Collective behaviour & diffusion: mathematical models and simulations – CBDif-fr

Recently a group of European scientists proposed an homonymous ESF project on Collective behaviour & diffusion (CBDif). French mathematicians were associated to the preparation the scientific project. This proposal is intended to fund the French participation to the project. French and other European participants have a long track of active and successful collaborations, through joint research project, bilateral contracts and networks. More specifically, the goal of this proposal is to • cover travel expenses for early stage researchers visiting French groups (1-2 months per visit), • cover travel expenses for French mathematicians participating to workshops organized in the framework of the ESF project, visiting a group of the ESF project or for invitations, • organize one workshop, in coordination with the ESF project. Although very strong, the ESF project is under evaluation and may eventually not be selected. In that case, the present project, CBDif-Fr, would have an even more important role, as it would structure the forces at the European level. The ESF project on 'Collective behaviour & Diffusion: mathematical models and simulations' (CBDif) aims at creating a multinational and multidisciplinary research network at the interface of mathematics with sociology, economy, life sciences, physics and engineering sciences. All partners feature a modern, application-oriented approach to ordinary and partial differential equations (ODEs and PDEs), incorporating core areas of physics and engineering, as well as emerging fields in social, economic and life sciences: asymptotic analysis is used for deriving and understanding hierarchies of models, while sound numerical analysis provides efficient predictive computer simulations for the targeted applications. Mathematical modelling using PDEs plays an increasing role in the aforementioned fields: multidimensional computations of complex multi-scale phenomena are now with-in reach; sophisticated nonlinear analysis deepens our understanding of increasingly complex models; computational results feed back into the modelling process, giving insight to detailed mechanisms which often cannot be studied by real life experiments. Among the numerous areas of applications, we will concentrate particularly on those examples which can be identified, at the modelling stage, as systems made out of a large number of 'individuals' which show a 'collective behaviour' and how to obtain from them 'averaged' information. The behaviour of the individuals can be typically modelled via stochastic ODEs or 'kinetic' type PDEs, while the average dynamics is usually described via continuum model systems of 'diffusion' or 'hydrodynamic' type. The interplay between the aggregate behaviour, (nonlocal, nonlinear) transport phenomena and nonlinear diffusion, is the main issue in the asymptotic analysis of these models. We emphasize that the mathematical methodologies described here serve to investigate important scientific problems sharing the common (PDE) modelling framework of this proposal. At the European level, the participants of CBDif were carefully chosen for their proven track record in successful participations and collaborations in projects of previous international and national framework programmes. French mathematicians have played an important role in the past collaborations among participants. This CBDif-Fr proposal covers the French contribution to the project and is proposed to the ANR in close coordination with the promoters of the ESF project (J. Carrillo, M. Burger and L. Pareschi).