Prof. J.A. Carrillo's Research Group

Aim and scope:

Our group focuses on the development of new mathematical tools to analyse theoretical, numerical and  modelling aspects of novel applications of nonlinear nonlocal aggregation-diffusion and kinetic equations in areas of Science, Engineering and Social Sciences. Among the numerous areas of applications of kinetic modelling, we concentrate on phenomena identified, at the modelling stage, as systems involving a large number of "individuals" showing "collective behaviour" and how to obtain "averaged" information from them. Individuals behavior can be modelled via stochastic/deterministic ODEs from which one obtains mesoscopic/macroscopic descriptions based on mean-field PDEs leading to continuum mechanics, hydrodynamic and/or kinetic systems. Understanding the interplay between the interaction behaviour (nonlocal, nonlinear), the diffusion (nonlinear), the transport phenomena, and the synchronization is our main mathematical goal.

 

The present research is centred on developing tools underpinning the analysis of long time asymptotics, phase transitions, stability of patterns, consensus and clustering, and qualitative properties of these models. On the other hand, designing numerical schemes to accurately solve these models is key not only to understand theoretical issues but also crucial in applications. Current research lines of the group include the Landau equation with applications in weakly nonlinear plasmas by means of the gradient flow techniques, zebra fish patterning formation as example of spontaneous self-organisation processes in developmental biology, and grid cells for navigation in mammals as prototype for the synchronization of neural networks. Our research connects with other areas of current interest in science and technology such as agent-based models in engineering: global optimization, clustering, and social sciences.

 

Our research is funded the Advanced Grant Nonlocal-CPD: "Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization" of the European Research Council Executive Agency (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 883363) and the projects EP/T022132/1 "Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging" and EP/V051121/1 "DMS-EPSRC: Stability Analysis for Nonlinear Partial Differential Equations across Multiscale Applications" of the Engineering and Physical Sciences Research Council (EPSRC, UK).