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Propagation of chaos for large Brownian particle system with Coulomb interaction. (English) Zbl 1355.82025

Summary: We investigate a system of \(N\) Brownian particles with the Coulomb interaction in any dimension \(d\geq 2\), and we assume that the initial data are independent and identically distributed with a common density \(\rho _0\) satisfying \(\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\text{d}x<\infty \) and \(\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\text{d}x)\). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For \(d=2\), we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When \(N\rightarrow \infty \), the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson-Nernst-Planck equation of single component.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60J65 Brownian motion
35Q82 PDEs in connection with statistical mechanics
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
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