Descrizione del progetto
Nuovi strumenti matematici per svelare le complesse dinamiche dei sistemi a molti corpi
Le equazioni differenziali alle derivate parziali sono il linguaggio di base attraverso cui può essere scritta la maggior parte delle leggi nel campo della fisica o dell’ingegneria, nonché uno degli strumenti matematici più importanti per la modellizzazione nelle scienze sociali. Tra le numerose aree di applicazione della modellizzazione cinetica nella biologia matematica, il progetto Nonlocal-CPD, finanziato dall’UE, concentrerà l’attenzione sulle equazioni differenziali alle derivate parziali per studiare un gran numero di «individui» che mostrano un «comportamento collettivo» e per ottenere da questi una «media» di informazioni. Gli studi si incentreranno inoltre sull’asintotica a lungo termine, sulle proprietà qualitative e sugli schemi numerici per la diffusione non lineare, nonché sulle equazioni idrodinamiche e cinetiche nella modellizzazione del comportamento collettivo dei sistemi a molti corpi.
Obiettivo
"This proposal focuses on the development of new mathematical tools to analyse theoretical, numerical and
modelling aspects of novel applications of nonlinear nonlocal aggregation-diffusion equations in Mathematical Biology and in classical problems of kinetic theory. Among the numerous areas of applications of kinetic modelling in Mathematical Biology, we will 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 my main mathematical goal.
The proposed 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. We will focus on the important case of the Landau equation with applications in weakly nonlinear plasmas by means of the gradient flow techniques. On the other hand, we showcase our tools in patterns and consensus by focusing on 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. This project connects with other areas of current interest in science and technology such as agent-based models in engineering: global optimization, clustering, and social sciences."
Campo scientifico
- natural sciencesbiological sciencesdevelopmental biology
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesbiological scienceszoologymammalogy
- natural sciencescomputer and information sciencesartificial intelligencecomputational intelligence
Parole chiave
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-ADG - Advanced GrantIstituzione ospitante
OX1 2JD Oxford
Regno Unito