Singular flows with time-varying weights
Immanuel Ben Porat, José A. Carrillo, Pierre-Emmanuel Jabin
TL;DR
This work extends the mean-field analysis of singular Coulomb-type flows to the setting where interaction weights evolve in time. By marrying a renormalized modulated-energy framework with novel functional inequalities tailored to time-dependent source terms, the authors establish global well-posedness for the mean-field PDE $\partial_t\mu - \nabla\cdot(\mu\mathbb{J}\nabla V\star\mu)=h[\mu]$ and global well-posedness for the associated particle system with evolving weights. The main technical advance is a time-dependent commutator estimate that controls off-diagonal interactions and a regularized-energy argument that bounds the cross-term involving $h[\mu]$, enabling a Grönwall-type convergence of the empirical measure $\mu_N$ to $\mu$ in the mean-field limit. Overall, the paper provides a rigorous framework for mean-field limits with evolving weights in high dimensions, contributing to the theory of nonlocal transport with source terms and informing models of time-dependent opinion or agent-influence dynamics.
Abstract
We study the mean field limit for singular dynamics with time evolving weights. Our results are an extension of the work of Serfaty \cite{duerinckx2020mean} and Bresch-Jabin-Wang \cite{bresch2019modulated}, which consider singular Coulomb flows with weights which are constant time. The inclusion of time dependent weights necessitates the commutator estimates of \cite{duerinckx2020mean,bresch2019modulated}, as well as a new functional inequality. The well-posedness of the mean field PDE and the associated system of trajectories is also proved.