Graph-based nonlocal gradient systems and their local limitsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk I will discuss the evolution of a system of two nonlocally interacting species possibly with nonlinear mobility on a graph, which may be infinite. This evolution, which is induced by an upwind interpolation, can be rigorously understood as a Finslerian gradient flow. Weakening the notion of Minkowski norm and nonlocal gradient, the geometric interpretations and the analysis can be carried over to non-quadratic settings. The analytical studies are accompanied by numerical simulations on finite graphs of different shapes, showcasing phenomena such as aggregation of a species or the separation of different species.

Furthermore, in the quadratic setting with a single species and linear mobility, I will indicate how our non-symmetric graph gradient structures approximate a symmetric Otto-Wasserstein gradient structure by means of evolutionary Gamma-convergence. In particular, this implies existence of solutions to the nonlocal interaction equation on Euclidean space.