Empirical measures and Vlasov hierarchies

François Golse; Clément Mouhot; Valeria Ricci

doi:10.3934/krm.2013.6.919

Article Contents

2013, Volume 6, Issue 4: 919-943. Doi: 10.3934/krm.2013.6.919

This issue Previous Article The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits Next Article Convex analysis and thermodynamics

Empirical measures and Vlasov hierarchies

1.
Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz, 91128 Palaiseau Cedex
2.
University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce road, Cambridge CB30WA
3.
Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo

Received: August 2013

Revised: September 2013

Published: December 2013

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Abstract

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the $N$-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent $1$ on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, H. Neunzert, Math. Meth. Appl. Sci. 3 (1981), 445--455].

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Mathematics Subject Classification: Primary: 82C05; Secondary: 35F25, 28A33.

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