Communications in Mathematical Physics

, Volume 256, Issue 3, pp 589–609 | Cite as

Existence of Self-Similar Solutions to Smoluchowski’s Coagulation Equation

  • Nicolas FournierEmail author
  • Philippe Laurençot


The existence of self-similar solutions to Smoluchowski’s coagulation equation has been conjectured for several years by physicists, and numerical simulations have confirmed the validity of this conjecture. Still, there was no existence result up to now, except for the constant and additive kernels for which explicit formulae are available. In this paper, the existence of self-similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Explicit Formula 
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  1. 1.
    Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation, coagulation) : a review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999)Google Scholar
  2. 2.
    Amann, H.: Ordinary differential equations. An introduction to nonlinear analysis. de Gruyter Studies in Mathematics 13, Berlin: Walter de Gruyter & Co., 1990Google Scholar
  3. 3.
    Bertoin, J.: Eternal solutions to Smoluchowski’s coagulation equation with additive kernel and their probabilistic interpretation. Ann. Appl. Probab. 12, 547–564 (2002)CrossRefGoogle Scholar
  4. 4.
    da Costa, F.P.: On the dynamic scaling behaviour of solutions to the discrete Smoluchowski equations. Proc. Edinburgh Math. Soc. (2) 39, 547–559 (1996)Google Scholar
  5. 5.
    Cueille, S., Sire, C.: Nontrivial polydispersity exponents in aggregation models. Phys. Rev. E 55, 5465–5478 (1997)CrossRefGoogle Scholar
  6. 6.
    Deaconu, M., Tanré, E.: Smoluchowski’s coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29, 549–579 (2000)Google Scholar
  7. 7.
    van Dongen, P.G.J., Ernst, M.H.: Scaling solutions of Smoluchowski’s coagulation equation. J. Statist. Phys. 50, 295–329 (1988)CrossRefGoogle Scholar
  8. 8.
    Drake, R.L.: A general mathematical survey of the coagulation equation. In: “Topics in Current Aerosol Research (part 2),” International Reviews in Aerosol Physics and Chemistry, Oxford: Pergamon Press, 1972, pp. 203–376Google Scholar
  9. 9.
    Escobedo, M., Mischler, S. Perthame, B.: Gelation in coagulation and fragmentation models. Commun. Math. Phys. 231, 157–188 (2002)CrossRefGoogle Scholar
  10. 10.
    Filbet, F., Laurençot, Ph.: Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Sci. Comput. 25, 2004–2028 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Friedlander, S.K., Wang, C.S.: The self-preserving particle size distribution for coagulation by brownian motion. J. Colloid Interface Sci. 22, 126–132 (1966)CrossRefGoogle Scholar
  12. 12.
    Jeon, I.: Existence of gelling solutions for coagulation-fragmentation equations, Commun. Math. Phys. 194, 541–567 (1998)CrossRefGoogle Scholar
  13. 13.
    Kreer, M., Penrose, O.: Proof of dynamical scaling in Smoluchowski’s coagulation equation with constant kernel. J. Statist. Phys. 75, 389–407 (1994)Google Scholar
  14. 14.
    Krivitsky, D.S.: Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function. J. Phys. A 28, 2025–2039 (1995)MathSciNetGoogle Scholar
  15. 15.
    Laurençot, Ph., Mischler, S.: On coalescence equations and related models. In: “Modeling and computational methods for kinetic equations”. P. Degond, L. Pareschi, G. Russo (eds.), Boston: Birkhäuser, 2004, pp. 321–356Google Scholar
  16. 16.
    Laurençot, Ph., Mischler, S.: Liapunov functionals for Smoluchowski’s coagulation equation and convergence to self-similarity. Monatsh. Math., to appearGoogle Scholar
  17. 17.
    Lee, M.H.: A survey of numerical solutions to the coagulation equation. J. Phys. A 34, 10219–10241 (2001)Google Scholar
  18. 18.
    Leyvraz, F.: Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Rep. 383, 95–212 (2003)CrossRefGoogle Scholar
  19. 19.
    Lushnikov, A.A., Kulmala, M.: Singular self-preserving regimes of coagulation processes. Phys. Rev. E 65, 041604, (2002)CrossRefGoogle Scholar
  20. 20.
    Meesters, A., Ernst, M.H.: Numerical evaluation of self-preserving spectra in Smoluchowski’s coagulation theory. J. Colloid Interface Sci. 119, 576–587 (1987)CrossRefGoogle Scholar
  21. 21.
    Menon, G., Pego, R.L.: Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence. SIAM J. Math. Anal., to appearGoogle Scholar
  22. 22.
    Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Comm. Pure Appl. Math. 57, 1197–1232 (2004)MathSciNetGoogle Scholar
  23. 23.
    Smoluchowski, M., Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physik. Zeitschr. 17, 557–599 (1916)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut Elie Cartan – NancyUniversité Henri Poincaré – Nancy IVandoeuvre-lès-Nancy cedexFrance
  2. 2.Mathématiques pour l’Industrie et la Physique, CNRS UMR 5640Université Paul Sabatier – Toulouse 3Toulouse cedex 4France

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