Uniqueness of post-gelation solutions of a class of coagulation equations

TL;DR

This paper establishes the well-posedness of global solutions for a class of coagulation equations that undergo gelation, using PDE techniques, and connects these solutions to random graph models.

Contribution

It proves the existence and uniqueness of solutions across the gelation phase transition for various coagulation models, including classical and recent symmetric models.

Findings

Well-posedness of solutions before and after gelation.

Explicit computation of limiting concentrations in symmetric models.

Connection between coagulation solutions and random graph models.

Abstract

We prove well-posedness of global solutions for a class of coagulation equations which exhibit the gelation phase transition. To this end, we solve an associated partial differential equation involving the generating functions before and after the phase transition. Applications include the classical Smoluchowski and Flory equations with multiplicative coagulation rate and the recently introduced symmetric model with limited aggregations. For the latter, we compute the limiting concentrations and we relate them to random graph models.