Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 5, pp. 1223-1257.

Nous démontrons l'existence des solutions auto-similaires avec queues lourdes pour l'équation de coagulation de Smoluchowski avec un noyau K satisfaisant C1(xayb+xbya)K(x,y)C2(xayb+xbya) avec a>0 et b<1. Cela contient en particulier le noyau classique de Smoluchowski K(x,y)=(x1/3+y1/3)(x1/3+y1/3).

Pour la démonstration de l'existence nous prenons une solution auto-similaire hε pour un noyau régularisé Kε et nous obtenons une solution pour le noyau original K en passant à la limite ε0. La difficulté principale consiste à établir une borne inférieure pour hε. 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.

We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying C1(xayb+xbya)K(x,y)C2(xayb+xbya) with a>0 and b<1. This covers especially the case of Smoluchowski's classical kernel K(x,y)=(x1/3+y1/3)(x1/3+y1/3).

For the proof of existence we take a self-similar solution hε for a regularized kernel Kε and pass to the limit ε0 to obtain a solution for the original kernel K. The main difficulty is to establish a uniform lower bound on hε. The basic idea for this is to consider the time-dependent problem and to choose a special test function that solves the dual problem.

DOI : 10.1016/j.anihpc.2015.04.002
Mots-clés : Smoluchowski's coagulation equation, Self-similar solution, Singular kernel, Fat tail, Dual problem
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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, Tome 33 (2016) no. 5, pp. 1223-1257. doi : 10.1016/j.anihpc.2015.04.002. https://www.numdam.org/articles/10.1016/j.anihpc.2015.04.002/

[1] Aldous, D.J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli, Volume 5 (1999) no. 1, pp. 3–48 MR 1673235 (2001c:60153) | DOI | MR | Zbl

[2] Cañizo, J.; Mischler, S. Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation, Rev. Mat. Iberoam., Volume 27 (2011) no. 3, pp. 503–564 | MR | Zbl

[3] Drake, R.-L. A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research (Part 2), International Reviews in Aerosol Physics and Chemistry, Oxford University Press, Oxford, 1972, pp. 203–376

[4] Escobedo, M.; Mischler, S. Dust and self-similarity for the Smoluchowski coagulation equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006) no. 3, pp. 331–362 MR 2217655 (2007a:82043) | DOI | Numdam | MR | Zbl

[5] Escobedo, M.; Mischler, S.; Rodriguez Ricard, M. On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005) no. 1, pp. 99–125 MR 2114413 (2006b:35034) | DOI | Numdam | MR | Zbl

[6] Fournier, N.; Laurençot, P. Existence of self-similar solutions to Smoluchowski's coagulation equation, Commun. Math. Phys., Volume 256 (2005) no. 3, pp. 589–609 MR 2161272 (2007a:82061) | DOI | MR | Zbl

[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, Volume 136 (2006) no. 3, pp. 485–508 MR 2227805 (2007f:45012) | DOI | MR | Zbl

[8] Gamba, I.M.; Panferov, V.; Villani, C. On the Boltzmann equation for diffusively excited granular media, Commun. Math. Phys., Volume 246 (2004) no. 3, pp. 503–541 MR 2053942 (2005b:82076) | DOI | MR | Zbl

[9] Menon, G.; Pego, R.L. Approach to self-similarity in Smoluchowski's coagulation equations, Commun. Pure Appl. Math., Volume 57 (2004) no. 9, pp. 1197–1232 MR 2059679 (2005i:82051) | DOI | MR | Zbl

[10] Niethammer, B.; Throm, S.; Velázquez, J.J.L. Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels (preprint, available at) | arXiv | Numdam | MR | Zbl

[11] Niethammer, B.; Velázquez, J.J.L. Self-similar solutions with fat tails for a coagulation equation with diagonal kernel, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 9–10, pp. 559–562 (MR 2802924) | MR | Zbl

[12] Niethammer, B.; Velázquez, J.J.L. Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels, Commun. Math. Phys., Volume 318 (2013), pp. 505–532 | MR

[13] Smoluchowski, M. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Phys. Z. Sowjetunion, Volume 17 (1916), pp. 557–599

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