On gelation for the Smoluchowski coagulation equation

Nicolas Fournier

doi:10.5802/crmath.738

Article de recherche - Équations aux dérivées partielles

On gelation for the Smoluchowski coagulation equation
[Sur la gélification pour l’équation de coagulation de Smoluchowski]

Nicolas Fournier ¹

¹ Sorbonne Université, LPSM-UMR 8001, Case courrier 158, 75252 Paris Cedex 05, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 583-591.

Abstract (VO)
Résumé

Motivated by the recent results of Andreis–Iyer–Magnanini [2], we provide a short proof, revisiting the one of Escobedo–Mischler–Perthame [7], that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation loses mass in finite time. The class of kernels we consider is essentially the same as the one of [2]: homogeneous kernels of degree $γ > 1$ not vanishing on the diagonal, or homogeneous kernels of degree $γ = 1$ not vanishing on the diagonal with some additional logarithmic factor. We also show that when $γ = 1$ , the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel.

Motivés par les résultats récents d’Andreis–Iyer–Magnanini [2], nous fournissons une courte preuve, revisitant celle d’Escobedo–Mischler–Perthame [7], que pour une grande classe de noyaux de coagulation, toute solution faible de l’équation de Smoluchowski perd de la masse en temps fini. La classe de noyaux que nous considérons est essentiellement la même que celle de [2] : les noyaux homogènes de degré $γ > 1$ ne s’annulant pas sur la diagonale, et les noyaux homogènes de degré $γ = 1$ ne s’annulant pas sur la diagonale avec un facteur logarithmique supplémentaire. Nous montrons également que lorsque $γ = 1$ , la puissance du facteur logarithmique assurant la gélification peut dépendre de la forme du noyau.

Reçu le : 2025-01-30
Accepté le : 2025-03-04
Publié le : 2025-06-05

DOI : 10.5802/crmath.738

Classification : 45K05, 45M99
Keywords: Coagulation, Smoluchowski equation, gelation, explosion
Mots-clés : Coagulation, équation de Smoluchowski, gélification, explosion

Affiliations des auteurs :

Nicolas Fournier ¹

¹ Sorbonne Université, LPSM-UMR 8001, Case courrier 158, 75252 Paris Cedex 05, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Nicolas Fournier},
     title = {On gelation for the {Smoluchowski} coagulation equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--591},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.738},
     language = {en},
}

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Nicolas Fournier. On gelation for the Smoluchowski coagulation equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 583-591. doi : 10.5802/crmath.738. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.738/

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