Convergence Analysis of a Stochastic Particle Approximation for Measure Valued Solutions of the 2D Keller-Segel System

The analysis of a stochastic interacting particle scheme for approximation of the measure valued solutions to the Keller-Segel system in 2D is continued. In previous work it has been shown that, in the limit of the regularized scheme when the number of particles N tends to infinity, solutions of the regularized Keller-Segel system are recovered. In the present work the limit is carried out, when the regularization parameter tends to zero, which requires an application of the framework of time dependent measures with defects, developed by Poupaud. The subsequent limit N → ∞ leads to a problem, which can be solved using a measure valued solutions of the KellerSegel system. However, for fundamental reasons it is impossible to make a rigorous statement about the equivalence of the two problems. Finally, a detailed description of the dynamics of a system comprising two particles is given in terms of strong solutions, i.e. sums of regular contributions and Delta distributions. It is shown that the particle distribution function is unbounded, and depending on the value of the assigned total mass, it either concentrates or it has an L∞ blow-up without concentration. Finally, there is some explanation, why strong solutions of systems with three or more particles remain not completely understood.