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.

翻译：我们研究了一类空间上具有张量迁移性的非局部偏微分方程，该方程从局部化无穷图上的非局部动力学渐近推导得出。我们的策略依赖于两个方程的变分结构，它们分别是黎曼流和芬斯勒流。更确切地说，我们证明了图上的非局部相互作用方程的弱解收敛到欧几里得空间中具有张量迁移性的上述非局部相互作用方程的弱解。这凸显了图的一个有趣性质，即可以作为所研究方程的一种空间离散化方案。