Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent

TL;DR

This paper establishes conditions for existence and uniqueness in Smoluchowski's coagulation equation, constructs a non-uniqueness example, and demonstrates the stochastic coalescent's convergence to the equation's solution.

Contribution

It provides comprehensive criteria for existence and uniqueness, presents a counterexample of non-uniqueness, and links stochastic coalescent processes to the deterministic equation.

Findings

Conditions for existence and uniqueness are identified.

A non-uniqueness example is constructed.

Stochastic coalescent converges to the deterministic solution.

Abstract

Sufficient conditions are given for existence and uniqueness in Smoluchowski's coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of non-uniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski's equation.