Rates of convergence for Smoluchowski's coagulation equations

TL;DR

This paper derives nearly optimal convergence rates to self-similar solutions for Smoluchowski's coagulation equations with specific kernels, using a probabilistic approach similar to the Berry-Esséen theorem, under minimal initial data assumptions.

Contribution

It introduces a simple probabilistic method to establish convergence rates for Smoluchowski's equations with three kernels, requiring only finite moments of initial data.

Findings

Achieves nearly optimal convergence rates for kernels 2, x + y, and xy.

Method applies to monodisperse initial data.

Requires minimal assumptions on initial data.

Abstract

We establish nearly optimal rates of convergence to self-similar solutions of Smoluchowski's coagulation equation with kernels $K = 2$, $x + y$, and $xy$. The method is a simple analogue of the Berry-Ess\'een theorem in classical probability and requires minimal assumptions on the initial data, namely that of an extra finite moment condition. For each kernel it is shown that the convergence rate is achieved in the case of monodisperse initial data.