Coagulation, diffusion and the continuous Smoluchowski equation

TL;DR

This paper derives the continuous Smoluchowski PDE from a microscopic Brownian particle model, introducing a new correlation bound that simplifies the derivation and discusses implications for solution blow-up.

Contribution

It presents a novel derivation of the continuous Smoluchowski PDE from microscopic models using a correlation bound, extending previous discrete derivations.

Findings

Derived the continuous Smoluchowski PDE as a kinetic limit.

Introduced a correlation-type bound simplifying the derivation.

Discussed scaling and potential blow-up of solutions.

Abstract

The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.