The proposal addresses complex and nonlocal systems, arising in a variety of contexts, ranging from complex fluids to social networks. The study of mixed systems in which microscopic objects influence their macroscopic environment is a fundamental component of nonlinear science. The project considers nonlinear PDE models in which the microscopic part is modelled by kinetic equations derived from underlying stochastic equations, coupled with macroscopic fluid equations. The proposal aims at basic questions of singularity formation, existence and uniqueness or lack thereof for weak solutions, mixing, long time behavior and statistical solutions. The proposal is concerned also with the study of active networks, natural or artificial networks that evolve in response to network-dependent processes. The overarching goal of the proposal is to study the dynamical effects of nonlocal interaction and feedback on evolving complex systems.
The proposal addresses problems that arise in the modeling, computation and theory of complex nonlocal systems. Such systems occur in a great variety of circumstances, ranging from complex matter, such as blood or intracellular fluids, to complex relational systems, such as social networks. Complex matter is characterized by the presence of complex, interacting networks of very small objects being embedded, influenced, and in turn influencing a matrix or a solvent. The basic understanding of these interactions is crucial for progress in applications such as the manufacturing of materials with extreme properties or the delivery of drugs at the cellular level. Diffusion of information with feedback through such networks is of great importance for the study of basic biological processes and also for the study of artificial networks. The aim of the proposal is to uncover fundamental mechanisms of system-size qualitative changes emerging from local and nonlocal interactions.
