Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels

Niethammer, B.; Throm, S.; Velázquez, J.J.L.

doi:10.1016/j.anihpc.2015.04.002

Niethammer, B.; Throm, S.; Velázquez, J.J.L.

Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 5, pp. 1223-1257.

Abstract
Résumé

We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying $C_{1} (x^{- a} y^{b} + x^{b} y^{- a}) \leq K (x, y) \leq C_{2} (x^{- a} y^{b} + x^{b} y^{- a})$ with $a > 0$ and $b < 1$ . This covers especially the case of Smoluchowski's classical kernel $K (x, y) = (x^{1 / 3} + y^{1 / 3}) (x^{- 1 / 3} + y^{- 1 / 3})$ .

For the proof of existence we take a self-similar solution $h_{ε}$ for a regularized kernel $K_{ε}$ and pass to the limit $ε \to 0$ to obtain a solution for the original kernel K. The main difficulty is to establish a uniform lower bound on $h_{ε}$ . The basic idea for this is to consider the time-dependent problem and to choose a special test function that solves the dual problem.

Nous démontrons l'existence des solutions auto-similaires avec queues lourdes pour l'équation de coagulation de Smoluchowski avec un noyau K satisfaisant $C_{1} (x^{- a} y^{b} + x^{b} y^{- a}) \leq K (x, y) \leq C_{2} (x^{- a} y^{b} + x^{b} y^{- a})$ avec $a > 0$ et $b < 1$ . Cela contient en particulier le noyau classique de Smoluchowski $K (x, y) = (x^{1 / 3} + y^{1 / 3}) (x^{- 1 / 3} + y^{- 1 / 3})$ .

Pour la démonstration de l'existence nous prenons une solution auto-similaire $h_{ε}$ pour un noyau régularisé $K_{ε}$ et nous obtenons une solution pour le noyau original K en passant à la limite $ε \to 0$ . La difficulté principale consiste à établir une borne inférieure pour $h_{ε}$ . La clé ici est de considérer le problème dépendant du temps et choisir une solution du problème dual comme fonction test dans la formulation faible de l'équation auto-similaire.

MR Zbl | 1 citation in Numdam

DOI: 10.1016/j.anihpc.2015.04.002

Keywords: Smoluchowski's coagulation equation, Self-similar solution, Singular kernel, Fat tail, Dual problem

@article{AIHPC_2016__33_5_1223_0,
     author = {Niethammer, B. and Throm, S. and Vel\'azquez, J.J.L.},
     title = {Self-similar solutions with fat tails for {Smoluchowski's} coagulation equation with singular kernels},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1223--1257},
     publisher = {Elsevier},
     volume = {33},
     number = {5},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.04.002},
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}

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Niethammer, B.; Throm, S.; Velázquez, J.J.L. Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 5, pp. 1223-1257. doi : 10.1016/j.anihpc.2015.04.002. http://www.numdam.org/articles/10.1016/j.anihpc.2015.04.002/

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