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

Anupama Ghorai, Jitraj Saha

TL;DR

The paper develops a rigorous weak-convergence framework to connect discrete and continuous condensing aggregation equations, addressing the coupled OHS and inverse aggregation dynamics. It constructs discrete approximations with carefully discretized kernels, proves a priori bounds, and establishes convergence to a weak solution of the continuous CA, while also proving existence for the discrete problem. The authors analyze long-time behavior, showing moment propagation under mild kernel growth and gelation under stronger lower bounds, and validate the theory with numerical experiments showing convergence as the discretization parameter vanishes. This work fills a gap in the theory of CA models by providing convergence guarantees for the coupled discrete-to-continuous transition and illustrates practical implications through simulations. The results have implications for accurately modeling particle growth, gelation phenomena, and the computational bridging of discrete and continuous formulations in complex aggregation systems.

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.

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

TL;DR

The paper develops a rigorous weak-convergence framework to connect discrete and continuous condensing aggregation equations, addressing the coupled OHS and inverse aggregation dynamics. It constructs discrete approximations with carefully discretized kernels, proves a priori bounds, and establishes convergence to a weak solution of the continuous CA, while also proving existence for the discrete problem. The authors analyze long-time behavior, showing moment propagation under mild kernel growth and gelation under stronger lower bounds, and validate the theory with numerical experiments showing convergence as the discretization parameter vanishes. This work fills a gap in the theory of CA models by providing convergence guarantees for the coupled discrete-to-continuous transition and illustrates practical implications through simulations. The results have implications for accurately modeling particle growth, gelation phenomena, and the computational bridging of discrete and continuous formulations in complex aggregation systems.

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 approaches zero.

Paper Structure

This paper contains 20 sections, 14 theorems, 144 equations, 5 figures.

Table of Contents

  1. Introduction
  2. State of the art and motivation
  3. Main results
  4. Continuous regime
  5. Construction and assumption
  6. Approximations and reformulations
  7. Discrete regime: approximations and weak reformulations
  8. Continuous formulations for the discrete approximations:
  9. Weak convergence of solution
  10. A priori estimates
  11. Continuous regime: convergence and weak solution
  12. Existence of solution in discrete regime
  13. Long-time behavior and occurrence of gelation
  14. Propagation of Moments:
  15. Occurrence of gelation
  16. ...and 5 more sections

Key Result

Lemma 3.1

For each $t\geq0$ and $\varepsilon\in(0,1)$,

Figures (5)

  • Figure 1: Numerical and exact solutions at times $t=1$ and $t=2.5$ for test case \ref{['t_1']}
  • Figure 2: Error with test case \ref{['t_1']}
  • Figure 3: Numerical and exact solutions for different values of $\lambda$ for test case \ref{['t_2']}
  • Figure 4: Numerical and exact solutions at times $t=1$ and $t=2.5$ for $M=3$ for test case \ref{['t_3']}
  • Figure 5: Error with test case \ref{['t_3']} for $M=3$

Theorems & Definitions (29)

  • Definition 2.1: Weak solution
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 19 more