Graph-to-local limit for the nonlocal interaction equation

TL;DR

This paper demonstrates that solutions of nonlocal interaction equations on graphs converge to solutions of a continuum nonlocal PDE with tensor-mobility, revealing the graph as a potential discretization tool.

Contribution

It establishes a rigorous link between graph-based nonlocal dynamics and continuum PDEs with tensor-mobility, using variational and gradient flow frameworks.

Findings

Weak solutions on graphs converge to continuum solutions

Graph structures can discretize nonlocal PDEs with tensor-mobility

Highlights the potential of graphs as discretization tools for PDEs

Abstract

We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.