# American Institute of Mathematical Sciences

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 and 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), 203-230. doi: 10.1007/BF01197880. [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), 3-48. doi: 10.2307/3318611. [3] J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation, J. Statist. Phys., 61 (1990), 203-234. doi: 10.1007/BF01013961. [4] J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, J. Math. Anal. Appl., 391 (2012), 312-322. doi: 10.1016/j.jmaa.2012.02.002. [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), 231-244. doi: 10.1017/S0308210500027888. [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), 892-914. doi: 10.1006/jmaa.1995.1210. [7] P. B. Dubovskiĭ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Meth. Appl. Sci., 19 (1996), 571-591. doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q. [8] 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. [9] A. K. Giri, "Mathematical and Numerical Analysis for Coagulation-Fragmentation Equations," Ph.D thesis, Otto-von-Guericke University Magdeburg, Germany, 2010. [10] A. K. Giri, J. Kumar and G. Warnecke, The continuous coagulation equation with multiple fragmentation, J. Math. Anal. Appl., 374 (2011), 71-87. doi: 10.1016/j.jmaa.2010.08.037. [11] A. K. Giri and G. Warnecke, Uniqueness for the continuous coagulation-fragmentation equation with strong fragmentation, Z. Angew. Math. Phys., 62 (2011), 1047-1063. doi: 10.1007/s00033-011-0129-0. [12] A. K. Giri, Ph. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation equation with multiple fragmentation, Nonlinear Analysis, 75 (2012), 2199-2208. doi: 10.1016/j.na.2011.10.021. [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), 745-759. [14] W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation, Math. Methods Appl. Sci., 27 (2004), 703-721. doi: 10.1002/mma.496. [15] Ph. Laurençot, On a class of continuous coagulation-fragmentation equations, J. Differential Equations, 167 (2000), 245-274. doi: 10.1006/jdeq.2000.3809. [16] Ph. Laurençot, The discrete coagulation equation with multiple fragmentation, Proc. Edinburgh Math. Soc. (2), 45 (2002), 67-82. doi: 10.1017/S0013091500000316. [17] Ph. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1219-1248. doi: 10.1017/S0308210500002080. [18] Ph. Laurençot and S. Mischler, On coalescence equations and related models, in "Modeling and Computational Methods for Kinetic Equations" (eds. P. Degond, L. Pareschi and G. Russo), Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, MA, (2004), 321-356. [19] E. D. McGrady and R. M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895. doi: 10.1103/PhysRevLett.58.892. [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), 1173-1190. doi: 10.1137/S0036141095291713. [21] Z. A. Melzak, A scalar transport equation, Trans. Amer. Math. Soc., 85 (1957), 547-560. doi: 10.1090/S0002-9947-1957-0087880-6. [22] T. W. Peterson, Similarity solutions for the population balance equation describing particle fragmentation, Aerosol. Sci. Technol., 5 (1986), 93-101. doi: 10.1080/02786828608959079. [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), 1025-1035. doi: 10.1016/0009-2509(94)80009-X. [24] I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels, Math. Methods Appl. Sci., 11 (1989), 627-648. doi: 10.1002/mma.1670110505. [25] I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Camb. Phil. Soc., 107 (1990), 573-578. doi: 10.1017/S0305004100068821.

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##### References:
 [1] M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers, Comm. Math. Phys., 65 (1979), 203-230. doi: 10.1007/BF01197880. [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), 3-48. doi: 10.2307/3318611. [3] J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation, J. Statist. Phys., 61 (1990), 203-234. doi: 10.1007/BF01013961. [4] J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, J. Math. Anal. Appl., 391 (2012), 312-322. doi: 10.1016/j.jmaa.2012.02.002. [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), 231-244. doi: 10.1017/S0308210500027888. [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), 892-914. doi: 10.1006/jmaa.1995.1210. [7] P. B. Dubovskiĭ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Meth. Appl. Sci., 19 (1996), 571-591. doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q. [8] 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. [9] A. K. Giri, "Mathematical and Numerical Analysis for Coagulation-Fragmentation Equations," Ph.D thesis, Otto-von-Guericke University Magdeburg, Germany, 2010. [10] A. K. Giri, J. Kumar and G. Warnecke, The continuous coagulation equation with multiple fragmentation, J. Math. Anal. Appl., 374 (2011), 71-87. doi: 10.1016/j.jmaa.2010.08.037. [11] A. K. Giri and G. Warnecke, Uniqueness for the continuous coagulation-fragmentation equation with strong fragmentation, Z. Angew. Math. Phys., 62 (2011), 1047-1063. doi: 10.1007/s00033-011-0129-0. [12] A. K. Giri, Ph. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation equation with multiple fragmentation, Nonlinear Analysis, 75 (2012), 2199-2208. doi: 10.1016/j.na.2011.10.021. [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), 745-759. [14] W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation, Math. Methods Appl. Sci., 27 (2004), 703-721. doi: 10.1002/mma.496. [15] Ph. Laurençot, On a class of continuous coagulation-fragmentation equations, J. Differential Equations, 167 (2000), 245-274. doi: 10.1006/jdeq.2000.3809. [16] Ph. Laurençot, The discrete coagulation equation with multiple fragmentation, Proc. Edinburgh Math. Soc. (2), 45 (2002), 67-82. doi: 10.1017/S0013091500000316. [17] Ph. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1219-1248. doi: 10.1017/S0308210500002080. [18] Ph. Laurençot and S. Mischler, On coalescence equations and related models, in "Modeling and Computational Methods for Kinetic Equations" (eds. P. Degond, L. Pareschi and G. Russo), Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, MA, (2004), 321-356. [19] E. D. McGrady and R. M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895. doi: 10.1103/PhysRevLett.58.892. [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), 1173-1190. doi: 10.1137/S0036141095291713. [21] Z. A. Melzak, A scalar transport equation, Trans. Amer. Math. Soc., 85 (1957), 547-560. doi: 10.1090/S0002-9947-1957-0087880-6. [22] T. W. Peterson, Similarity solutions for the population balance equation describing particle fragmentation, Aerosol. Sci. Technol., 5 (1986), 93-101. doi: 10.1080/02786828608959079. [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), 1025-1035. doi: 10.1016/0009-2509(94)80009-X. [24] I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels, Math. Methods Appl. Sci., 11 (1989), 627-648. doi: 10.1002/mma.1670110505. [25] I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Camb. Phil. Soc., 107 (1990), 573-578. doi: 10.1017/S0305004100068821.
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