Propagation Of Chaos For The Large Brownian Particle Systems With Newton And Coulomb Interactions

This thesis consists of three parts for the study of propagation of chaos for the large Brownian particle systems with Newton and Coulomb interactions in the Rd (d≥2) space.The first part considers the N-particles interacting system with Newton potential aggregation and Brownian motions. Assuming that the initial data are independent and identically distributed (i.i.d.) with common probability density function ρ0∈= L∞(Rd)∩ L1(Rd, (1+|x|)dx), this thesis proves the following results:(a) The "propagation of chaos" for this interacting system with a cut-off parameter ε～ (lnN)-1/d and the corresponding mean-field equation is the Keller-Segel equation. (b) The Yudovich type uniqueness and stability in Wasserstein metric with initial data for the weak solution to Keller-Segel equation.(c) For d= 2, if 8πv> 1, the "propagation of chaos" is valued globally in time. On the other hand, if 8πv<1, the expectation of the collision time for the interacting particle system is bounded by 2πVar{X10}/1-8πv. For d≥3, if ρ0(x) is bounded by a univer-sal constant in Ld/2 norm, then the "propagation of chaos" is also valued globally in time. The second part is concerned with the numerical method for solving the Keller-Segel equation. A random particle blob scheme is introduced and a rigorous analysis of the convergence for this scheme is given which is mainly based on the stability of Wasserstein metric for weak solutions to the Keller-Segel equation.The third part presents a simple proof of the non-collision between particles al-most surely (a.s.) for a system of N Brownian particles with Coulomb interaction s-ince the potential is positive, and then the wellposedness of this system is obtained. In two dimension, assuming that the initial data are i.i.d. and have a common densi-ty ρ0∈= L∞(Rd)∩L1(Rd, (1+|x|2)dx),it is also rigorously proved the "Propagation of chaos" for this interacting system and showed that the mean-field equation is the Poisson-Nernst-Planck (PNP) equation.