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October  2018, 11(5): 1125-1138. doi: 10.3934/krm.2018043

A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation

Prasanta Kumar Barik and  Ankik Kumar Giri ,

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India

* Corresponding author: A. K. Giri is supported by Faculty Initiation Grant: MTD/FIG/100680, Indian Institute of Technology Roorkee, Roorkee-247667, India

Received  May 2017 Revised  October 2017 Published  May 2018

Fund Project: The first author is supported by University Grant Commission: 6405/11/44, India

In general, the non-conservative approximation of coagulation-fragmentation equations (CFEs) may lead to the occurrence of gelation phenomenon. In this article, it is shown that the non-conservative approximation of CFEs can also provide the existence of mass conserving solutions to CFEs for large classes of unbounded coagulation and fragmentation kernels.

Keywords: Particles, coagulation, fragmentation, mass conserving solution, existence.
Mathematics Subject Classification: Primary: 45K05, 45J05; Secondary: 34K30.
Citation: Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043
References:
[1]

R. B. Ash, Measure, Integration and Functional Analysis, Academic Press, New York-London, 1972. Google Scholar

[2]

J. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation, J. Stat. Phys., 61 (1990), 203-234. doi: 10.1007/BF01013961. Google Scholar

[3]

J. Banasiak and M. M. Kharroubi, Evolutionary Equations with Applications in Natural Sciences, Springer Cham Heidelberg New York Dordrecht London, 2015. doi: 978-3-319-11321-0;978-3-319-11322-7. Google Scholar

[4]

J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations, Math. Comp., 77 (2008), 851-882. doi: 10.1090/S0025-5718-07-02054-6. Google Scholar

[5]

C. Dellacherie and P. A. Mayer, Probabilitiés et Potentiel, Chapitres I à IV, Paris, 1975. Google Scholar

[6]

P. B. Dubovskii and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Methods Appl. Sci., 19 (1996), 571-591. doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q. Google Scholar

[7]

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

[8]

M. Escobedo, S. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys., 231 (2002), 157-188. doi: 10.1007/s00220-002-0680-9. Google Scholar

[9]

F. Filbet and Ph. Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation, Archiv der Mathematik, 83 (2004), 558-567. doi: 10.1007/s00013-004-1060-9. Google Scholar

[10]

F. Filbet and Ph. Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132. Google Scholar

[11]

A. K. Giri, On the uniqueness for coagulation and multiple fragmentation equation, Kinet. Relat. Models, 6 (2013), 589-599. doi: 10.3934/krm.2013.6.589. Google Scholar

[12]

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. Google Scholar

[13]

A. K. Giri, Ph. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation with multiple fragmentation, Nonlinear Anal., 75 (2012), 2199-2208. doi: 10.1016/j.na.2011.10.021. Google Scholar

[14]

A. K. Giri and G. Warnecke, Uniqueness for the coagulation-fragmentation equation with strong fragmentation, Z. Angew. Math. Phys., 62 (2011), 1047-1063. doi: 10.1007/s00033-011-0129-0. Google Scholar

[15]

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/S0308210502000598. Google Scholar

[16]

Ph. Laurençot, The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci., 11 (2001), 731-748. doi: 10.1142/S0218202501001070. Google Scholar

[17]

F. Leyvraz, Existence and properties of post-gel solutions for the kinetic equations of coagulation, J. Phys. A, 16 (1983), 2861-2873. doi: 10.1088/0305-4470/16/12/032. Google Scholar

[18]

F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A, 14 (1981), 3389-3405. doi: 10.1088/0305-4470/14/12/030. Google Scholar

[19]

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. Google Scholar

[20]

I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Cambridge. Philos. Soc., 107 (1990), 573-578. doi: 10.1017/S0305004100068821. Google Scholar

show all references

References:
[1]

R. B. Ash, Measure, Integration and Functional Analysis, Academic Press, New York-London, 1972. Google Scholar

[2]

J. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation, J. Stat. Phys., 61 (1990), 203-234. doi: 10.1007/BF01013961. Google Scholar

[3]

J. Banasiak and M. M. Kharroubi, Evolutionary Equations with Applications in Natural Sciences, Springer Cham Heidelberg New York Dordrecht London, 2015. doi: 978-3-319-11321-0;978-3-319-11322-7. Google Scholar

[4]

J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations, Math. Comp., 77 (2008), 851-882. doi: 10.1090/S0025-5718-07-02054-6. Google Scholar

[5]

C. Dellacherie and P. A. Mayer, Probabilitiés et Potentiel, Chapitres I à IV, Paris, 1975. Google Scholar

[6]

P. B. Dubovskii and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Methods Appl. Sci., 19 (1996), 571-591. doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q. Google Scholar

[7]

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

[8]

M. Escobedo, S. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys., 231 (2002), 157-188. doi: 10.1007/s00220-002-0680-9. Google Scholar

[9]

F. Filbet and Ph. Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation, Archiv der Mathematik, 83 (2004), 558-567. doi: 10.1007/s00013-004-1060-9. Google Scholar

[10]

F. Filbet and Ph. Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132. Google Scholar

[11]

A. K. Giri, On the uniqueness for coagulation and multiple fragmentation equation, Kinet. Relat. Models, 6 (2013), 589-599. doi: 10.3934/krm.2013.6.589. Google Scholar

[12]

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. Google Scholar

[13]

A. K. Giri, Ph. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation with multiple fragmentation, Nonlinear Anal., 75 (2012), 2199-2208. doi: 10.1016/j.na.2011.10.021. Google Scholar

[14]

A. K. Giri and G. Warnecke, Uniqueness for the coagulation-fragmentation equation with strong fragmentation, Z. Angew. Math. Phys., 62 (2011), 1047-1063. doi: 10.1007/s00033-011-0129-0. Google Scholar

[15]

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/S0308210502000598. Google Scholar

[16]

Ph. Laurençot, The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci., 11 (2001), 731-748. doi: 10.1142/S0218202501001070. Google Scholar

[17]

F. Leyvraz, Existence and properties of post-gel solutions for the kinetic equations of coagulation, J. Phys. A, 16 (1983), 2861-2873. doi: 10.1088/0305-4470/16/12/032. Google Scholar

[18]

F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A, 14 (1981), 3389-3405. doi: 10.1088/0305-4470/14/12/030. Google Scholar

[19]

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. Google Scholar

[20]

I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Cambridge. Philos. Soc., 107 (1990), 573-578. doi: 10.1017/S0305004100068821. Google Scholar

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