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

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.

Nous démontrons l'existence des solutions auto-similaires avec queues lourdes pour l'équation de coagulation de Smoluchowski avec un noyau K satisfaisant $C_1(x^{-a}{y}^b+x^b{y}^{-a})\le K(x,y)\le C_2(x^{-a}{y}^b+x^b{y}^{-a})$ 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 $h_{\epsilon }$ pour un noyau régularisé $K_{\epsilon }$ et nous obtenons une solution pour le noyau original K en passant à la limite $\epsilon \to 0$. La difficulté principale consiste à établir une borne inférieure pour $h_{\epsilon }$. 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 $C_1(x^{-a}{y}^b+x^b{y}^{-a})\le K(x,y)\le C_2(x^{-a}{y}^b+x^b{y}^{-a})$ 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 $h_{\epsilon }$ for a regularized kernel $K_{\epsilon }$ and pass to the limit $\epsilon \to 0$ to obtain a solution for the original kernel K. The main difficulty is to establish a uniform lower bound on $h_{\epsilon }$. The basic idea for this is to consider the time-dependent problem and to choose a special test function that solves the dual problem.

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