Smoluchowski's Coagulation Equation: Probabilistic Interpretation of Solutions for Constant, Additive and Multiplicative Kernels

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Volumn 29, Issue 3, 2000, Pages 549-579

  Deaconu, Madalina ^a   Tanré, Etienne ^a  

^a UNIVERSITÉ HENRI POINCARÉ NANCY 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0012548294     PISSN: 0391173X     EISSN: 20362145     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. [Ald99] D. J. ALDOUS, Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): A Review ofthe Mean-Field TheoryforProbabilists, Bernoulli 5 (1999), 3-48.
2. [AN72] K. B. A T H R E Y A - P. E. N E Y, "Branching Processes", Springer, 1972.
3. [Bab99] H. BABOVSKY, On a Monte Carlo schemefor SmoluchowskVs coagulation equation, Monte Carlo Methods Appi. 5 (1999), 1-18.
4. [BC90] J. M. B A L L - J. CARR, The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation, J. Stat. Phys. 61 (1990), 203-234.
5. [CdC92] J. CARR - F. P. DA COSTA, Instantaneous gelation in coagulation dynamics, Z. Angew. Math. Phys. 43 (1992), 974-983.
6. [Dra72] R. D R A K E, " A general mathematical survey of the coagulation equation", vol. 3, Pergamon Press, Oxford, 1972, 201-376.
7. [Dub94] P. B. DuBOVSKii, "Mathematical theory of coagulation". Global Analysis Research Center, Seoul National University, vol. 23, 1994.
8. [EP98] S. N. EVANS - J. PITMAN, Construction ofMarkovian coalescents, Ann. Inst. H. Poincaré Probab. Statis. 34 (1998), 339-383.
9. [EZH84] M. H. ERNST - R. M. Z I F F - E. M. HENDRIKS, Coagulation Processes with phase transition, J. Colloid Interface Sci. 97 (1984), 266-277.
10. [Fel66] W. FELLER, "An Introduction to Probability Theory and Its Apphcations", John Wiley and Sons, 1966.
11. [Flo41a] P. J. FLORY, Molecular size distribution in three dimensional polymers. I. Gelation, J. Amer. Chem. Soc. 63 (1941), 3091-3096.
12. [Flo41b] P. J. FLORY, Molecular size distribution in three dimensional polymers. II. Trifunctional branching units, J. Amer. Chem. Soc. 63 (1941), 3083-3090.
13. [Flo41c] P. J. F L O R Y, Molecular size distribution in three dimensional polymers. III. Tetra-functional branching units, J. Amer. Chem. Soc. 63 (1941), 3096-3100.
14. [Gol63] A. M. GoLOViN, The solution of the coagulating equation for cloud droplets in a rising air current, Izv. Geophys. 5 (1963), 482-487.
15. [Gor62] M. GORDON, Good's theory ofcascade processes applied to the statistics ofpolymer distribution, Proc. Royal Soc. London Ser. A 268 (1962), 240-259.
16. [Gui98] F. GUIAS, Coagulation-fragmentation processes: relations between finite particle models and differential equations, PhD thesis, Universität Heidelberg, 1998.
17. [Hei92] O. J. HEILMANN, Analytical solutions of Smoluchowski's coagulation equation, J. Phys. A 25 (1992), 3763-3771.
18. [Jeo98] I. JEON, Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys. 194 (1998), 541-567.
19. [Kok88] N. J. KoKHOLM, On SmoluchowskVs coagulation equation, J. Phys. A 21 (1988), 839-842.
20. [Lau99] P. LAURENpoT, The discrete coagulation equations: existence ofsolutionsandgelation. Private Communication, 1999.
21. [Ley84] F. L E Y V R A Z, Phys. Rev. A 29 (1984), 854.
22. [LN80] R. LANG - X. X. NGUYEN, SmoluchowskVs theory ofcoagulation in colloids holds rigorously in the Boltzmann-Grad-Limit, Z. Wahrscheinlichkeitstheorie verw. Gebiete 54(1980), 227-280.
23. [LT81] F. LEYVRAZ - H. R. TSCHUDI, Singularities in the kinetics ofcoagulationprocesses, J. Phys. A: Math. Gen. 14 (1981), 3389-3405.
24. [Mc62] J. B. MCLEOD, On an infinite set ofnon-linear differential equations, Quart. J. Math. Oxford Ser. (2), 13 (1962), 119-128.
25. [Mel57] Z. A. MELZAK, A scalar transport equation. Trans. Amer. Math. Soc. 85 (1957), 547-560.
26. [Nor99] J. R. NORRIS, Smoluchowski*s coagulation equation: uniqueness, non-uniqueness and hydrodynamic limitfor the stochastic coalescent, Ann. Appi. Probab. 9 (1999), 78-109.
27. [Smo 16] M. V. SMOLUCHOWSKI, Drei Vortage über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen, Physik 17 (1916), 557-585.
28. [Spo83a] J. L. SPOUGE, Asymmetrie bonding of identical units: a general AgRBf-g polymer model, Macromolecules 16 (1983), 831-835.
29. [Spo83b] J. L. SPOUGE, Equilibrium polymer distributions, Macromolecules 16 (1983), 121¬ 127.
30. [Spo83c] J. L. SPOUGE, The size distributionfor the AgRBf-g model ofpolymerization, J. Stat. Phys. 31 (1983), 363-378.
31. [Spo84] J. L. SPOUGE, A branching-process solution ofthe polydisperse coagulation equation. Adv. Appi. Prob. 16 (1984), 56-69.
32. [vDE85] P. G. J VAN DoNGEN - M. H. ERNST, Cluster size distribution in irreversible aggregation at large times, J. Phys. A: Math. Gen. 18 (1985), 2779-2793.