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A note on Smoluchowski's equations with diffusion


Herrero, Miguel A. and Rodrigo, Marianito R. (2005) A note on Smoluchowski's equations with diffusion. Applied Mathematics Letters, 18 (9). pp. 969-975. ISSN 0893-9659

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We consider an infinite system of reaction-diffusion equations that models aggregation of particles. Under suitable assumptions on the diffusion coefficients and aggregation rates, we show that this system can be reduced to a scalar equation, for which an explicit self-similar solution is obtained. In addition, pointwise bounds for the solutions of associated initial and initial-boundary value problems are provided.

Item Type:Article
Uncontrolled Keywords:Particle aggregation; reaction-diffusion; explicit solutions; supersolution; subsolution; kpp-fisher equation; coagulation equations; kinetics; aggregation; existence; discrete; gelation; model; dynamics; behavior
Subjects:Sciences > Mathematics > Differential equations
ID Code:16342

M. von Smoluchowski, Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen, Physik Z. 17 (1916) 557–585.

S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943) 1–91.

M. Slemrod, Coagulation–diffusion systems: derivation and existence of solutions for the diffuse interface structure equations, Physica D 46 (1990) 351–366.

E.M. Hendriks, M.H. Ernst, R.M. Ziff, Coagulation equations with gelation, J. Stat. Phys. 31 (1983) 519–563.

R.M. Ziff, G. Stell, Kinetics of polymer gelation, J. Chem. Phys. 73 (1980) 3492–3499.

R.M. Ziff, Kinetics of polymerization, J. Stat. Phys. 23 (1980) 241–263.

F. Leyvraz, Existence and properties of post-gel solutions for the kinetic equations of coagulation, J. Phys. A 16 (1983) 2861–2873.

F. Leyvraz, H.R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A 14 (1981) 3389–3405.

J.B. McLeod, On an infinite set of non-linear differential equations, Q. J. Math. Oxford 2 (1962) 119–128.

F.P. da Costa, A finite-dimensional dynamical model for gelation in coagulation processes, J. Nonlinear Sci. 8 (1998) 619–653.

P.G.J. van Dongen, M.H. Ernst, Cluster size distribution in irreversible aggregation at large times, J. Phys. A 1985 (1985) 2779–2793.

F. Leyvraz, Large-time behavior of the Smoluchowski equations of coagulation, Phys. Rev. A 29 (1984) 854–858.

K. Binder, Theory for the dynamics of clusters, II. Critical diffusion in binary systems for phase separation, Phys. Rev. B 15 (1977) 4425–4447.

R. Lang, N.X. Xanh, Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann–Grad limit, Z. Wahrsch. verw. Gebiete 54 (1980) 227–280.

P.G.J. van Dongen, Spatial fluctuations in reaction-limited aggregation, J. Stat. Phys. 54 (1989) 221–271.

Ph. Benilan, D. Wrzosek, On an infinite system of reaction–diffusion equations, Adv. Math. Sci. Appl. 7 (1997) 351–366.

M.A. Herrero, J.J.L. Velázquez, D. Wrzosek, Sol–gel transition in a coagulation–diffusion model, Physica D 141 (2000) 221–247.

H. Amann, Coagulation–fragmentation processes, Arch. Ration Mech. Anal. 151 (2000) 339–366.

J.F. Collet, F. Poupaud, Existence of solutions to coagulation–fragmentation systems with diffusion, Transport Theory Statist. Phys. 25 (1996) 503–513.

M. Deaconu, N. Fournier, Probabilistic approach of some discrete and continuous coagulation equations with diffusion, Stochastic Process. Appl. 101 (2002) 83–111.

Ph. Laurencot, S. Mischler, Global existence for the discrete diffusive coagulation–fragmentation equations in L1, Rev. Mat. Iberoamericana 18 (2002) 731–745.

D. Wrzosek, Existence of solutions for the discrete diffusive coagulation–fragmentation model with diffusion, Topol. Methods Nonlinear Anal. 9 (1997) 279–296.

D. Wrzosek, Mass-conserving solutions to the discrete coagulation–fragmentation model with diffusion, Nonlinear Anal. 49 (2002) 297–314.

M. Rodrigo, M. Mimura, Annihilation dynamics in the KPP–Fisher equation, European J. Appl.Math. 13 (2002) 195–204.

M. Rodrigo, Evolution of bounding functions for the solution of the KPP–Fisher equation in bounded domains, Stud. Appl. Math. 110 (2003) 49–61.

D.J. Aldous, Deterministic and stochastic models for coagulation (coalescence and aggregation): a review of themean-field theory for probabilists, Bernoulli 5 (1) (1999) 3–48.

S.C. Davies, J.R. King, J.A.D. Wattis, Self-similar behavior in the coagulation equations, J. Eng. Math. 36 (1999) 57–88.

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