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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 & 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

R. M. Ziff and E. D. McGrady, Kinetics of polymer degradation, Macromolecules, 19 (1986), 2513-2519. doi: 10.1021/ma00164a010.  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, J. Math. Biol., 35 (1997), 480-502. doi: 10.1007/s002850050062.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

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