Graph‐to‐local limit for a multi‐species nonlocal cross‐interaction system

Abstract

In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of Benamou and Brenier to an arbitrary number of species. Our analysis relies on the observation that the graph dynamics form a gradient flow with respect to a non‐symmetric Finslerian gradient structure. Keeping the nonlocal interaction energy fixed, while localizing the graph structure, we are able to prove evolutionary Γ‐convergence to an Otto‐Wassertein‐type gradient flow with a tensor‐weighted, yet symmetric, inner product. As a byproduct this implies the existence of solutions to the multi‐species non‐local (cross‐)interaction system on the tensor‐weighted Euclidean space.

On evolution PDEs on co-evolving graphs

“…The result follows from taking the supremum with respect to t ∈ [0, T ] in ( 17) and (18), and letting n to ∞.…”

“…) (set of antisymmetric velocity fields), including possible dependence on the mass ρ. Among others, a driving example is given by V [21,19,18]. We shall focus on…”

“…Graphs give rise to interesting discrete-to-continuum problems, such as those considered in [25], for the total variation flow and the Allen-Cahn flow. Recently, [19,18] study graphs as nonlocal approximation of nonlocal interaction equations on graphs. This is linked to discrete-to-continuum evolution problems using tessellations, [16,24,30,29].…”