Abstract
The discrete coagulation equations with collisional breakage describe the dynamics of cluster growth when clusters undergo binary collisions resulting either in coalescence or breakup with possible transfer of matter. Each of these two events may happen with an a priori prescribed probability depending for instance on the sizes of the colliding clusters. We study the existence, density conservation and uniqueness of solutions. We also consider the large time behaviour and discuss the possibility of the occurrence of gelation in some particular cases.
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REFERENCES
D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists, Bernoulli 5:3–48 (1999).
J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Statist. Phys. 61:203–234(1990).
J. M. Ball, J. Carr, and O. Penrose, The Becker–Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Comm. Math. Phys. 104:657–692 (1986).
J. Carr, Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case, Proc. Roy. Soc. Edinburgh Sect. A 121:231–244 (1992).
J. Carr and F. P. da Costa, Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Statist. Phys. 77:89–123 (1994).
F. P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation, J. Math. Anal. Appl. 192:892–914 (1995).
F. P. da Costa, Convergence to equilibria of solutions to the coagulation-fragmentation equations, in Nonlinear Evolution Equations and Their Applications (Macau, 1998), pp. 45–56 (World Sci. Publishing, River Edge, NJ, 1999).
Z. Cheng and S. Redner, Scaling theory of fragmentation, Phys. Rev. Lett. 60:2450–2453 (1988).
Z. Cheng and S. Redner, Kinetics of fragmentation, J. Phys. A 23:1233–1258 (1990).
C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Chapitres I–IV (Hermann, Paris, 1975).
R. L. Drake, A general mathematical survey of the coagulation equation, in Topics in Current Aerosol Research (part 2), International Reviews in Aerosol Physics and Chemistry (Pergamon Press, Oxford, 1972), pp. 203–376.
I. Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194:541–567 (1998).
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (Prentice-Hall, Englewood Cliffs, 1970).
M. Kostoglou and A. J. Karabelas, A study of the nonlinear breakage equation: Analytical and asymptotic solutions, J. Phys. A 33:1221–1232 (2000).
Ph. Laurençot, Global solutions to the discrete coagulation equations, Mathematika, to appear.
Ph. Laurençot, The discrete coagulation equations with multiple fragmentation, Proc. Edinburgh Math. Soc. (2), to appear.
Ph. Laurençot and D. Wrzosek, Time asymptotics for a Becker–Döring-type model of aggregation with nonlinear breakage, in preparation.
Ph. Laurençot and D. Wrzosek, in preparation.
Lê Châu-Hoân, Etude de la classe des opérateurs m-accrétifs de L 1(Ω) et accrétifs dans L∞(Ω), Thèse de 3 ème cycle, Université de Paris VI, 1977.
V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets (Israel Program for Scientific Translations Ltd., Jerusalem, 1972).
M. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschr. 17:557–599 (1916).
M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschrift f. physik. Chemie 92:129–168 (1917).
J. L. Spouge, An existence theorem for the discrete coagulation-fragmentation equations, Math. Proc. Cambridge Philos. Soc. 96:351–357 (1984).
R. C. Srivastava, A simple model of particle coalescence and breakup, J. Atmos. Sci. 39:1317–1322 (1982).
W. H. Stockmayer, Theory of molecular size distribution and gel formation in branchedchain polymers, J. Chem. Phys. 11:45–55 (1943).
D. Wilkins, A geometrical interpretation of the coagulation equation, J. Phys. A 15:1175–1178 (1982).
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Laurençot, P., Wrzosek, D. The Discrete Coagulation Equations with Collisional Breakage. Journal of Statistical Physics 104, 193–220 (2001). https://doi.org/10.1023/A:1010309727754
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DOI: https://doi.org/10.1023/A:1010309727754