voraussichtliche Laufzeit: 01.12.2024–30.11.2025
Schrödinger Fellowship
Project leader: Tobias WÖHRER

The project focuses on global dynamics for interacting particles on large networks, studied through entropy methods. The models under consideration have applications in areas such as synchronization phenomena, opinion dynamics, biomechanics and the analysis of artificial neural network algorithms. We investigate the mean-field limits of such dynamics which are described by diffusive nonlinear and nonlocal partial differential equations (PDEs). In real-world scenarios – such as social networks, power grids or neural networks – heterogeneous interaction structures are ubiquitous. These structures range from sparse to intermediate to dense graphs, each requiring distinct mathematical treatments in traditional approaches. To unify the approach for a wide range of such structures, we apply the graph limit framework of graphons and the recently developed generalization to graph operators (graphops). The project demonstrates how these graph limit theories integrate seamlessly into existing PDE analysis techniques. In particular, we employ entropy methods to establish global stability and derive explicit large-time convergence estimates. An emerging challenge in this context is the regularity of solutions in the graph variable for such graph-limit PDEs.