Open Access

Georg Heinze

• In this thesis, we study systems of multiple nonlocally interacting species on a large class of graphs, ranging from graphs to continuous graphs or graphons. The proposed model is based on the theory of metric gradient flows and provides a unified upwind-based framework, including not only concave mobilities but also non-1-homogeneous kinetic relations. Exploiting the system’s gradient flow nature, we prove the existence of measure-valued solutions by establishing a rigorous link to a variational formulation in a quasi-metric Finslerian setting. In addition, the behavior of the arising dynamics is explored in numerical and analytical case studies, displaying phenomena such as the formation of patterns, the aggregation of one species, or the separation of different species. Finally, employing ideas of evolutionary Γ-convergence, we prove for linear mobilities and 1-homogeneous kinetic relations that solutions of nonlocal systems, defined on a suitable family of graphons, converge toIn this thesis, we study systems of multiple nonlocally interacting species on a large class of graphs, ranging from graphs to continuous graphs or graphons. The proposed model is based on the theory of metric gradient flows and provides a unified upwind-based framework, including not only concave mobilities but also non-1-homogeneous kinetic relations. Exploiting the system’s gradient flow nature, we prove the existence of measure-valued solutions by establishing a rigorous link to a variational formulation in a quasi-metric Finslerian setting. In addition, the behavior of the arising dynamics is explored in numerical and analytical case studies, displaying phenomena such as the formation of patterns, the aggregation of one species, or the separation of different species. Finally, employing ideas of evolutionary Γ-convergence, we prove for linear mobilities and 1-homogeneous kinetic relations that solutions of nonlocal systems, defined on a suitable family of graphons, converge to solutions of a system of nonlocal interaction equations in Euclidean space. What is more, a similar approximation is possible even after introducing tensor-valued anisotropies into the limiting geometry. Since the upwind-induced prelimit geometries are of non-symmetric Finslerian nature, while the limiting geometry is of Riemannian type, our graphon-to-local limit can also be viewed as a non-symmetric-to-symmetric limit of gradient structures.…