Skip to main content
SpringerLink
Log in

Self-similarity in Smoluchowski’s Coagulation Equation

  • Survey Article
  • Published:
Jahresbericht der Deutschen Mathematiker-Vereinigung Aims and scope Submit manuscript

Abstract

Smoluchowski’s coagulation equation is one of the fundamental deterministic models that describe mass aggregation phenomena. A key question in the analysis of this equation is whether the large-time behaviour of solutions is universal and described by special self-similar solutions. This issue is however only well-understood for the small class of solvable kernels while the analysis for non-solvable kernels still poses many challenging problems.

Our main focus in this article will be to describe recent progress in the analysis of self-similar solutions of Smoluchowski’s equation for non-solvable kernels. Existence results for self-similar solutions with finite mass have been available for some time for a large class of kernels. In contrast, the uniqueness of such solutions has been an open problem for some time. A first uniqueness result has recently been obtained for kernels that are in a certain sense close to constant. We present here a shorter proof under an additional assumption on the kernels that makes the analysis significantly simpler. We also give an overview of recent results on the existence of fat tail solutions for non-solvable kernels.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Ball, J.M., Carr, J.: The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Stat. Phys. 61(1–2), 203–234 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cañizo, J., Mischler, S.: Regularity, asymptotic behavior and partial uniqueness for Smoluchowski’s coagulation equation. Rev. Mat. Iberoam. 27(3), 503–564 (2011)

    Google Scholar 

  4. Da Costa, F.P.: On the dynamic scaling behaviour of solutions to the discrete Smoluchowski equations. Proc. Edinb. Math. Soc. 39(3), 547–559 (1996)

    Article  MATH  Google Scholar 

  5. Deaconu, M., Tanré, E.: Smoluchowski’s coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 29(3), 549–579 (2000).

    MATH  Google Scholar 

  6. 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, pp. 203–376. Oxford University Press, Oxford (1972)

    Google Scholar 

  7. Escobedo, M., Mischler, S.: Dust and self-similarity for the Smoluchowski coagulation equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23(3), 331–362 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Escobedo, M., Mischler, S., Perthame, B.: Gelation in coagulation and fragmentation models. Commun. Math. Phys. 231(1), 157–188 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Escobedo, M., Mischler, S., Rodriguez Ricard, M.: On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22(1), 99–125 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fournier, N., Laurençot, P.: Existence of self-similar solutions to Smoluchowski’s coagulation equation. Commun. Math. Phys. 256(3), 589–609 (2005)

    Article  MATH  Google Scholar 

  11. Fournier, N., Laurençot, P.: Local properties of self-similar solutions to Smoluchowski’s coagulation equation with sum kernels. Proc. R. Soc. Edinb., Sect. A, Math. 136(3), 485–508 (2006)

    Article  MATH  Google Scholar 

  12. Fournier, N., Laurençot, P.: Well-posedness of Smoluchowski’s coagulation equation for a class of homogeneous kernels. J. Funct. Anal. 233(2), 351–379 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jeon, I.: Existence of gelling solutions for coagulation-fragmentation equations. Commun. Math. Phys. 194(3), 541–567 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kreer, M., Penrose, O.: Proof of dynamical scaling in Smoluchowski’s coagulation equation with constant kernel. J. Stat. Phys. 75(3–4), 389–407 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Laurençot, P., Mischler, S.: From the discrete to the continuous coagulation-fragmentation equations. Proc. R. Soc. Edinb., Sect. A, Math. 132(5), 1219–1248 (2002)

    Article  MATH  Google Scholar 

  16. Laurençot, P., Mischler, S.: On coalescence equations and related models. In: Modeling and Computational Methods for Kinetic Equations. Model. Simul. Sci. Eng. Technol., pp. 321–356. Birkhäuser, Boston (2004)

    Chapter  Google Scholar 

  17. Laurençot, P., Mischler, S.: Lyapunov functionals for Smoluchowski’s coagulation equation and convergence to self-similarity. Monatshefte Math. 146(2), 127–142 (2005)

    Article  MATH  Google Scholar 

  18. Leyvraz, R.: Scaling theory and exactly solvable models in the kinetics of irreversible aggregation. Phys. Rep. 383, 95–212 (2003)

    Article  Google Scholar 

  19. Leyvraz, F.: Rigorous results in the scaling theory of irreversible aggregation kinetics. J. Nonlinear Math. Phys. 12(suppl. 1), 449–465 (2005)

    Article  MathSciNet  Google Scholar 

  20. Leyvraz, F., Tschudi, H.R.: Singularities in the kinetics of coagulation processes. J. Phys. A 14(12), 3389–3405 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  21. McLeod, J.B.: On an infinite set of non-linear differential equations. Q. J. Math. 13, 119–128 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  22. McLeod, J.B., Niethammer, B., Velázquez, J.J.L.: Asymptotics of self-similar solutions to coagulation equations with product kernel. J. Stat. Phys. 144, 76–100 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Commun. Pure Appl. Math. 57(9), 1197–1232 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mulholland, G.W., Samson, R.J., Mountain, R.D., Ernst, M.H.: Cluster size distribution for free molecular agglomeration. Energy Fuels 2, 481–486 (1988)

    Article  Google Scholar 

  25. Niethammer, B., Velázquez, J.J.L.: Optimal bounds for self-similar solutions to coagulation equations with product kernel. Commun. Partial Differ. Equ. 36(12), 2049–2061 (2011)

    Article  MATH  Google Scholar 

  26. Niethammer, B., Velázquez, J.J.L.: Self-similar solutions with fat tails for a coagulation equation with diagonal kernel. C. R. Math. Acad. Sci. Paris 349(9–10), 559–562 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Niethammer, B., Velázquez, J.J.L.: Exponential decay of self-similar profiles to Smoluchowski’s coagulation equation (2013). Preprint, arXiv:1310.4732

  28. Niethammer, B., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels. Commun. Math. Phys. 318, 505–532 (2013)

    Article  MATH  Google Scholar 

  29. Niethammer, B., Velázquez, J.J.L.: Uniqueness of self-similar solutions to Smoluchowski’s coagulation equations for kernels that are close to constant (2013). Preprint, arXiv:1309.4621

  30. Niethammer, B., Throm, S., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels (2013, in preparation)

  31. Norris, J.R.: Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9(1), 78–109 (1999)

    MATH  MathSciNet  Google Scholar 

  32. Silk, J., Takamasa, T.: A statistical model for the initial stellar mass function. Astrophys. J. 229, 242–256 (1979)

    Article  Google Scholar 

  33. Smoluchowski, M.: Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Phys. Z. 17, 557–599 (1916)

    Google Scholar 

  34. van Dongen, P.G.J., Ernst, M.H.: Scaling solutions of Smoluchowski’s coagulation equation. J. Stat. Phys. 50(1–2), 295–329 (1988)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

I am indebted to Philippe Laurençot and Juan Velázquez for numerous discussions and collaborations on the subject of this article. Support through the Cooperative Research Centre (CRC) 1060 “The Mathematics of Emergent Effects” at the University of Bonn is also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Barbara Niethammer

Authors
  1. Barbara Niethammer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barbara Niethammer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Niethammer, B. Self-similarity in Smoluchowski’s Coagulation Equation. Jahresber. Dtsch. Math. Ver. 116, 43–65 (2014). https://doi.org/10.1365/s13291-014-0085-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1365/s13291-014-0085-7

Keywords

Search

Navigation