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Diffusive Partial Differential Equations with Nonlocal Interaction in Biology and Social Sciences

Periodic Report Summary 1 - DIFNONLOC (Diffusive Partial Differential Equations with Nonlocal Interaction in Biology and Social Sciences)

The “DifNonLoc” project is focused on the mathematical theory of interdisciplinary models arising in biology, social sciences, and real world applications. Animal swarming and chemotactical movements in biology, pedestrian movements and opinion formation in sociology are the main examples. These models are mathematically posed as partial differential equations (PDEs) of transport type, seen as the “macroscopic” counterpart of a system of many agents interacting at a micro-scale. Such models are “nonlocal” in that they describe long-range interactions among the agents. Moreover, they also feature “diffusion” terms, for example as a result of a volume-filling effect. For those models, “DifNonLoc”
- addresses mathematical problems, such as “existence of solutions” and “long-time behaviour”,
- intersects with many areas of applied mathematics , such as “optimal transport theory” and “nonlinear conservation laws”,
- provides an interpretation of the results in their applied setting.

The Researcher in charge of the project (Dr. Marco Di Francesco) has achieved several scientific results in the first two years of “DifNonLoc”, in collaboration with leading figures in the field such as J. A. Carrillo (Imperial College) and M. Burger (University of Muenster) among others. He has also supervised the PhD dissertation of Dr. Simone Fagioli (University of L’Aquila). The overall work carried out in this period led to 9 scientific papers published in highly-rated scientific journals in the field, with the joint paper with M.D. Rosini (ICM Warsaw) published on “Archive for Rational Mechanics and Analysis” standing out as the main one. A thematic week on “Microscopic descriptions and mean-field equations in physics and social sciences” was organised at the University of Bath in May 2014.

The main scientific achievements of this project in the first two years are
- A systematic theory for nonlocal interaction systems with applications to predator-swarm systems and to “kinetic” models in opinion formation, focused on the emergence of “collective behaviour” for systems with many species. These results are contained in the PhD dissertation of Simone Fagioli.
- A new rigorous mathematical framework on a “particle aproximation” for nonlinear conservation laws, which poses new perspectives in the numerical study of transport models in traffic flow and pedestrian movements.
- The analysis of the interplay between mathematical models for pedestrians movements and the theory of “mean field games” by Lasry and Lions.

The Researcher worked as Reader at the University of Bath during the first two years of this project. In recognition of his scientific profile, and in acknowledgment of the strong impact of his research work on the local scientific community, in 2013 the University of Bath asked the Researcher to direct the local MSc programme “Modern Applications of Mathematics”. Due to the significant improvement of his scientific and academic record in these two years, the Researcher received the “Italian National Scientific Qualification” (“Abilitazione Scientifica Nazionale”) as Full Professor in Mathematical Analysis and Mathematical Pysics, and was hired as Associate Professor at the University of L’Aquila later on.