# American Institute of Mathematical Sciences

November  2020, 40(11): 6115-6133. doi: 10.3934/dcds.2020272

## Weak solutions to the continuous coagulation model with collisional breakage

 1 Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore-560065, Karnataka, India 2 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India

* Corresponding author: Ankik Kumar Giri, Tel. +91-1332-284818 (O); Fax: +91-1332-273560

Received  June 2019 Revised  June 2020 Published  July 2020

Fund Project: Partially supported by Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India through the project YSS/2015/001306. The first author thanks to University Grant Commission (UGC), 6405/11/44, India for funding his PhD

A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernel and distribution functions. The model describes the dynamics of particle growth when binary collisions occur to form either a single particle via coalescence or two/more particles via breakup with possible transfer of mass. Each of these processes may take place with a suitably assigned probability depending on the volume of particles participating in the collision.

Citation: Prasanta Kumar Barik, Ankik Kumar Giri. Weak solutions to the continuous coagulation model with collisional breakage. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6115-6133. doi: 10.3934/dcds.2020272
##### References:
 [1] R. B. Ash, Measure, Integration and Functional Analysis, Academic Press, New York, 1972. [2] P. K. Barik, Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation, Evolution Equations and Control Theory, 9 (2020), 431-416.  doi: 10.3934/eect.2020012. [3] P. K. Barik and A. K. Giri, A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation, Kinet. Relat. Models, 11 (2018), 1125-1138.  doi: 10.3934/krm.2018043. [4] P. K. Barik, A. K. Giri and Ph. Laurençot, Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel, Proc. Roy. Soc. Edinb. Sec. A: Math., (2019). doi: 10.1017/prm.2018.158. [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [6] P. S. Brown, Structural stability of the coalescence/breakup equations, J. Atmospheric Sci., 52 (1995), 3857-3865. [7] Z. Cheng and S. Redner, Kinetics of fragmentation, J. Phys., 23 (1990), 1233-1258.  doi: 10.1088/0305-4470/23/7/028. [8] Z. Cheng and S. Redner, Scaling theory of fragmentation, Phys. Rev. Lett., 60 (1988), 2450-2453.  doi: 10.1103/PhysRevLett.60.2450. [9] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York, 1965. [10] M. Ernst and I. Pagonabarraga, The nonlinear fragmentation equation, J. Phys. A, 40 (2007), F331–F337. doi: 10.1088/1751-8113/40/17/F03. [11] A. K. Giri, Mathematical and Numerical Analysis for Coagulation-Fragmentation Equations, Ph.D thesis, Otto-von-Guericke University Magdeburg, Germany, 2010. [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 equation with multiple fragmentation, Nonlinear Analysis, 75 (2012), 2199-2208.  doi: 10.1016/j.na.2011.10.021. [14] M. Kostoglou and A. J. Karabelas, A study of the nonlinear breakage equation: Analytical and asymptotic solutions, J. Phys. A. Math. Gen., 33 (2000), 1221-1232.  doi: 10.1088/0305-4470/33/6/309. [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, Weak compactness techniques and coagulation equations, Evolutionary Equations with Applications in Natural Sciences, J. Banasiak & M. Mokhtar-Kharroubi (eds.), Lecture Notes Math. 2126 Springer, 199–253, (2015). [17] Ph. Laurençot and D. Wrzosek, The discrete coagulation equations with collisional breakage, J. Statist. Phys., 104 (2001), 193-220.  doi: 10.1023/A:1010309727754. [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] V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, (Israel Program for Scientific Translations Ltd., Jerusalem), 1972. [20] 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. [21] I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Comb. Phil. Soc., 107 (1990), 573-578.  doi: 10.1017/S0305004100068821. [22] R. D. Vigil, I. Vermeersch and R. O. Fox, Destructive aggregation: Aggregation with collision-induced breakage, Journal of Colloid and Interface Science, 302 (2006), 149-158.  doi: 10.1016/j.jcis.2006.05.066. [23] D. Wilkins, A geometrical interpretation of the coagulation equation, J. Phys. A., 15 (1982), 1175-1178.  doi: 10.1088/0305-4470/15/4/020.

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##### References:
 [1] R. B. Ash, Measure, Integration and Functional Analysis, Academic Press, New York, 1972. [2] P. K. Barik, Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation, Evolution Equations and Control Theory, 9 (2020), 431-416.  doi: 10.3934/eect.2020012. [3] P. K. Barik and A. K. Giri, A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation, Kinet. Relat. Models, 11 (2018), 1125-1138.  doi: 10.3934/krm.2018043. [4] P. K. Barik, A. K. Giri and Ph. Laurençot, Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel, Proc. Roy. Soc. Edinb. Sec. A: Math., (2019). doi: 10.1017/prm.2018.158. [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [6] P. S. Brown, Structural stability of the coalescence/breakup equations, J. Atmospheric Sci., 52 (1995), 3857-3865. [7] Z. Cheng and S. Redner, Kinetics of fragmentation, J. Phys., 23 (1990), 1233-1258.  doi: 10.1088/0305-4470/23/7/028. [8] Z. Cheng and S. Redner, Scaling theory of fragmentation, Phys. Rev. Lett., 60 (1988), 2450-2453.  doi: 10.1103/PhysRevLett.60.2450. [9] R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York, 1965. [10] M. Ernst and I. Pagonabarraga, The nonlinear fragmentation equation, J. Phys. A, 40 (2007), F331–F337. doi: 10.1088/1751-8113/40/17/F03. [11] A. K. Giri, Mathematical and Numerical Analysis for Coagulation-Fragmentation Equations, Ph.D thesis, Otto-von-Guericke University Magdeburg, Germany, 2010. [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 equation with multiple fragmentation, Nonlinear Analysis, 75 (2012), 2199-2208.  doi: 10.1016/j.na.2011.10.021. [14] M. Kostoglou and A. J. Karabelas, A study of the nonlinear breakage equation: Analytical and asymptotic solutions, J. Phys. A. Math. Gen., 33 (2000), 1221-1232.  doi: 10.1088/0305-4470/33/6/309. [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, Weak compactness techniques and coagulation equations, Evolutionary Equations with Applications in Natural Sciences, J. Banasiak & M. Mokhtar-Kharroubi (eds.), Lecture Notes Math. 2126 Springer, 199–253, (2015). [17] Ph. Laurençot and D. Wrzosek, The discrete coagulation equations with collisional breakage, J. Statist. Phys., 104 (2001), 193-220.  doi: 10.1023/A:1010309727754. [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] V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, (Israel Program for Scientific Translations Ltd., Jerusalem), 1972. [20] 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. [21] I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Comb. Phil. Soc., 107 (1990), 573-578.  doi: 10.1017/S0305004100068821. [22] R. D. Vigil, I. Vermeersch and R. O. Fox, Destructive aggregation: Aggregation with collision-induced breakage, Journal of Colloid and Interface Science, 302 (2006), 149-158.  doi: 10.1016/j.jcis.2006.05.066. [23] D. Wilkins, A geometrical interpretation of the coagulation equation, J. Phys. A., 15 (1982), 1175-1178.  doi: 10.1088/0305-4470/15/4/020.
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