American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling
  • KRM Home
  • This Issue
  • Next Article
    Boundary layers and stabilization of the singular Keller-Segel system
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

[1]

Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020012
[2]

Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589
[3]

Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445
[4]

Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563
[5]

Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014
[6]

Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009
[7]

Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541
[8]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126
[9]

Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177
[10]

Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic & Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040
[11]

Jacek Banasiak. Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020161
[12]

Mihai Bostan, Claudia Negulescu. Mathematical models for strongly magnetized plasmas with mass disparate particles. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 513-544. doi: 10.3934/dcdsb.2011.15.513
[13]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[14]

Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070
[15]

Gerhard Rein. Galactic dynamics in MOND---Existence of equilibria with finite mass and compact support. Kinetic & Related Models, 2015, 8 (2) : 381-394. doi: 10.3934/krm.2015.8.381
[16]

Jorge Cortés. Energy conserving nonholonomic integrators. Conference Publications, 2003, 2003 (Special) : 189-199. doi: 10.3934/proc.2003.2003.189
[17]

Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136
[18]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213
[19]

Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117
[20]

Jifeng Chu, Zaitao Liang, Pedro J. Torres, Zhe Zhou. Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2669-2685. doi: 10.3934/dcdsb.2017130

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (69)
  • HTML views (119)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences