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Asymptotic behaviour of solutions to the coagulation–fragmentation equations. I. The strong fragmentation case

Published online by Cambridge University Press:  14 November 2011

J. Carr
J. Carr
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, U.K.
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Synopsis

The discrete coagulation-fragmentation equations are a model for the time-evolution of cluster growth. The processes described by the model are the coagulation of clusters via binary interactions and the fragmentation of clusters. The assumptions made on the fragmentation coefficients in this paper have the physical interpretation that surface effects are not important, i.e. it is unlikely that a large cluster will fragment into two large pieces. Since solutions of the initial-value problem are not unique, we have to restrict the class of solutions. With this restriction, we prove that the fragmentation acts as a strong damping mechanism and we obtain results on the asymptotic behaviour of solutions. The main tool used is an estimate on the moments of admissible solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 3-4 , 1992 , pp. 231 - 244
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Aizenman, M. and Bak, T. A.. Convergence to equilibrium in a system of reacting polymers. Comm. Math. Phys. 65 (1979) 203–230.CrossRefGoogle Scholar
2Ball, J. M., Carr, J. and Penrose, O.. The Becker-Doring cluster equations: Basic properties and asymptotic behaviour of solutions. Comm. Math. Phys. 104 (1986), 657–692.CrossRefGoogle Scholar
3Ball, J. M. and Carr, J.. Asymptotic behaviour of solutions to the Becker-Doring equations for arbitrary initial data. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 109–116.CrossRefGoogle Scholar
4Ball, J. M. and Carr, J.. The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation. J. Stat. Phys. 61 (1990), 203–234.CrossRefGoogle Scholar
5Ernst, M. H.. Exact Solutions of the Nonlinear Boltzmann Equation and Related Kinetic Equations. In Studies in Statistical Mechanics, Vol. X, eds Montroll, E. W. and Lebowitz, J. L., 51–120 (Amsterdam: North-Holland, 1983).Google Scholar
6Fisher, M. E.. The Nature of Critical Points. In Lectures in Theoretical Physics, VIIC, 1–159 (Boulder, Colorado: University of Colorado Press, 1965).Google Scholar
7Leyvraz, F.. Existence and properties of post-gel solutions for the kinetic equations of coagulation. J. Phys. A: Math. Gen. 16 (1983), 2861–2873.CrossRefGoogle Scholar