# Mean-field limit and quantitative estimates with singular attractive kernels

@article{Bresch2020MeanfieldLA, title={Mean-field limit and quantitative estimates with singular attractive kernels}, author={Didier Bresch and Pierre-Emmanuel Jabin and Zhenfu Wang}, journal={arXiv: Analysis of PDEs}, year={2020} }

This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the…

## 21 Citations

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## References

SHOWING 1-10 OF 40 REFERENCES

### Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

- MathematicsANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2021

We propose a new approach to obtain quantitative convergence of moderately interacting
particle systems to solutions of nonlinear Fokker-Planck equations with singular kernels. Our
result only…

### Mean field limit for Coulomb-type flows

- MathematicsDuke Mathematical Journal
- 2020

We establish the mean-field convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-coulombic Riesz…

### WASSERSTEIN DISTANCES FOR VORTICES APPROXIMATION OF EULER-TYPE EQUATIONS

- Mathematics
- 2009

We establish the convergence of a vortex system towards equations similar to the 2D Euler equation in vorticity formulation. The only but important difference is that we use singular kernel of the…

### Modulated free energy and mean field limit

- MathematicsSéminaire Laurent Schwartz — EDP et applications
- 2020

This is the document corresponding to the talk the first author gave at IH{E}S for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in…

### Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions

- Mathematics
- 2006

The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion…

### Quantitative estimates of propagation of chaos for stochastic systems with $$W^{-1,\infty }$$W-1,∞ kernels

- MathematicsInventiones mathematicae
- 2018

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the…

### Propagation of chaos for the 2D viscous vortex model

- Mathematics
- 2014

We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of…

### On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak–Keller–Segel model

- MathematicsComptes Rendus Mathematique
- 2019

### Quantitative estimates of propagation of chaos for stochastic systems with kernels

- Mathematics
- 2017

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the…