-
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
A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, 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.
References:
[1] |
R. B. Ash,
Measure, Integration and Functional Analysis, Academic Press, New York-London, 1972. |
[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. |
[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. |
[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. Dellacherie and P. A. Mayer,
Probabilitiés et Potentiel, Chapitres I à IV, Paris, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. 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 with multiple fragmentation, Nonlinear Anal., 75 (2012), 2199-2208.
doi: 10.1016/j.na.2011.10.021. |
[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. |
[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. |
[16] |
Ph. Laurençot,
The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci., 11 (2001), 731-748.
doi: 10.1142/S0218202501001070. |
[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. |
[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] |
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. |
[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. |
show all references
References:
[1] |
R. B. Ash,
Measure, Integration and Functional Analysis, Academic Press, New York-London, 1972. |
[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. |
[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. |
[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. Dellacherie and P. A. Mayer,
Probabilitiés et Potentiel, Chapitres I à IV, Paris, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. 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 with multiple fragmentation, Nonlinear Anal., 75 (2012), 2199-2208.
doi: 10.1016/j.na.2011.10.021. |
[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. |
[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. |
[16] |
Ph. Laurençot,
The Lifshitz-Slyozov equation with encounters, Math. Models Methods Appl. Sci., 11 (2001), 731-748.
doi: 10.1142/S0218202501001070. |
[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. |
[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] |
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. |
[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. |
[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
Other articles
by authors
[Back to Top]