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A uniqueness theorem for the coagulation-fragmentation equation

Published online by Cambridge University Press:  24 October 2008

I. W. Stewart
I. W. Stewart
Affiliation:
Mathematics Department, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS
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Extract

This paper presents a uniqueness result for solutions to the general nonlinear coagulation-fragmentation equation

where

Equation (1·1) has many applications in the applied sciences (cf. [1, 3, 8, 13, 15]) and a brief physical interpretation can be found in Melzak [12] or the survey article by Drake[7]. c(x, t), for x ≥ 0, t ≥ 0, denotes the number of particles of size x at time t and the non-negative kernels K and F describe, respectively, the rates at which particles of size x coalesce with those of size y and particles of size (x + y) break-up into those of sizes x and y.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 3 , May 1990 , pp. 573 - 578
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Aizenman, M. and Bak, T. A.. Convergence to equilibrium in a system of reacting polymers. Comm. Math. Phys. 65 (1979), 203–320.CrossRefGoogle Scholar
[2]Ball, J. M. and Carr, J.. Coagulation-fragmentation dynamics. In Dynamics of Infinite-dimensional Systems (Springer-Verlag, 1987), pp. 23–30.CrossRefGoogle Scholar
[3]Barrow, J. D.. Coagulation with fragmentation. J. Phys. A 14 (1981), 729–733.CrossRefGoogle Scholar
[4]Burobin, A. V.. Existence and Uniqueness of a Solution of a Cauchy Problem for the Inhomogeneous Three Dimensional Coagulation Equation. (English translation.) Differential Equations 10 (1983), 1568–1579.Google Scholar
[5]Burobin, A. V. and Galkin, V. A.. Solutions of the Coagulation Equation. (English translation.) Differential Equations 17 (1981), 669–677.Google Scholar
[6]Carr, J.. (Private communication.)Google Scholar
[7]Drake, R.. A general mathematical survey of the coagulation equation. In Topics in Current Aerosol Research, International Reviews in Aerosol Physics and Chemistry, vol. 2 (ed. Hidy, C. M. and Brock, J. R.) (Pergamon, 1972), pp. 203–368.Google Scholar
[8]Ernst, M. H., Ziff, R. M. and Hendriks, E. M.. Coagulation processes with a phase transition. J. Coll. Interf. Sci. 97 (1984), 266–277.CrossRefGoogle Scholar
[9]Galkin, V. A.. Existence and uniqueness of a solution of the coagulation equation. (English translation.) Differential Equations 13 (1977), 1460–1470.Google Scholar
[10]Galkin, V. A. and Dubovskii, P. B.. Solution of the coagulation equation with unbounded kernels. (English translation.) Differential Equations 22 (1986), 504–509.Google Scholar
[11]Mcleod, J. B.. On the scalar transport equation. Proc. London Math. Soc. (3) 14 (1964), 445–458.CrossRefGoogle Scholar
[12]Melzak, Z. A.. A Scalar Transport Equation. Trans. Amer. Math. Soc. 85 (1957), 547–560.CrossRefGoogle Scholar
[13]Piskunov, V. N.. The steady spectra of particles in dispersible systems with coagulation and fragmentation. (English translation.) J. Appi. Math. Mech. 49 (1985), 789–793.CrossRefGoogle Scholar
[14]Stewart, I. W.. A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11(1989), 627–648.CrossRefGoogle Scholar
[15]Ziff, R. M. and McGrady, E. D.. The kinetics of cluster fragmentation and depolymerisation. J. Phys. A 18 (1985), 3027–3037.CrossRefGoogle Scholar