September  2013, 6(3): 589-599. doi: 10.3934/krm.2013.6.589

On the uniqueness for coagulation and multiple fragmentation equation

1. 

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria

Received  January 2013 Revised  March 2013 Published  May 2013

In this article, the uniqueness of weak solutions to the continuous coagulation and multiple fragmentation equation is proved for a large range of unbounded coagulation and multiple fragmentation kernels. The multiple fragmentation kernels may have a singularity at origin. This work generalizes the preceding ones, by including some physically relevant coagulation and fragmentation kernels which were not considered before.
Citation: Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589
References:
[1]

M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers,, Comm. Math. Phys., 65 (1979), 203.  doi: 10.1007/BF01197880.  Google Scholar

[2]

D. J. Aldous, Deterministic and stochastic model for coalescence (aggregation and coagulation): A review of the mean-field theory and probabilists,, Bernoulli, 5 (1999), 3.  doi: 10.2307/3318611.  Google Scholar

[3]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation,, J. Statist. Phys., 61 (1990), 203.  doi: 10.1007/BF01013961.  Google Scholar

[4]

J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates,, J. Math. Anal. Appl., 391 (2012), 312.  doi: 10.1016/j.jmaa.2012.02.002.  Google Scholar

[5]

J. Carr, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case,, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 231.  doi: 10.1017/S0308210500027888.  Google Scholar

[6]

F. P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equation with strong fragmentation,, J. Math. Anal. Appl., 192 (1995), 892.  doi: 10.1006/jmaa.1995.1210.  Google Scholar

[7]

P. B. Dubovskiĭ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation,, Math. Meth. Appl. Sci., 19 (1996), 571.  doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q.  Google Scholar

[8]

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

[9]

A. K. Giri, "Mathematical and Numerical Analysis for Coagulation-Fragmentation Equations,", Ph.D thesis, (2010).   Google Scholar

[10]

A. K. Giri, J. Kumar and G. Warnecke, The continuous coagulation equation with multiple fragmentation,, J. Math. Anal. Appl., 374 (2011), 71.  doi: 10.1016/j.jmaa.2010.08.037.  Google Scholar

[11]

A. K. Giri and G. Warnecke, Uniqueness for the continuous coagulation-fragmentation equation with strong fragmentation,, Z. Angew. Math. Phys., 62 (2011), 1047.  doi: 10.1007/s00033-011-0129-0.  Google Scholar

[12]

A. K. Giri, Ph. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation equation with multiple fragmentation,, Nonlinear Analysis, 75 (2012), 2199.  doi: 10.1016/j.na.2011.10.021.  Google Scholar

[13]

J. Koch, W. Hackbusch and K. Sundmacher, H-matrix methods for linear and quasi-linear integral operators appearing in population balances,, Comput. Chem. Eng., 31 (2007), 745.   Google Scholar

[14]

W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation,, Math. Methods Appl. Sci., 27 (2004), 703.  doi: 10.1002/mma.496.  Google Scholar

[15]

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

[16]

Ph. Laurençot, The discrete coagulation equation with multiple fragmentation,, Proc. Edinburgh Math. Soc. (2), 45 (2002), 67.  doi: 10.1017/S0013091500000316.  Google Scholar

[17]

Ph. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1219.  doi: 10.1017/S0308210500002080.  Google Scholar

[18]

Ph. Laurençot and S. Mischler, On coalescence equations and related models,, in, (2004), 321.   Google Scholar

[19]

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

[20]

D. J. McLaughlin, W. Lamb and A. C. McBride, An existence and uniqueness result for a coagulation and multiple-fragmentation equation,, SIAM J. Math. Anal., 28 (1997), 1173.  doi: 10.1137/S0036141095291713.  Google Scholar

[21]

Z. A. Melzak, A scalar transport equation,, Trans. Amer. Math. Soc., 85 (1957), 547.  doi: 10.1090/S0002-9947-1957-0087880-6.  Google Scholar

[22]

T. W. Peterson, Similarity solutions for the population balance equation describing particle fragmentation,, Aerosol. Sci. Technol., 5 (1986), 93.  doi: 10.1080/02786828608959079.  Google Scholar

[23]

D. J. Smit, M. J. Hounslow and W. R. Paterson, Aggregation and gelation-I. Analytical solutions for cst and batch operation,, Chem. Eng. Sci., 49 (1994), 1025.  doi: 10.1016/0009-2509(94)80009-X.  Google Scholar

[24]

I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels,, Math. Methods Appl. Sci., 11 (1989), 627.  doi: 10.1002/mma.1670110505.  Google Scholar

[25]

I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation,, Math. Proc. Camb. Phil. Soc., 107 (1990), 573.  doi: 10.1017/S0305004100068821.  Google Scholar

show all references

References:
[1]

M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers,, Comm. Math. Phys., 65 (1979), 203.  doi: 10.1007/BF01197880.  Google Scholar

[2]

D. J. Aldous, Deterministic and stochastic model for coalescence (aggregation and coagulation): A review of the mean-field theory and probabilists,, Bernoulli, 5 (1999), 3.  doi: 10.2307/3318611.  Google Scholar

[3]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation,, J. Statist. Phys., 61 (1990), 203.  doi: 10.1007/BF01013961.  Google Scholar

[4]

J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates,, J. Math. Anal. Appl., 391 (2012), 312.  doi: 10.1016/j.jmaa.2012.02.002.  Google Scholar

[5]

J. Carr, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case,, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 231.  doi: 10.1017/S0308210500027888.  Google Scholar

[6]

F. P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equation with strong fragmentation,, J. Math. Anal. Appl., 192 (1995), 892.  doi: 10.1006/jmaa.1995.1210.  Google Scholar

[7]

P. B. Dubovskiĭ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation,, Math. Meth. Appl. Sci., 19 (1996), 571.  doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q.  Google Scholar

[8]

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

[9]

A. K. Giri, "Mathematical and Numerical Analysis for Coagulation-Fragmentation Equations,", Ph.D thesis, (2010).   Google Scholar

[10]

A. K. Giri, J. Kumar and G. Warnecke, The continuous coagulation equation with multiple fragmentation,, J. Math. Anal. Appl., 374 (2011), 71.  doi: 10.1016/j.jmaa.2010.08.037.  Google Scholar

[11]

A. K. Giri and G. Warnecke, Uniqueness for the continuous coagulation-fragmentation equation with strong fragmentation,, Z. Angew. Math. Phys., 62 (2011), 1047.  doi: 10.1007/s00033-011-0129-0.  Google Scholar

[12]

A. K. Giri, Ph. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation equation with multiple fragmentation,, Nonlinear Analysis, 75 (2012), 2199.  doi: 10.1016/j.na.2011.10.021.  Google Scholar

[13]

J. Koch, W. Hackbusch and K. Sundmacher, H-matrix methods for linear and quasi-linear integral operators appearing in population balances,, Comput. Chem. Eng., 31 (2007), 745.   Google Scholar

[14]

W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation,, Math. Methods Appl. Sci., 27 (2004), 703.  doi: 10.1002/mma.496.  Google Scholar

[15]

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

[16]

Ph. Laurençot, The discrete coagulation equation with multiple fragmentation,, Proc. Edinburgh Math. Soc. (2), 45 (2002), 67.  doi: 10.1017/S0013091500000316.  Google Scholar

[17]

Ph. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1219.  doi: 10.1017/S0308210500002080.  Google Scholar

[18]

Ph. Laurençot and S. Mischler, On coalescence equations and related models,, in, (2004), 321.   Google Scholar

[19]

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

[20]

D. J. McLaughlin, W. Lamb and A. C. McBride, An existence and uniqueness result for a coagulation and multiple-fragmentation equation,, SIAM J. Math. Anal., 28 (1997), 1173.  doi: 10.1137/S0036141095291713.  Google Scholar

[21]

Z. A. Melzak, A scalar transport equation,, Trans. Amer. Math. Soc., 85 (1957), 547.  doi: 10.1090/S0002-9947-1957-0087880-6.  Google Scholar

[22]

T. W. Peterson, Similarity solutions for the population balance equation describing particle fragmentation,, Aerosol. Sci. Technol., 5 (1986), 93.  doi: 10.1080/02786828608959079.  Google Scholar

[23]

D. J. Smit, M. J. Hounslow and W. R. Paterson, Aggregation and gelation-I. Analytical solutions for cst and batch operation,, Chem. Eng. Sci., 49 (1994), 1025.  doi: 10.1016/0009-2509(94)80009-X.  Google Scholar

[24]

I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels,, Math. Methods Appl. Sci., 11 (1989), 627.  doi: 10.1002/mma.1670110505.  Google Scholar

[25]

I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation,, Math. Proc. Camb. Phil. Soc., 107 (1990), 573.  doi: 10.1017/S0305004100068821.  Google Scholar

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