On the discrete to continuous condensing aggregation equation: A weak convergence approach

TL;DR

This paper investigates the transition from discrete to continuous condensing aggregation equations, establishing convergence criteria, analyzing long-term behavior, and validating results through numerical experiments.

Contribution

It provides a generalized framework for discrete-to-continuous convergence in condensing aggregation models, including new existence, bounds, and dynamics results.

Findings

Convergence of discrete models to continuous equations under suitable conditions.

Existence and uniform bounds for solutions over finite time.

Numerical experiments confirm accuracy and convergence as discretization parameter approaches zero.

Abstract

In this article, we study the passage of limits from discrete to continuous condensing aggregation equation which comprises of Oort-Hulst-Safronov (OHS) equation together with inverse aggregation process. We establish the relation between discrete and continuous condensing aggregation equations in its most generalized form, where kinetic-kernels with respect to OHS and inverse aggregation equations are not always equal. Convergence criterion is proved under suitable a priori estimates by approximating the continuous equation through a sequence of discrete equations, which subsequently converges towards the solution of the continuous equation by weak compactness principles. Existence of solution to the discrete model and uniform bounds on different order moments over finite time under particular conditions on kinetic-kernels are investigated. We analyze long-time dynamics and blowup of the solution leading to mass-loss or gelation for specific kernels. Three numerical experiments show the accuracy and convergence of approximated solutions to the exact solution of the continuous equation when $\varepsilon$ approaches zero.