Mean-field Limit For The Keller-Segel System And The Theory Of Propagation Of Chaos

This thesis consists of two parts,in which we study the propagation of chaos for the N-particle chemotaxis system subject to Brownian diffusion in the Rd(d ? 2)space.The first part is to present a probabilistic proof of the distance between the exact microscopic and the approximate mean-filed dynamics,which leads to a derivation of the Keller-Segel equation from the microscopic-particle system.Specifically,With a blob size ? = hK(1/2<k<G),we prove a rate h|ln h| of convergence in lhp(p>d/1-k)norm up to a probability 1-hC|lnh|,where h is the initial grid size.In this case,the initial positions of the particles are taken on the lattice points.As for the second part,by assuming the initial data are identically independent distributed,we analyze the discrete-in-time method of approximating solutions of the Keller-Segel equation by the interacting particles subject to Brownian diffusion.This work contains some technical results on the degree of the approximation of solutions to the mean-field equation by random evolving systems.More precisely,with a blob size?-N-1/d(d+1)log(N),we prove the convergence rate between the solution to the Keller-Segel equation and the empirical measure of the random particle method under L2 norm in probability,where N is the number of the particles.