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, 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
[2]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
[3]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001
[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[5]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[6]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
[7]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[8]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099
[9]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
[10]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011
[11]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024
[12]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[13]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[14]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[15]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[16]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[17]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[18]

Melis Alpaslan Takan, Refail Kasimbeyli. Multiobjective mathematical models and solution approaches for heterogeneous fixed fleet vehicle routing problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2073-2095. doi: 10.3934/jimo.2020059
[19]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405
[20]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (127)
  • HTML views (160)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences