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Global solutions to the discrete coagulation equations

Part of: General theory in ordinary differential equations

Published online by Cambridge University Press:  26 February 2010

Philippe Laurençot
Philippe Laurençot
Affiliation:
CNRS and Institut Elie Cartan-Nancy, Universitç de Nancy I, BP 239, F-54506 Vandœuvre-lès-Nancy cedex, France.
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Abstract

The existence of global solutions to the discrete coagulation equations is investigated for a class of coagulation rates of the form ai, j = rirj + αi, j with αi, j≤Krirj. In particular, global solutions are shown to exist when the sequence (ri) increases linearly or superlinearly with respect to i. In this case also, the failure of density conservation (indicating the occurrence of the gelation phenomenon) is studied.

MSC classification

Secondary: 34A34: Nonlinear equations and systems, general
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 433 - 442
Copyright
Copyright © University College London 1999

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References

1. Aldous, D. J.. Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli, 5 (1999), 3–48.Google Scholar
2. Bak, T. A. and Heilmann, O.. A finite version of Smoluchowski's coagulation, equation, J. Phys. A, 24 (1991), 4889–4893.Google Scholar
3. Ball, J. M. and Carr, J.. The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Statist. Phys., 61 (1990), 203–234.Google Scholar
4. Bénilan, Ph. and Wrzosek, D.. On an infinite system of reaction-diffusion equations. Adv. Math. Sci. Appl, 7 (1997), 349–364.Google Scholar
5. Costa, F. P. da. A finite dimensional dynamical model for gelation in coagulation processes. J. Nonlinear Sci., 8 (1998), 619–653.Google Scholar
6. Heilmann, O. J.. Analytical solutions of Smoluchowski's coagulation equation. J. Phys. A, 25 (1992), 3763–3771.Google Scholar
7. Hendriks, E. M., Ernst, M. H. and Ziff, R. M.. Coagulation equations with gelation. J. Statist. Phys., 31 (1983), 519–563.Google Scholar
8. Jeon, I.. Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys., 194 (1998), 541–567.Google Scholar
9. Kokholm, N. J.. On Smoluchowski's coagulation equation. J. Phys. A, 21 (1988), 839–842.CrossRefGoogle Scholar
10. Leyvraz, F. and Tschudi, H. R.. Singularities in the kinetics of coagulation processes. J. Phys. A, 14 (1981), 3389–3405.CrossRefGoogle Scholar
11. Leyvraz, F. and Tschudi, H. R.. Critical kinetics near gelation. J. Phys. A, 15 (1982), 1951–1964.CrossRefGoogle Scholar
12. McLeod, J. B.. On an infinite set of non-linear differential equations. Quart. J. Math. Oxford Ser. (2). 13 (1962), 119–128.CrossRefGoogle Scholar
13. McLeod, J. B.. On an infinite set of non-linear differential equations (II). Quart. J. Math. Oxford Ser. (2), 13 (1962), 193–205.CrossRefGoogle Scholar
14. McLeod, J. B.. On a recurrence formula in differential equations. Quart. J. Math. Oxford Ser. (2), 13 (1962), 283–284.Google Scholar
15. Smoluchowski, M.. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physik. Zeitschr., 17 (1916), 557–599.Google Scholar
16. White, W. H.. A global existence theorem for Smoluchowski's coagulation equations. Proc. Amer. Math. Soc, 80 (1980), 273–176.Google Scholar