no. 9-10
Partial Differential Equations
Self-similar solutions with fat tails for a coagulation equation with diagonal kernel
[Solutions auto-similaires avec queues épaisses dʼune équation de coagulation à noyau diagonal]

Présenté par : the Editorial Board

Niethammer, Barbara1 ; Velázquez, Juan J.L.2
1 Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, 24-29 St. Gilesʼ, Oxford, OX1 3LB, United Kingdom
2 ICMAT, C/ Nicolas Cabrera 15, Cantoblanco, 28049 Madrid, Spain
Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 559-562.

Nous considérons des solutions autosimilaires de lʼéquation de coagulation de Smoluchowski avec un noyau diagonal dʼhomogénéité γ<1. Nous prouvons lʼexistence dʼune famille de solutions autosimilaires de deuxième type avec comportement à lʼinfini en puissance x−(1+ρ), ρ∈(γ,1). A notre connaissance, ceci constitue le premier exemple dʼexistence dʼune telle famille pour un noyau non explicitement résoluble.

We consider self-similar solutions of Smoluchowskiʼs coagulation equation with a diagonal kernel of homogeneity γ<1. We show that there exists a family of second-kind self-similar solutions with power-law behavior x−(1+ρ) as x→∞ with ρ∈(γ,1). To our knowledge this is the first example of a non-solvable kernel for which the existence of such a family has been established.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.03.017
Niethammer, Barbara 1 ; Velázquez, Juan J.L. 2

1 Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, 24-29 St. Gilesʼ, Oxford, OX1 3LB, United Kingdom
2 ICMAT, C/ Nicolas Cabrera 15, Cantoblanco, 28049 Madrid, Spain 
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Niethammer, Barbara; Velázquez, Juan J.L. Self-similar solutions with fat tails for a coagulation equation with diagonal kernel. Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 559-562. doi : 10.1016/j.crma.2011.03.017. http://www.numdam.org/articles/10.1016/j.crma.2011.03.017/

[1] Leyvraz, F. Rigorous results in the scaling theory of irreversible aggregation kinetics, J. Nonl. Math. Phys., Volume 47 (2006) no. 4, p. 042901

[2] Menon, G.; Pego, R.L. Approach to self-similarity in Smoluchowskiʼs coagulation equations, Comm. Pure Appl. Math., Volume 57 (2004), pp. 1197-1232

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