Quantitative estimates for the mean field limit of systems with moderate interaction

• Applicant: Professorin Li Chen, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 547277619

The classical mean-field theory from statistical mechanics has been applied to analyze the collective behavior of dynamical systems comprising a large number of interacting particles, which appears in mathematical models in biology and social sciences. The introduction of moderate interactions makes it possible to rigorously derive the mean-field partial differential equations with singular force. This project aims to provide quantitative estimates in the mean-field limit discussion for systems with moderate interactions, and their further applications on both mean-field control problems and mean field systems coupled with fluid dynamical systems. The convergence rate estimate is going to be given through the relative entropy inequality and the convergence in probability of the particle trajectories. This helps also in establishing the further fluctuation analysis of mean-field system. Additional estimates are going to be developed in proving that the particle control problems converge to the control problem of the corresponding partial differential equations. In the case of coupling with fluid equations, uniform estimates from the fluid system play important roles. The focus of this project is to establish quantitative estimates on the partial differential equation level. Enough regularity of the PDE solution compromises the singularity induced by the interaction potential. This proposed project will enrich the theoretical results of mean-field limit with singular interactions.

• DFG Programme: Research Grants