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Tail Behaviour of Self-Similar Profiles with Infinite Mass for Smoluchowski’s Coagulation Equation

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Abstract

In this article, we consider self-similar profiles to Smoluchowski’s coagulation equation for which we derive the precise asymptotic behaviour at infinity. More precisely, we look at so-called fat-tailed profiles which decay algebraically and as a consequence have infinite total mass. The results only require mild assumptions on the coagulation kernel and thus cover a large class of rate kernels.

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References

  1. Cañizo, J.A., Mischler, S.: Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation. Rev. Mat. Iberoam. 27(3), 803–839 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Drake, R.: A general mathematical survey of the coagulation equation. In: Hidy, G.M. (ed.) Topics in current aerosol research (part 2). International Reviews in Aerosol Physics and Chemistry, pp. 203–376. Pergamon Press, Oxford (1972)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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  MathSciNet  MATH  Google Scholar 

  9. Herrmann, M., Niethammer, B., Velázquez, J.J.L.: Instabilities and oscillations in coagulation equations with kernels of homogeneity one. Q. Appl. Math. 75(1), 105–130 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Laurençot, P., Mischler, S.: On coalescence equations and related models. In: Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology, pp. 321–356. Birkhäuser Boston, Boston, MA (2004)

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Niethammer, B., Throm, S., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(5), 1223–1257 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Niethammer, B., Throm, S., Velázquez, J.J.L.: A uniqueness result for self-similar profiles to Smoluchowski’s coagulation equation revisited. J. Stat. Phys. 164(2), 399–409 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. 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  MathSciNet  MATH  Google Scholar 

  15. 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(2), 505–532 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Niethammer, B., Velázquez, J.J.L.: Exponential tail behavior of self-similar solutions to Smoluchowski’s coagulation equation. Commun. Partial Differ. Equ. 39(12), 2314–2350 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Niethammer, B., Velázquez, J.J.L.: Uniqueness of self-similar solutions to Smoluchowski’s coagulation equations for kernels that are close to constant. J. Stat. Phys. 157(1), 158–181 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Throm, S.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation. PhD thesis, University of Bonn (2016)

  20. 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  ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

  22. von Smoluchowski, M.: Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift für physikalische Chemie 92, 129–168 (1917)

    Google Scholar 

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Acknowledgements

The author thanks the two anonymous reviewers for their valuable comments which helped to improve this article.

Funding

This work has been supported through the CRC 1060 The mathematics of emergent effects at the University of Bonn that is funded through the German Science Foundation (DFG). Moreover, partial support through a Lichtenberg Professorship Grant of the VolkswagenStiftung awarded to Christian Kühn is acknowledged.

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  1. Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748, Garching bei München, Germany

    Sebastian Throm

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  1. Sebastian Throm
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Correspondence to Sebastian Throm.

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Throm, S. Tail Behaviour of Self-Similar Profiles with Infinite Mass for Smoluchowski’s Coagulation Equation. J Stat Phys 170, 1215–1241 (2018). https://doi.org/10.1007/s10955-018-1980-6

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