June  2020, 9(2): 431-446. doi: 10.3934/eect.2020012

Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation

1. 

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

2. 

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore-560065, Karnataka, India

Received  March 2019 Revised  April 2019 Published  June 2020 Early access  August 2019

In this paper we study the continuous coagulation and multiple fragmentation equation for the mean-field description of a system of particles taking into account the combined effect of the coagulation and the fragmentation processes in which a system of particles growing by successive mergers to form a bigger one and a larger particle splits into a finite number of smaller pieces. We demonstrate the global existence of mass-conserving weak solutions for a wide class of coagulation rate, selection rate and breakage function. Here, both the breakage function and the coagulation rate may have algebraic singularity on both the coordinate axes. The proof of the existence result is based on a weak $ L^1 $ compactness method for two different suitable approximations to the original problem, namely, the conservative and non-conservative approximations. Moreover, the mass-conservation property of solutions is established for both approximations.

Citation: Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations and Control Theory, 2020, 9 (2) : 431-446. doi: 10.3934/eect.2020012
References:
[1]

D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernouli, 5 (1999), 3-48.  doi: 10.2307/3318611.

[2]

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.

[3]

P. K. Barik, A. K. Giri and P. Laurençot, Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel, Proc. Royal Soc. Edinburgh Sec. A: Math., (2019), 1–21. doi: 10.1017/prm.2018.158.

[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. 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.

[6]

J. M. C. Clark and V. Katsouros, Stably coalescent stochastic froths, Adv. Appl. Probab., 31 (1999), 199-219.  doi: 10.1239/aap/1029954273.

[7]

P. B. Dubovskiǐ 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.

[8]

M. EscobedoP. LaurençotS. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Diff. Eqs., 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7.

[9]

F. Filbet and P. 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 P. 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. GiriJ. 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. GiriP. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation equation with multiple fragmentation, Nonlinear Anal., 75 (2012), 2199-2208.  doi: 10.1016/j.na.2011.10.021.

[14]

P. C. Kapur, Kinetics of granulation by non-random coalescence mechanism, Chem. Eng. Sci., 27 (1972), 1863-1869.  doi: 10.1016/0009-2509(72)85048-6.

[15]

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

[16]

P. Laurençot, Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels, J. Statist. Phys., 171 (2018), 484-492.  doi: 10.1007/s10955-018-2018-9.

[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.

[18]

P. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Royal Soc. Edinburgh Sect. A, 132 (2002), 1219-1248.  doi: 10.1017/S0308210502000598.

[19]

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.

[20]

D. J. McLaughlinW. 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]

J. R. Norris, Smoluchowski's coagulation equation: Uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab., 9 (1999), 78-109.  doi: 10.1214/aoap/1029962598.

[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.

[24]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd edition, Pitman Monogr. Surveys Pure Appl. Math., 75, Longman, 1995.

show all references

References:
[1]

D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernouli, 5 (1999), 3-48.  doi: 10.2307/3318611.

[2]

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.

[3]

P. K. Barik, A. K. Giri and P. Laurençot, Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel, Proc. Royal Soc. Edinburgh Sec. A: Math., (2019), 1–21. doi: 10.1017/prm.2018.158.

[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. 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.

[6]

J. M. C. Clark and V. Katsouros, Stably coalescent stochastic froths, Adv. Appl. Probab., 31 (1999), 199-219.  doi: 10.1239/aap/1029954273.

[7]

P. B. Dubovskiǐ 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.

[8]

M. EscobedoP. LaurençotS. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Diff. Eqs., 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7.

[9]

F. Filbet and P. 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 P. 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. GiriJ. 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. GiriP. Laurençot and G. Warnecke, Weak solutions to the continuous coagulation equation with multiple fragmentation, Nonlinear Anal., 75 (2012), 2199-2208.  doi: 10.1016/j.na.2011.10.021.

[14]

P. C. Kapur, Kinetics of granulation by non-random coalescence mechanism, Chem. Eng. Sci., 27 (1972), 1863-1869.  doi: 10.1016/0009-2509(72)85048-6.

[15]

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

[16]

P. Laurençot, Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels, J. Statist. Phys., 171 (2018), 484-492.  doi: 10.1007/s10955-018-2018-9.

[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.

[18]

P. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Royal Soc. Edinburgh Sect. A, 132 (2002), 1219-1248.  doi: 10.1017/S0308210502000598.

[19]

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.

[20]

D. J. McLaughlinW. 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]

J. R. Norris, Smoluchowski's coagulation equation: Uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab., 9 (1999), 78-109.  doi: 10.1214/aoap/1029962598.

[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.

[24]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd edition, Pitman Monogr. Surveys Pure Appl. Math., 75, Longman, 1995.

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