• Previous Article
    A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication
  • KRM Home
  • This Issue
  • Next Article
    Differentiability in perturbation parameter of measure solutions to perturbed transport equation
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 & 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. Google Scholar

[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. Google Scholar

[3]

J. BanasiakL. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7] J. BanasiakW. Lamb and P. Laurençot, Analytic Methods for Coagulation–Fragmentation Models, I & II, CRC Press, Boca Raton, 2019. Google Scholar
[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. Google Scholar

[9]

J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin-New York, 1976. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

J. A. CañizoL. 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. Google Scholar

[13]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press, Oxford, 1998. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar

[19]

M. EscobedoP. LaurençotS. 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. Google Scholar

[20]

A. K. GiriJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

M. Smoluchowski, Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen, Zeitschrift Für Physik, 17 (1916), 557-585. Google Scholar

[29]

M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen, Zeitschrift für Physik, 92 (1917), 129-168. doi: 10.1515/zpch-1918-9209. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

J. BanasiakL. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7] J. BanasiakW. Lamb and P. Laurençot, Analytic Methods for Coagulation–Fragmentation Models, I & II, CRC Press, Boca Raton, 2019. Google Scholar
[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. Google Scholar

[9]

J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin-New York, 1976. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

J. A. CañizoL. 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. Google Scholar

[13]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press, Oxford, 1998. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar

[19]

M. EscobedoP. LaurençotS. 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. Google Scholar

[20]

A. K. GiriJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

M. Smoluchowski, Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen, Zeitschrift Für Physik, 17 (1916), 557-585. Google Scholar

[29]

M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen, Zeitschrift für Physik, 92 (1917), 129-168. doi: 10.1515/zpch-1918-9209. Google Scholar

[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. Google Scholar

[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. Google Scholar

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)
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)
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)
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)
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)
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)
[1]

Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 529-542. doi: 10.3934/dcdsb.2005.5.529

[2]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577

[3]

José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195

[4]

Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67

[5]

Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223

[6]

Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445

[7]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963

[8]

Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563

[9]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361

[10]

Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177

[11]

Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009

[12]

Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595

[13]

Katarzyna Pichór, Ryszard Rudnicki. Applications of stochastic semigroups to cell cycle models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2365-2381. doi: 10.3934/dcdsb.2019099

[14]

Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014

[15]

Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic & Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253

[16]

Lavinia Roncoroni. Exact lumping of feller semigroups: A $C^{\star}$-algebras approach. Conference Publications, 2015, 2015 (special) : 965-973. doi: 10.3934/proc.2015.0965

[17]

Hilla Behar, Alexandra Agranovich, Yoram Louzoun. Diffusion rate determines balance between extinction and proliferation in birth-death processes. Mathematical Biosciences & Engineering, 2013, 10 (3) : 523-550. doi: 10.3934/mbe.2013.10.523

[18]

Florian De Vuyst, Francesco Salvarani. Numerical simulations of degenerate transport problems. Kinetic & Related Models, 2014, 7 (3) : 463-476. doi: 10.3934/krm.2014.7.463

[19]

Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457

[20]

Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (32)
  • HTML views (70)
  • Cited by (0)

Other articles
by authors

[Back to Top]