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, Tome 33 (2016) no. 5, pp. 1223-1257.

Résumé
Abstract

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.

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.

MR Zbl | 1 citation dans Numdam

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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     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},
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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. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.04.002/

Bibliographie
Cité par

[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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