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arXiv · 2108.09878

Global-in-time mean-field convergence for singular Riesz-type diffusive flows

Matthew Rosenzweig·Sylvia Serfaty
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Abstract

We consider the mean-field limit of systems of particles with singular interactions of the type $-\log|x|$ or $|x|^{-s}$, with $0< s 0$, the convergence is global in time, and it is the first such result valid for both conservative and gradient flows in a singular setting on $\mathbb{R}^d$. The proof relies on an adaptation of an argument of Carlen-Loss to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in arXiv:1508.03377, arXiv:1803.08345, arXiv:2107.02592 making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution.

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Matthew Rosenzweig, Sylvia Serfaty. 2021-08-23. Global-in-time mean-field convergence for singular Riesz-type diffusive flows. https://arxiv.org/abs/2108.09878

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