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arXiv · 2402.13977

Relative entropy and modulated free energy without confinement via self-similar transformation

Matthew Rosenzweig·Sylvia Serfaty
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Abstract

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.

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Matthew Rosenzweig, Sylvia Serfaty. 2024-02-21. Relative entropy and modulated free energy without confinement via self-similar transformation. https://arxiv.org/abs/2402.13977

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