Quantitative approximation of the Keller-Segel and Burgers equations by moderately interacting particles

Abstract

In this work we derive the rate of convergence for the empirical measure of a moderately interacting stochastic particle system towards the Keller-Segel and Burgers equations. In the case of the Keller-Segel equation on a $d$-dimensional torus, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some $L^1\cap L^p$ spaces with a rate of order $N^{-\frac{1}{2d+1}}$. The same holds for the genuine empirical measure in Wasserstein distance. The result holds in both subcritical and supercritical cases for the sensitivity parameter $\chi$ of the equation up to the maximal existence time of the PDE. This extends our previous work to moderately interacting particle systems without cutoff in the drift term. Then, we extend our method to Burgers equation in $\mathbb{R}$ and obtain a rate of convergence of the order $N^{-1/8}$.Comment: This is a preliminary version of the pape