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October  2019, 12(5): 1069-1092. doi: 10.3934/krm.2019040

## 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

Received  March 2018 Published  July 2019

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

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.

Citation: Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic and Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040
##### References:
 [1] A. S. Ackleh and B. G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, Journal of Mathematical Biology, 35 (1997), 480-502.  doi: 10.1007/s002850050062. [2] J. Banasiak, Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates, Nonlinear Analysis. Real World Applications, 13 (2012), 91-105.  doi: 10.1016/j.nonrwa.2011.07.016. [3] J. Banasiak, L. O. Joel and S. Shindin, Analysis and simulations of the discrete fragmentation equation with decay, Mathematical Methods in the Applied Sciences, 41 (2018), 6530-6545.  doi: 10.1002/mma.4666. [4] J. Banasiak, Analytic fragmentation semigroups and classical solutions to coagulation ragmentation equations – a survey, Acta Mathematica Sinica, English Series, 35 (2019), 83-104.  doi: 10.1007/s10114-018-7435-9. [5] J. Banasiak, L. O. Joel and S. Shindin, Discrete growth-decay-fragmentation equation - well-posedness and long term dynamics, Journal of Evolution Equations, 2019, in print. doi: 10.1007/s00028-019-00499-4. [6] J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, Journal of Mathematical Analysis and Applications, 391 (2012), 312-322.  doi: 10.1016/j.jmaa.2012.02.002. [7] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation–Fragmentation Models, I & II, CRC Press, Boca Raton, 2019. [8] R. Becker and W. Döring, Kinetische behandlung der keimbildung in übersättigten dämpfen, Annalen der Physik, 416 (1935), 719-752.  doi: 10.1002/andp.19354160806. [9] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin-New York, 1976. [10] A. T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications, McGraw-Hill Series in Probability and Statistics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. [11] P. J. Blatz and A. V. Tobolsky, Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena, The Journal of Physical Chemistry, 49 (1945), 77-80.  doi: 10.1021/j150440a004. [12] J. A. Cañizo, L. Desvillettes and K. Fellner, Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion, Annales de L'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 639-654.  doi: 10.1016/j.anihpc.2009.10.001. [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press, Oxford, 1998. [14] J.-F. Collet, Some modelling issues in the theory of fragmentation-coagulation systems, Communications in Mathematical Sciences, 2 (2004), 35-54.  doi: 10.4310/CMS.2004.v2.n5.a3. [15] J.-F. Collet and F. Poupaud, Existence of solutions to coagulation-fragmentation systems with diffusion, Transport Theory and Statistical Physics, 25 (1996), 503-513.  doi: 10.1080/00411459608220717. [16] F. P. Da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, Journal of Mathematical Analysis and Applications, 192 (1995), 892-914.  doi: 10.1006/jmaa.1995.1210. [17] J. A. David, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli, 5 (1999), 3-48.  doi: 10.2307/3318611. [18] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [19] M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, Journal of Differential Equations, 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7. [20] A. K. Giri, J. Kumar and G. Warnecke, The continuous coagulation equation with multiple fragmentation, Journal of Mathematical Analysis and Applications, 374 (2011), 71-87.  doi: 10.1016/j.jmaa.2010.08.037. [21] S. Gueron and S. A. Levin, The dynamics of group formation, Mathematical Biosciences, 128 (1995), 243-264.  doi: 10.1016/0025-5564(94)00074-A. [22] G. A. Jackson, A model of the formation of marine algal flocs by physical coagulation processes, Deep Sea Research Part A. Oceanographic Research Papers, 37 (1990), 1197-1211.  doi: 10.1016/0198-0149(90)90038-W. [23] A. Lunardi., Analytic Semigroups and Optimal Regularity in Parabolic Problems, volume 16 of Progress in Nonlinear Differential Equations and their Applications., Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [24] I. Mirzaev and D. M. Bortz, On the existence of non-trivial steady-state size-distributions for a class of flocculation equations, preprint, arXiv: 1804.00977. [25] A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Advances in Biophysics, 22 (1986), 1-94.  doi: 10.1016/0065-227X(86)90003-1. [26] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14 of Interdisciplinary Applied Mathematics, 2nd edition, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [28] M. Smoluchowski, Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen, Zeitschrift Für Physik, 17 (1916), 557-585. [29] M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen, Zeitschrift für Physik, 92 (1917), 129-168.  doi: 10.1515/zpch-1918-9209. [30] J. A. D. Wattis, An introduction to mathematical models of coagulation–fragmentation processes: a discrete deterministic mean-field approach, Physica D: Nonlinear Phenomena, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024. [31] D. Wrzosek, Existence of solutions for the discrete coagulation-fragmentation model with diffusion, Topological Methods in Nonlinear Analysis, 9 (1997), 279-296.  doi: 10.12775/TMNA.1997.014.

show all references

##### References:
 [1] A. S. Ackleh and B. G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, Journal of Mathematical Biology, 35 (1997), 480-502.  doi: 10.1007/s002850050062. [2] J. Banasiak, Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates, Nonlinear Analysis. Real World Applications, 13 (2012), 91-105.  doi: 10.1016/j.nonrwa.2011.07.016. [3] J. Banasiak, L. O. Joel and S. Shindin, Analysis and simulations of the discrete fragmentation equation with decay, Mathematical Methods in the Applied Sciences, 41 (2018), 6530-6545.  doi: 10.1002/mma.4666. [4] J. Banasiak, Analytic fragmentation semigroups and classical solutions to coagulation ragmentation equations – a survey, Acta Mathematica Sinica, English Series, 35 (2019), 83-104.  doi: 10.1007/s10114-018-7435-9. [5] J. Banasiak, L. O. Joel and S. Shindin, Discrete growth-decay-fragmentation equation - well-posedness and long term dynamics, Journal of Evolution Equations, 2019, in print. doi: 10.1007/s00028-019-00499-4. [6] J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, Journal of Mathematical Analysis and Applications, 391 (2012), 312-322.  doi: 10.1016/j.jmaa.2012.02.002. [7] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation–Fragmentation Models, I & II, CRC Press, Boca Raton, 2019. [8] R. Becker and W. Döring, Kinetische behandlung der keimbildung in übersättigten dämpfen, Annalen der Physik, 416 (1935), 719-752.  doi: 10.1002/andp.19354160806. [9] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin-New York, 1976. [10] A. T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications, McGraw-Hill Series in Probability and Statistics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. [11] P. J. Blatz and A. V. Tobolsky, Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena, The Journal of Physical Chemistry, 49 (1945), 77-80.  doi: 10.1021/j150440a004. [12] J. A. Cañizo, L. Desvillettes and K. Fellner, Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion, Annales de L'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 639-654.  doi: 10.1016/j.anihpc.2009.10.001. [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press, Oxford, 1998. [14] J.-F. Collet, Some modelling issues in the theory of fragmentation-coagulation systems, Communications in Mathematical Sciences, 2 (2004), 35-54.  doi: 10.4310/CMS.2004.v2.n5.a3. [15] J.-F. Collet and F. Poupaud, Existence of solutions to coagulation-fragmentation systems with diffusion, Transport Theory and Statistical Physics, 25 (1996), 503-513.  doi: 10.1080/00411459608220717. [16] F. P. Da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, Journal of Mathematical Analysis and Applications, 192 (1995), 892-914.  doi: 10.1006/jmaa.1995.1210. [17] J. A. David, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli, 5 (1999), 3-48.  doi: 10.2307/3318611. [18] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [19] M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, Journal of Differential Equations, 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7. [20] A. K. Giri, J. Kumar and G. Warnecke, The continuous coagulation equation with multiple fragmentation, Journal of Mathematical Analysis and Applications, 374 (2011), 71-87.  doi: 10.1016/j.jmaa.2010.08.037. [21] S. Gueron and S. A. Levin, The dynamics of group formation, Mathematical Biosciences, 128 (1995), 243-264.  doi: 10.1016/0025-5564(94)00074-A. [22] G. A. Jackson, A model of the formation of marine algal flocs by physical coagulation processes, Deep Sea Research Part A. Oceanographic Research Papers, 37 (1990), 1197-1211.  doi: 10.1016/0198-0149(90)90038-W. [23] A. Lunardi., Analytic Semigroups and Optimal Regularity in Parabolic Problems, volume 16 of Progress in Nonlinear Differential Equations and their Applications., Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [24] I. Mirzaev and D. M. Bortz, On the existence of non-trivial steady-state size-distributions for a class of flocculation equations, preprint, arXiv: 1804.00977. [25] A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Advances in Biophysics, 22 (1986), 1-94.  doi: 10.1016/0065-227X(86)90003-1. [26] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, vol. 14 of Interdisciplinary Applied Mathematics, 2nd edition, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [28] M. Smoluchowski, Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen, Zeitschrift Für Physik, 17 (1916), 557-585. [29] M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen, Zeitschrift für Physik, 92 (1917), 129-168.  doi: 10.1515/zpch-1918-9209. [30] J. A. D. Wattis, An introduction to mathematical models of coagulation–fragmentation processes: a discrete deterministic mean-field approach, Physica D: Nonlinear Phenomena, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024. [31] D. Wrzosek, Existence of solutions for the discrete coagulation-fragmentation model with diffusion, Topological Methods in Nonlinear Analysis, 9 (1997), 279-296.  doi: 10.12775/TMNA.1997.014.
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)
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)
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)
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)
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)
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)
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