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A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, 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.
References:
| [1] |
R. B. Ash,
Measure, Integration and Functional Analysis, Academic Press, New York-London, 1972. |
| [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. |
| [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. |
| [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. |
| [5] |
C. Dellacherie and P. A. Mayer,
Probabilitiés et Potentiel, Chapitres I à IV, Paris, 1975. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [16] |
Ph. Laurençot,
The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci., 11 (2001), 731-748.
doi: 10.1142/S0218202501001070. |
| [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. |
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
R. B. Ash,
Measure, Integration and Functional Analysis, Academic Press, New York-London, 1972. |
| [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. |
| [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. |
| [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. |
| [5] |
C. Dellacherie and P. A. Mayer,
Probabilitiés et Potentiel, Chapitres I à IV, Paris, 1975. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [16] |
Ph. Laurençot,
The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci., 11 (2001), 731-748.
doi: 10.1142/S0218202501001070. |
| [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. |
| [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. |
| [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. |
| [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. |
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