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

Abstract

AbstractIn this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of Esposito, Heinze and Schlichting 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.