Modulated free energy and mean field limit
Didier Bresch1; Pierre-Emmanuel Jabin2; Zhenfu Wang3
1 LAMA CNRS UMR5127, Univ. Savoie Mont-Blanc 73376 Le Bourget du Lac, France
2 CSCAMM and departement of Mathematics, Univ. of Maryland College Park, MD, USA
3 Department of Mathematics, Univ. of Pennsylvania, Philadelphia PA, USA
Séminaire Laurent Schwartz — EDP et applications (2019-2020), Talk no. 2, 22 p.

This is the document corresponding to the talk the first author gave at IHÉS for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in mean-field theory in [4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in mean field limit theory by P.–E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the mean field limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates in [8] and [20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced in [4] first in the case σ N →0 when N→+∞ and then if σ>0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R. Acad Science Paris 357, Issue 9, (2019), 708–720] by the authors for more details.

Published online:
DOI: 10.5802/slsedp.135
Didier Bresch 1; Pierre-Emmanuel Jabin 2; Zhenfu Wang 3

1 LAMA CNRS UMR5127, Univ. Savoie Mont-Blanc 73376 Le Bourget du Lac, France
2 CSCAMM and departement of Mathematics, Univ. of Maryland College Park, MD, USA
3 Department of Mathematics, Univ. of Pennsylvania, Philadelphia PA, USA 
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Didier Bresch; Pierre-Emmanuel Jabin; Zhenfu Wang. Modulated free energy and mean field limit. Séminaire Laurent Schwartz — EDP et applications (2019-2020), Talk no. 2, 22 p. doi : 10.5802/slsedp.135. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.135/

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