ANR-08-BLAN-0333-01

Scientific project

A group of scientist has emerged, who aims at creating a multinational and multidisciplinary research network at the interface of mathematics with sociology, economy, life sciences, physics and engineering sciences.

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.

**Keywords:** Collective behaviour phenomena; Nonlinear diffusion; Non-local interactions; Model Coupling; Kinetic/Hydrodynamic equations.

I - Socio-economic models

I.1. Models for opinion formation

Microscopic models of social and political phenomena describing collective behaviours and self-organization in a society can be framed within the methodologies of statistical mechanics. The basic graph model in opinion formation is represented by cellular automata, where lattice points are the agents of a community. Recently, other attempts have been developed to describe opinion formation by means of mean-field diffusive PDEs taking into account two aspects: the human tendency to settle conflicts, in which pairs of agents reach a compromise after exchanging opinions, and the diffusion process, accounting for the possibility of changing opinion through global access to information. The impact of specific community structures on large-scale behaviour is an important effect. In coarse-graining techniques such as homogenization, fine details of networks are lost, so the study of this connection is a non-trivial aspect. Problems of this kind also play a role in computational neuroscience, financial markets, voting dynamics and wireless networks.I.2. Agent based and mean field modelling in financial markets

Various strategies were recently proposed to go beyond equilibrium and completeness assumptions for financial markets and related contexts. A key concept that can be found in these approaches is to include in the modelling heterogeneity of traders and their interaction. This can be done via several microscopic approaches: differential games, transition rates, Brownian agents and coarse-grained to mean-field PDEs. Several previously unexplained macroscopic observations in financial markets, often called stylized facts, were captured by such models and further strong impact is to be expected, in particular due to improved analysis and simulation. Systems of nonlinear PDEs with possibly non-local interactions should be considered as in transport-reaction-diffusion systems. A strictly related problem concerns the dynamic of simple market economies and their influence on wealth distribution. The study of wealth distributions goes back to Pareto who studied the distribution of income among the population in western countries, finding an inverse power law. Mesoscopic models of an open market economy, using the binary interaction mechanism of trades, were recently introduced with average wealth not conserved due to random dynamics describing the spontaneous change of wealth via speculative investments. Different models have been considered starting from kinetic equations to asymptotic models of Fokker-Planck type.II - Models in life sciences

II.1. Aggregation phenomena of cell, animal, and human populations

Many phenomena in the life sciences, ranging from the microscopic level in cell biology (transport through membranes, chemotaxis, angiogenesis, tissue properties determined by intra-cellular network interactions) to the macroscopic level (swarming, herding of animal populations, motion of human crowds), exhibit a surprisingly similar structure, composed of a long-range attractive force (due to electrical, chemical or social interaction, for example) and short-range repulsion (due to finite size, for example). Various model approaches (random walks, cellular automata, Brownian agents) have been used to describe these phenomena, leading to mean-field PDEs, where the long-range attraction is incorporated in a nonlocal term and the short-range repulsion leads to nonlinear diffusions. On the other hand, the coupling of mathematical models in the life sciences and in the economic field remains to be fully explored. For instance, the question of fish stock handling in the oceans, where the coupling betweenII.2. Models in computational neuroscience

Modern computational neuroscience tries to explain the overall behaviour of neurons which interact through synaptic signalling. The modelling starts with systems of ODEs, both deterministic and stochastic, describing single cell dynamics in each neuron. On the other hand, collective behaviour of neuron ensembles is obtained by deriving PDEs as usually done in kinetic theory. The understanding of such collective behaviour is still at its early stages and the asymptotic analysis and numerical comparison to simplified models based on averaged quantities needs to be carried out. These techniques will lead to further understanding of complex questions in computational neuroscience, such as mean and variance conductance estimates in neurons, and average integrate-and-fire neural networks.III - Transport models with application in physics and engineering

III.1. Dynamics of granular materials

The qualitative behaviour of large ensembles of inelastic interacting particles is still far from being completely understood. Recently, the advances in the mathematical analysis of different models, their numerical simulation and the close contact with experiments have fostered interactions of team members with several physicists in this topic. New mathematical tools are being devised by various groups to tackle more realistic situations like hard-spheres dissipative Boltzmann-type equations and their hydrodynamic approximations. An interesting direction to explore is the sedimentation and erosion modelling in shallow-water models with applications to geophysical problems. Fluid-kinetic coupling naturally appears to try to cope with these highly nonlinear problems.III.2. Coagulation/fragmentation processes

Coalescence and break-up of particles is of fundamental importance in several physical and technological problems such as fuel droplets in Diesel engines, air pollution modelling, cloud formation, gas bubbles in volcanic magma, cell division modelling and blood thrombi aggregation in human arteries, to name but a few. These phenomena are usually modelled by kinetic type equations describing the particleIII.3. Fluid-particles interaction models

Sprays (droplets evolving in an underlying gas) and polymers (spring like molecules in a fluid) can be found in many industrial devices (pipelines, liquid crystals, etc.). They also constitute an important category of flows appearing in nature (clouds, viscoelastic fluids, etc.). The progress in the mathematical analysis and the numerical simulation of the equations of fluid mechanics (Euler or Navier-Stokes) and of the kinetic equations make possible the study of these complex flows. In particular, the interaction force-coupling between the microscopic and the macroscopic equations is decisive in many contexts. Macroscopic (fluid equations) and microscopic (kinetics) equations are also coupled in polymer complex fluids through the stress tensor, whose further understanding by numerical and analytical techniques is needed.III.4. Hydrodynamic and diffusive models in electromagnetism

The collective behaviour of ensembles with a large number of charged particles can be described by macroscopic models in several regimes. In the case of dominant scattering, hydrodynamic or diffusive models can be derived using the moment method. We will examine two classes of such moment models: the magneto-hydrodynamic (MHD) equations, which are based on a coupling of the magnetic field with the compressible Navier-Stokes or Euler equations; and energy-transport-type models, which describe the evolution of the density of charged carriers and their thermal energy, coupled to an electric field. Applications include astrophysical jets, plasma boundary-layer control and nuclear fusion. Both model classes represent daunting challenges when it comes to the development of mathematical theory and numerical methods.III.5. Nonlinear and nonlocal diffusion models in fluid mechanics

Flow in porous media is a truly multiscale phenomenon of great importance within the earth sciences. In a porous medium the physical flow occurs in a network of pore throats between individual sand grains on a microscale. However, when studying petroleum reservoirs, groundwater flow, and CO2 deposition, one is typically interested in flow patterns occurring on a scale of hundred to thousand meters and model equations must be derived based on the collective behaviour of all pore throats within representative elementary volumes. This leads to convection-diffusion equations with individual terms depending on the scales. Models with nonlocal diffusive or dispersive terms, arising for instance in shallow water theory or in high temperature ionized gases, will be also studied.