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March  2012, 17(2): 445-472. doi: 10.3934/dcdsb.2012.17.445

Transport processes with coagulation and strong fragmentation

Jacek Banasiak 1,

1. 

School of Mathematical Sciences, University of KwaZulu-Natal, Durban

Received  July 2010 Revised  December 2010 Published  December 2011

In this paper we deal with equations describing fragmentation and coagulation processes with growth or decay, where the latter are modelled by first order transport equations. Our main interest lies in processes with strong fragmentation and thus we carry out the analysis in spaces ensuring that higher moments of the solution exist. We prove that the linear part, consisting of the transport and fragmentation terms, generates a strongly continuous semigroup in such spaces and characterize its generator as the closure of the sum (and in some cases the sum itself) of the operators describing the transport and fragmentation, defined on their natural domains. These results allow us to prove the existence of global in time strict solutions to the full nonlinear fragmentation-coagulation-transport equation.
Keywords: semilinear Cauchy problem, Semigroups of operators, structured population models., coagulation, fragmentation, transport processes.
Mathematics Subject Classification: Primary: 35F25, 47D06; Secondary: 45K05, 80A30, 92D2.
Citation: Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445
References:
[1]

A. S. Ackleh and B. G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: Analysis and computations, J. Math. Biol., 35 (1997), 480-502. doi: 10.1007/s002850050062.

[2]

O. Arino and R. Rudnicki, Phytoplankton dynamics, Comptes Rendus Biologies, 327 (2004), 961-969. doi: 10.1016/j.crvi.2004.03.013.

[3]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.

[4]

J. Banasiak, On non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556. doi: 10.1002/mma.301.

[5]

J. Banasiak, Conservative and shattering solutions for some classes of fragmentation models, Math. Models Methods Appl. Sci., 14 (2004), 483-501. doi: 10.1142/S0218202504003325.

[6]

J. Banasiak, On conservativity and shattering for an equation of phytoplankton dynamics, Comptes Rendus Biologies, 337 (2004), 1025-1036. doi: 10.1016/j.crvi.2004.07.017.

[7]

J. Banasiak, Shattering and non-uniqueness in fragmentation models--an analytic approach, Physica D, 222 (2006), 63-72. doi: 10.1016/j.physd.2006.07.025.

[8]

J. Banasiak and W. Lamb, On a coagulation and fragmentation equation with mass loss, Proc. Roy. Soc. Edinburgh Sec. A, 136 (2006), 1157-1173. doi: 10.1017/S0308210500004923.

[9]

J. Banasiak and W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Sys. Ser. B, 11 (2009), 563-585. doi: 10.3934/dcdsb.2009.11.563.

[10]

J. Banasiak and M. Mokhtar-Kharroubi, Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes, Discrete Contin. Dyn. Sys. B, 5 (2005), 524-542.

[11]

J. Banasiak, S. C. Oukoumi Noutchie and R. Rudnicki, Global solvability of a fragmentation-coagulation eqation with growth and restricted coagulation, J. Nonlinear Math. Phys., 16 (2009), 13-26. doi: 10.1142/S1402925109000297.

[12]

J. Banasiak and W. Lamb, Global strict solutions to continuous coagulation-fragmentation equations with strong fragmentation, Proc. Roy. Soc. Edinburgh Sec. A, 141 (2011), 465-480. doi: 10.1017/S0308210509001255.

[13]

M. Cai, B. F. Edwards and H. Han, Exact and asymptotic scaling solutions for fragmentation with mass loss, Phys. Rev. A (3), 43 (1991), 656-662. doi: 10.1103/PhysRevA.43.656.

[14]

J. Carr and F. P. da Costa, Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Stat. Phys., 77 (1994), 89-123. doi: 10.1007/BF02186834.

[15]

B. F. Edwards, M. Cai and H. Han, Rate equation and scaling for fragmentation with mass loss, Phys. Rev. A, 41 (1990), 5755-5757. doi: 10.1103/PhysRevA.41.5755.

[16]

M. Escobedo, Ph. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations, 195 (2003), 143-174. doi: 10.1016/S0022-0396(03)00134-7.

[17]

B. Haas, Loss of mass in deterministic and random fragmentations, Stochastic Process. Appl., 106 (2003), 245-277. doi: 10.1016/S0304-4149(03)00045-0.

[18]

J. Huang, B. E. Edwards and A. D. Levine, General solutions and scaling violation for fragmentation with mass loss, J. Phys. A: Math. Gen., 24 (1991), 3967-3977. doi: 10.1088/0305-4470/24/16/031.

[19]

J. Huang, X. P. Guo, B. F. Edwards and A. D. Levine, Cut-off model and exact general solutions for fragmentation with mass loss, J. Phys. A: Math. Gen., 29 (1996), 7377-7388. doi: 10.1088/0305-4470/29/23/008.

[20]

G. A. Jackson, A model of formation of marine algal flocks by physical coagulation processes, Deep-Sea Research, 37 (1990), 1197-1211. doi: 10.1016/0198-0149(90)90038-W.

[21]

M. Kostoglou and A. J. Karabelas, On the breakage problem with a homogeneous erosion type kernel, J. Phys. A, 34 (2001), 1725-1740. doi: 10.1088/0305-4470/34/8/316.

[22]

P. Laurençot, On a class of continuous coagulation-fragmentation equations, J. Differential Equations, 167 (2000), 245-274. doi: 10.1006/jdeq.2000.3809.

[23]

E. D. McGrady and R. M. Ziff, "Shattering'' transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895. doi: 10.1103/PhysRevLett.58.892.

[24]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.

[25]

B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[26]

J. Prüss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 225-235.

[27]

R. Rudnicki and R. Wieczorek, Phytoplankton dynamics: From the behaviour of cells to a transport equation, Math. Model. Nat. Phenom., 1 (2006), 83-100. doi: 10.1051/mmnp:2006005.

[28]

R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerization, J. Phys. A, 18 (1985), 3027-3037. doi: 10.1088/0305-4470/18/15/026.

[29]

R. M. Ziff and E. D. McGrady, Kinetics of polymer degradation, Macromolecules, 19 (1986), 2513-2519. doi: 10.1021/ma00164a010.

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, J. Math. Biol., 35 (1997), 480-502. doi: 10.1007/s002850050062.

[2]

O. Arino and R. Rudnicki, Phytoplankton dynamics, Comptes Rendus Biologies, 327 (2004), 961-969. doi: 10.1016/j.crvi.2004.03.013.

[3]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications," Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.

[4]

J. Banasiak, On non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556. doi: 10.1002/mma.301.

[5]

J. Banasiak, Conservative and shattering solutions for some classes of fragmentation models, Math. Models Methods Appl. Sci., 14 (2004), 483-501. doi: 10.1142/S0218202504003325.

[6]

J. Banasiak, On conservativity and shattering for an equation of phytoplankton dynamics, Comptes Rendus Biologies, 337 (2004), 1025-1036. doi: 10.1016/j.crvi.2004.07.017.

[7]

J. Banasiak, Shattering and non-uniqueness in fragmentation models--an analytic approach, Physica D, 222 (2006), 63-72. doi: 10.1016/j.physd.2006.07.025.

[8]

J. Banasiak and W. Lamb, On a coagulation and fragmentation equation with mass loss, Proc. Roy. Soc. Edinburgh Sec. A, 136 (2006), 1157-1173. doi: 10.1017/S0308210500004923.

[9]

J. Banasiak and W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Sys. Ser. B, 11 (2009), 563-585. doi: 10.3934/dcdsb.2009.11.563.

[10]

J. Banasiak and M. Mokhtar-Kharroubi, Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes, Discrete Contin. Dyn. Sys. B, 5 (2005), 524-542.

[11]

J. Banasiak, S. C. Oukoumi Noutchie and R. Rudnicki, Global solvability of a fragmentation-coagulation eqation with growth and restricted coagulation, J. Nonlinear Math. Phys., 16 (2009), 13-26. doi: 10.1142/S1402925109000297.

[12]

J. Banasiak and W. Lamb, Global strict solutions to continuous coagulation-fragmentation equations with strong fragmentation, Proc. Roy. Soc. Edinburgh Sec. A, 141 (2011), 465-480. doi: 10.1017/S0308210509001255.

[13]

M. Cai, B. F. Edwards and H. Han, Exact and asymptotic scaling solutions for fragmentation with mass loss, Phys. Rev. A (3), 43 (1991), 656-662. doi: 10.1103/PhysRevA.43.656.

[14]

J. Carr and F. P. da Costa, Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Stat. Phys., 77 (1994), 89-123. doi: 10.1007/BF02186834.

[15]

B. F. Edwards, M. Cai and H. Han, Rate equation and scaling for fragmentation with mass loss, Phys. Rev. A, 41 (1990), 5755-5757. doi: 10.1103/PhysRevA.41.5755.

[16]

M. Escobedo, Ph. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations, 195 (2003), 143-174. doi: 10.1016/S0022-0396(03)00134-7.

[17]

B. Haas, Loss of mass in deterministic and random fragmentations, Stochastic Process. Appl., 106 (2003), 245-277. doi: 10.1016/S0304-4149(03)00045-0.

[18]

J. Huang, B. E. Edwards and A. D. Levine, General solutions and scaling violation for fragmentation with mass loss, J. Phys. A: Math. Gen., 24 (1991), 3967-3977. doi: 10.1088/0305-4470/24/16/031.

[19]

J. Huang, X. P. Guo, B. F. Edwards and A. D. Levine, Cut-off model and exact general solutions for fragmentation with mass loss, J. Phys. A: Math. Gen., 29 (1996), 7377-7388. doi: 10.1088/0305-4470/29/23/008.

[20]

G. A. Jackson, A model of formation of marine algal flocks by physical coagulation processes, Deep-Sea Research, 37 (1990), 1197-1211. doi: 10.1016/0198-0149(90)90038-W.

[21]

M. Kostoglou and A. J. Karabelas, On the breakage problem with a homogeneous erosion type kernel, J. Phys. A, 34 (2001), 1725-1740. doi: 10.1088/0305-4470/34/8/316.

[22]

P. Laurençot, On a class of continuous coagulation-fragmentation equations, J. Differential Equations, 167 (2000), 245-274. doi: 10.1006/jdeq.2000.3809.

[23]

E. D. McGrady and R. M. Ziff, "Shattering'' transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895. doi: 10.1103/PhysRevLett.58.892.

[24]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.

[25]

B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[26]

J. Prüss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 225-235.

[27]

R. Rudnicki and R. Wieczorek, Phytoplankton dynamics: From the behaviour of cells to a transport equation, Math. Model. Nat. Phenom., 1 (2006), 83-100. doi: 10.1051/mmnp:2006005.

[28]

R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerization, J. Phys. A, 18 (1985), 3027-3037. doi: 10.1088/0305-4470/18/15/026.

[29]

R. M. Ziff and E. D. McGrady, Kinetics of polymer degradation, Macromolecules, 19 (1986), 2513-2519. doi: 10.1021/ma00164a010.

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