Weak compactness techniques and coagulation equations

TL;DR

This paper reviews the development of weak $L^1$-compactness techniques applied to Smoluchowski's coagulation equation, highlighting the progress in understanding weak solutions over the past twenty years.

Contribution

It summarizes the mathematical tools and results achieved in establishing a mature theory of weak solutions for coagulation equations using compactness methods.

Findings

Development of weak solution theory for coagulation equations

Application of weak $L^1$-compactness techniques

Comprehensive overview of mathematical tools used

Abstract

Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a blossoming of results in the last twenty years. In particular, the use of weak $L^1$-compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used.