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The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. (English) Zbl 1217.82050
Summary: The discrete coagulation-fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. These models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys. Our results relate to existence, uniqueness, density conservation and continuous dependence and they generalize the corresponding results for the Becker-Döring equations for which the processes are restricted to clusters gaining or shedding one particle. Examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation-fragmentation equations.

82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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[1] M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers,Commun. Math. Phys. 65:203?230 (1979). · Zbl 0458.76062
[2] J. M. Ball, J. Carr, and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions,Commun. Math. Phys. 104:657?692 (1986). · Zbl 0594.58063
[3] J. M. Ball and J. Carr, Asymptotic behaviour of solutions to the Becker-Döring equations for arbitrary initial data,Proc. R. Soc. Edinburgh 108A:109?116 (1988). · Zbl 0656.58021
[4] J. M. Ball and J. Carr, In preparation.
[5] K. Binder, Theory for the dynamics of clusters. II. Critical diffusion in binary systems and the kinetics of phase separation,Phys. Rev. B 15:4425?4447 (1977).
[6] P. G. J. van Dongen, Spatial fluctuations in reaction-limited aggregation,J. Stat. Phys. 54:221?271 (1989).
[7] R. Drake, InTopics in Current Aerosol research, G. M. Hidy and J. R. Brock, eds. (Pergamon Press, Oxford, 1972).
[8] E. M. Hendricks, M. H. Ernst, and R. M. Ziff, Coagulation equations with gelation,J. Stat. Phys. 31:519?563 (1983).
[9] F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes,J. Phys. A: Math. Gen. 14:3389?3405 (1981). · Zbl 0481.92020
[10] F. Leyvraz and H. R. Tschudi, Critical kinetics near gelation,J. Phys. A: Math. Gen. 15:1951?1964 (1982). · Zbl 0481.92020
[11] J. B. McLeod, On an infinite set of non-linear differential equations,Q. J. Math. Ser. (2) 13:119?128 (1962). · Zbl 0109.31501
[12] O. Penrose, Metastable states for the Becker-Döring cluster equations,Commun. Math. Phys. 121:527?540 (1989). · Zbl 0701.58007
[13] G. E. H. Reuter and W. Ledermann, On the differential equations for the transitional probabilities of Markov processes with enumerably many states,Proc. Camb. Phil. Soc. 49:247?262 (1953). · Zbl 0053.27202
[14] M. Slemrod, Trend to equilibrium in the Becker-Döring cluster equations,Nonlinearity 2:429?443 (1989). · Zbl 0709.60528
[15] J. L. Spouge, An existence theorem for the discrete coagulation-fragmentation equations,Math. Proc. Camb. Phil. Soc. 96:351?357 (1984). · Zbl 0541.92029
[16] I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels,Math. Meth. Appl. Sci. 11:627?648 (1989). · Zbl 0683.45006
[17] W. H. White, A global existence theorem for Smoluchowski’s coagulation equations,Proc. Am. Math. Soc. 80:273?276 (1980). · Zbl 0442.34003
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