Coagulation equations for aerosol dynamics

TL;DR

This paper analyzes the mathematical properties of Smoluchowski's coagulation equation in aerosol dynamics, establishing well-posedness, stationary solutions, and exploring multi-component particle systems with explicit solutions and mass localization phenomena.

Contribution

It provides new well-posedness results, existence and nonexistence of stationary solutions, and explicit solutions for multi-component coagulation equations with constant kernels.

Findings

Well-posedness for a broad class of kernels

Conditions for existence and nonexistence of stationary solutions

Explicit solutions and mass localization in multi-component systems

Abstract

Binary coagulation is an important process in aerosol dynamics by which two particles merge to form a larger one. The distribution of particle sizes over time may be described by the so-called Smoluchowski's coagulation equation. This integrodifferential equation exhibits complex non-local behaviour that strongly depends on the coagulation rate considered. We first discuss well-posedness results for the Smoluchowski's equation for a large class of coagulation kernels as well as the existence and nonexistence of stationary solutions in the presence of a source of small particles. The existence result uses Schauder fixed point theorem, and the nonexistence result relies on a flux formulation of the problem and on power law estimates for the decay of stationary solutions with a constant flux. We then consider a more general setting. We consider that particles may be constituted by different chemicals, which leads to multi-component equations describing the distribution of particle compositions. We obtain explicit solutions in the simplest case where the coagulation kernel is constant by using a generating function. Using an approximation of the solution we observe that the mass localizes along a straight line in the size space for large times and large sizes.