2024/02/21 by Matthew Rosenzweig, Sylvia Serfaty, Rosenzweig, Matthew +1 · 2 citations
Physics and Astronomy · #35Q70 #35Q82 #35Q92 #39B62 #82C22 #82C31 #82C40 #94A17 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.2402.13977
openalex publication_date 2024/02/21 · openalex created_date 2024/02/24 · openalex updated_date 2026/07/28
This note extends the modulated entropy and free energy methods for proving mean-field limits/propagation of chaos to the whole space without any confining potential, in contrast to previous work limited to the torus or requiring confinement in the whole space, for all log/Riesz flows. Our novel idea is a scale transformation, sometimes called self-similar coordinates in the PDE literature, which converts the problem to one with a quadratic confining potential, up to a time-dependent renormalization of the interaction potential. In these self-similar coordinates, one can then establish a Grönwall relation for the relative entropy or modulated free energy, conditional on bounds for the Hessian of the mean-field log density. This generalizes recent work of Feng-Wang arXiv:2310.05156, which extended the Jabin-Wang relative entropy method to the whole space for the viscous vortex model. Moreover, in contrast to previous work, our approach allows to obtain uniform-in-time propagation of chaos and even polynomial-in-time generation of chaos in the whole space without confinement, provided one has suitable decay estimates for the mean-field log density. The desired regularity bounds and decay estimates are the subject of a companion paper.