  • Definition 2.3: EDP solution to gradient system
  • Corollary 2.6
  • proof
  • Remark 3.2: Poincaré inequality
  • Definition 3.3: Continuity equation on metric graphs with node reservoirs
  • Lemma 3.4: Well-posedness of ${\mathsf{CE}}$
  • proof
  • Definition 3.5: Abstract formulation of ${\mathsf{CE}}$
  • ...and 69 more