Comptes Rendus
no. 15-16
Partial Differential Equations
Self-similar solutions with fat tails for a coagulation equation with nonlocal drift
[Solutions auto-similaires à décroissance lente pour une équation de coagulation avec transport non local]

Présenté par : Haïm Brezis

Michael Herrmann1 ; Philippe Laurençot2 ; Barbara Niethammer1
1 Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, United Kingdom
2 Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS UMR 5219, 118, route de Narbonne, F-31062 Toulouse cedex 9, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 909-914.

L'existence de solutions auto-similaires préservant le volume total est établie pour une équation de coagulation incluant un terme de transport non local. Bien que cette équation admette des profils auto-similaires décroissant exponentiellement à l'infini, des profils auto-similaires avec une décroissance algébrique à l'infini sont construits lorsque le volume total est suffisamment petit.

We investigate the existence of self-similar solutions for a coagulation equation with nonlocal drift. In addition to explicitly given exponentially decaying solutions we establish the existence of self-similar profiles with algebraic decay.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.05.006

Michael Herrmann 1 ; Philippe Laurençot 2 ; Barbara Niethammer 1

1 Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, United Kingdom
2 Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS UMR 5219, 118, route de Narbonne, F-31062 Toulouse cedex 9, France 
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Michael Herrmann; Philippe Laurençot; Barbara Niethammer. Self-similar solutions with fat tails for a coagulation equation with nonlocal drift. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 909-914. doi : 10.1016/j.crma.2009.05.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.006/

[1] J. Carr Stability of self-similar solutions in a simplified LSW model, Phys. D, Volume 222 (2006) no. 1–2, pp. 73-79

[2] J. Carr; O. Penrose Asymptotic behaviour in a simplified Lifshitz–Slyozov equation, Phys. D, Volume 124 (1998), pp. 166-176

[3] M. Herrmann, B. Niethammer, J.J.L. Velázquez, Self-similar solutions for the LSW model with encounters, preprint, 2008

[4] Ph. Laurençot The Lifshitz–Slyozov equation with encounters, Math. Models Methods Appl. Sci., Volume 11 (2001) no. 4, pp. 731-748

[5] I.M. Lifshitz; V.V. Slyozov The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids, Volume 19 (1961), pp. 35-50

[6] G. Menon; R.L. Pego Approach to self-similarity in Smoluchowski's coagulation equations, Comm. Pure Appl. Math., Volume 57 (2004) no. 9, pp. 1197-1232

[7] B. Niethammer; R.L. Pego Non-self-similar behavior in the LSW theory of Ostwald ripening, J. Statist. Phys., Volume 95 (1999) no. 5/6, pp. 867-902

[8] C. Wagner Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung), Z. Elektrochem., Volume 65 (1961), pp. 581-591

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  • D. V. Alexandrov A transient distribution of particle assemblies at the concluding stage of phase transformations, Journal of Materials Science, Volume 52 (2017) no. 12, p. 6987 | DOI:10.1007/s10853-017-0931-y
  • D V Alexandrov Kinetics of particle coarsening with allowance for Ostwald ripening and coagulation, Journal of Physics: Condensed Matter, Volume 28 (2016) no. 3, p. 035102 | DOI:10.1088/0953-8984/28/3/035102
  • D. V. Alexandrov The large-time behaviour of coarsening of a particulate assemblage due to Ostwald ripening and coagulation, Philosophical Magazine Letters, Volume 96 (2016) no. 9, p. 355 | DOI:10.1080/09500839.2016.1225996
  • Giancarlo Breschi; Marco A Fontelos Self-similar solutions of the second kind representing gelation in finite time for the Smoluchowski equation, Nonlinearity, Volume 27 (2014) no. 7, p. 1709 | DOI:10.1088/0951-7715/27/7/1709
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  • Shangbing Ai Self-similar solutions with algebraic decay for a non-local coagulation equation, Nonlinearity, Volume 23 (2010) no. 3, p. 579 | DOI:10.1088/0951-7715/23/3/008
  • B. Niethammer The influence of encounters on domain coarsening, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 90 (2010) no. 4, p. 259 | DOI:10.1002/zamm.200900384
  • M. Herrmann; B. Niethammer; J.J.L. Velázquez Self-similar solutions for the LSW model with encounters, Journal of Differential Equations, Volume 247 (2009) no. 8, p. 2282 | DOI:10.1016/j.jde.2009.07.021

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