The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation

Jacek Banasiak; Luke O. Joel; Sergey Shindin

doi:10.3934/krm.2019040

Article Contents

2019, Volume 12, Issue 5: 1069-1092. Doi: 10.3934/krm.2019040

This issue Previous Article A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication Next Article Differentiability in perturbation parameter of measure solutions to perturbed transport equation

The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation

1.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa & Institute of Mathematics, Technical University of Łódź, Łódź, Poland
2.
School of Mathematics, Statistics and Computer Science, University of Kwazulu-Natal, University of Kwazulu-Natal, Durban, South Africa

^* Corresponding author: J. Banasiak
^* Corresponding author: J. Banasiak

Received: March 2018

Published: July 2019

The research was supported by the NRF grants N00317 and N102275, and the National Science Centre, Poland, grant 2017/25/B /ST1/00051.

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Abstract

In this paper we study the discrete coagulation–fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly weakening the assumptions. Theoretical conclusions are supported by numerical simulations.

Keywords:

Discrete fragmentation–coagulation models,
birth-and-death processes,
$ C_0 $-semigroups,
analytic semigroups,
semilinear problems,
numerical simulations.

Mathematics Subject Classification: 34G20, 47D05, 47H07, 47H14, 47H20, 65J15, 82D, 70F45, 92D25.

Citation:

Full Text(HTML)

Figure 1. Evolution of the pure coagulation-fragmentation model (2) with the coagulation kernel (39a) and the fragmentation kernel (38a): number of clusters $ u_n(t) $ (top left); distribution of cluster masses $ nu_n(t) $ (top right); the total number of particles (middle left); the total mass (middle right) and the higher order moments (bottom)

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Figure 2. Evolution of the pure coagulation-fragmentation model (2) with the coagulation kernel (39b) and the fragmentation kernel (38b): number of clusters $ u_n(t) $ (top left); distribution of cluster masses $ nu_n(t) $ (top right); the total number of particles (middle left); the total mass (middle right) and the higher order moments (bottom)

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Figure 3. Evolution of the growth-decay-coagulation-fragmentation model (2) with the coagulation kernel (39a) and the fragmentation kernel (38a): number of clusters $ u_n(t) $ (top left); distribution of cluster masses $ nu_n(t) $ (top right); the total number of particles (middle left); the total mass (middle right) and the higher order moments (bottom)

Download: Full-size image PowerPoint slide

Figure 4. Evolution of the growth-decay-coagulation-fragmentation model (2) with the coagulation kernel (39b) and the fragmentation kernel (38b): number of clusters $ u_n(t) $ (top left); distribution of cluster masses $ nu_n(t) $ (top right); the total number of particles (middle left); the total mass (middle right) and the higher order moments (bottom)

Download: Full-size image PowerPoint slide

Figure 5. Evolution of the decay-sedimentation-coagulation-fragmentation model (2) with the coagulation kernel (39a) and the fragmentation kernel (38a): number of clusters $ u_n(t) $ (top left); distribution of cluster masses $ nu_n(t) $ (top right); the total number of particles (bottom left) and the total mass (bottom right)

Download: Full-size image PowerPoint slide

Figure 6. Evolution of the decay-sedimentation-coagulation-fragmentation model (2) with the coagulation kernel (39b) and the fragmentation kernel (38b): number of clusters $ u_n(t) $ (top left); distribution of cluster masses $ nu_n(t) $ (top right); the total number of particles (bottom left) and the total mass (bottom right)

Download: Full-size image PowerPoint slide

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