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April  2021, 14(2): 389-406. doi: 10.3934/krm.2021009

Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates

Prasanta Kumar Barik 1,, , Ankik Kumar Giri 2, and  Rajesh Kumar 3,

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
3. 

Department of Mathematics, Birla Institute of Technology and Science, Pilani, Pilani-333031, Rajasthan, India

* Corresponding author: Prasanta Kumar Barik

Received  March 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

In this article, the existence of mass-conserving solutions is investigated to the continuous coagulation and collisional breakage equation with singular collision kernels. Here, the probability distribution function attains singularity near the origin. The existence result is constructed by using both conservative and non-conservative truncations to the continuous coagulation and collisional breakage equation. The proof of the existence result relies on a classical weak $ L^1 $ compactness method.

Keywords: Weak compactness, particle, mass-conserving solution, existence.
Mathematics Subject Classification: Primary: 45K05; Secondary: 34K30.
Citation: Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, 2021, 14 (2) : 389-406. doi: 10.3934/krm.2021009
References:
[1]

P. K. Barik, Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation, Evol. Equ. Control Theory, 9 (2020), 431-446.  doi: 10.3934/eect.2020012.  Google Scholar

[2]

P. K. Barik and A. K. Giri, A note on mass-conserving solutions to the coagulation and fragmentation equation by using non-conservative approximation, Kinet. Relat. Models, 11 (2018), 1125-1138.  doi: 10.3934/krm.2018043.  Google Scholar

[3]

P. K. Barik and A. K. Giri, Existence and uniqueness of weak solutions to the singular kernels coagulation equation with collisional breakage, arXiv: 1806.03911, (2018). Google Scholar

[4]

P. K. Barik and A. K. Giri, Global classical solutions to the continuous coagulation equation with collisional breakage, Z. Angew. Math. Phys., 71 (2020), 1-23.  doi: 10.1007/s00033-020-1261-5.  Google Scholar

[5]

P. K. Barik and A. K. Giri, Weak solutions to the continuous coagulation model with collisional breakage, Discrete Contin. Dyn. Syst., 40 (2020), 6115-6133.  doi: 10.3934/dcds.2020272.  Google Scholar

[6]

P. K. Barik, A. K. Giri and P. Laurençot, Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1805-1825.  doi: 10.1017/prm.2018.158.  Google Scholar

[7]

P. S. Brown, Structural stability of the coalescence/breakage equations, J. Atmosph. Sci., 52 (1995), 3857-3865.   Google Scholar

[8]

C. C. Camejo and G. Warnecke, The singular kernel coagulation equation with multifragmentation, Math. Methods Appl. Sci., 38 (2015), 2953-2973.  doi: 10.1002/mma.3272.  Google Scholar

[9]

Z. Cheng and S. Redner, Kinetics of fragmentation, J. Phys. A. Math. Gen., 23 (1990), 1233-1258.  doi: 10.1088/0305-4470/23/7/028.  Google Scholar

[10]

Z. Cheng and S. Redner, Scaling theory of fragmentation, Phys. Rev. Lett., 60 (1988), 2450-2453.  doi: 10.1103/PhysRevLett.60.2450.  Google Scholar

[11]

M. H. Ernst and I. Pagonabarraga, The nonlinear fragmentation equation, J. Phys. A. Math. Theor., 40 (2007), F331–F337. doi: 10.1088/1751-8113/40/17/F03.  Google Scholar

[12]

M. Escobedo, P. 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

[13]

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

[14]

A. K. Giri, P. 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

[15]

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.  Google Scholar

[16]

P. Laurençot, Mass-conserving solutions to coagulation-fragmentation equations with nonintegrable fragment distribution function, Quart. Appl. Math., 76 (2018), 767-785.  doi: 10.1090/qam/1511.  Google Scholar

[17]

P. Laurençot, Weak compactness techniques and coagulation equations, Evolutionary Equations with Applications in Natural Sciences, J. Banasiak & M. Mokhtar-Kharroubi (eds.), Lecture Notes Math., 2126 (2015), 199–253. doi: 10.1007/978-3-319-11322-7_5.  Google Scholar

[18]

P. 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

[19]

P. Laurençot and D. Wrzosek, The discrete coagulation equations with collisional breakage, J. Statist. Phys., 104 (2001), 193-220.  doi: 10.1023/A:1010309727754.  Google Scholar

[20]

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

[21]

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.  Google Scholar

[22]

V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations Ltd. Jerusalem, 1972. Google Scholar

[23]

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

[24]

R. D. Vigil, I. Vermeersch and R. O. Fox, Destructive aggregation: aggregation with collision-induced breakage, Colloid and Interface Science, 302 (2006), 149-158.  doi: 10.1016/j.jcis.2006.05.066.  Google Scholar

[25]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd edition, Pitman Monogr. Surveys Pure Appl. Math., Longman, 1995.  Google Scholar

[26]

D. Wilkins, A geometrical interpretation of the coagulation equation, J. Phys. A, 15 (1982), 1175-1178.  doi: 10.1088/0305-4470/15/4/020.  Google Scholar

show all references

References:
[1]

P. K. Barik, Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation, Evol. Equ. Control Theory, 9 (2020), 431-446.  doi: 10.3934/eect.2020012.  Google Scholar

[2]

P. K. Barik and A. K. Giri, A note on mass-conserving solutions to the coagulation and fragmentation equation by using non-conservative approximation, Kinet. Relat. Models, 11 (2018), 1125-1138.  doi: 10.3934/krm.2018043.  Google Scholar

[3]

P. K. Barik and A. K. Giri, Existence and uniqueness of weak solutions to the singular kernels coagulation equation with collisional breakage, arXiv: 1806.03911, (2018). Google Scholar

[4]

P. K. Barik and A. K. Giri, Global classical solutions to the continuous coagulation equation with collisional breakage, Z. Angew. Math. Phys., 71 (2020), 1-23.  doi: 10.1007/s00033-020-1261-5.  Google Scholar

[5]

P. K. Barik and A. K. Giri, Weak solutions to the continuous coagulation model with collisional breakage, Discrete Contin. Dyn. Syst., 40 (2020), 6115-6133.  doi: 10.3934/dcds.2020272.  Google Scholar

[6]

P. K. Barik, A. K. Giri and P. Laurençot, Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1805-1825.  doi: 10.1017/prm.2018.158.  Google Scholar

[7]

P. S. Brown, Structural stability of the coalescence/breakage equations, J. Atmosph. Sci., 52 (1995), 3857-3865.   Google Scholar

[8]

C. C. Camejo and G. Warnecke, The singular kernel coagulation equation with multifragmentation, Math. Methods Appl. Sci., 38 (2015), 2953-2973.  doi: 10.1002/mma.3272.  Google Scholar

[9]

Z. Cheng and S. Redner, Kinetics of fragmentation, J. Phys. A. Math. Gen., 23 (1990), 1233-1258.  doi: 10.1088/0305-4470/23/7/028.  Google Scholar

[10]

Z. Cheng and S. Redner, Scaling theory of fragmentation, Phys. Rev. Lett., 60 (1988), 2450-2453.  doi: 10.1103/PhysRevLett.60.2450.  Google Scholar

[11]

M. H. Ernst and I. Pagonabarraga, The nonlinear fragmentation equation, J. Phys. A. Math. Theor., 40 (2007), F331–F337. doi: 10.1088/1751-8113/40/17/F03.  Google Scholar

[12]

M. Escobedo, P. 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

[13]

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

[14]

A. K. Giri, P. 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

[15]

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.  Google Scholar

[16]

P. Laurençot, Mass-conserving solutions to coagulation-fragmentation equations with nonintegrable fragment distribution function, Quart. Appl. Math., 76 (2018), 767-785.  doi: 10.1090/qam/1511.  Google Scholar

[17]

P. Laurençot, Weak compactness techniques and coagulation equations, Evolutionary Equations with Applications in Natural Sciences, J. Banasiak & M. Mokhtar-Kharroubi (eds.), Lecture Notes Math., 2126 (2015), 199–253. doi: 10.1007/978-3-319-11322-7_5.  Google Scholar

[18]

P. 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

[19]

P. Laurençot and D. Wrzosek, The discrete coagulation equations with collisional breakage, J. Statist. Phys., 104 (2001), 193-220.  doi: 10.1023/A:1010309727754.  Google Scholar

[20]

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

[21]

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.  Google Scholar

[22]

V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations Ltd. Jerusalem, 1972. Google Scholar

[23]

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

[24]

R. D. Vigil, I. Vermeersch and R. O. Fox, Destructive aggregation: aggregation with collision-induced breakage, Colloid and Interface Science, 302 (2006), 149-158.  doi: 10.1016/j.jcis.2006.05.066.  Google Scholar

[25]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd edition, Pitman Monogr. Surveys Pure Appl. Math., Longman, 1995.  Google Scholar

[26]

D. Wilkins, A geometrical interpretation of the coagulation equation, J. Phys. A, 15 (1982), 1175-1178.  doi: 10.1088/0305-4470/15/4/020.  Google Scholar

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