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**Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation.**
*(English)*
Zbl 1242.82031

The paper is concerned with the analysis of the Smoluchowski equation which models irreversible processes of aggregation of particles in clusters. The authors consider a continuum setting, where the size \(y\) of the cluster may be any positive number, and assume a homogeneous kernel depending on two parameters \(\alpha\) and \(\beta\), subjected to some limitations. It is proved that the self-similar solutions are infinitely differentiable and sharp results on the behaviour of the self-similar profiles at \(y = 0\) are given in the case \(\alpha < 0\). Moreover, partial uniqueness results are presented. In particular, when \(\alpha = 0\), it is proved that two profiles with same mass and same momentum of order \(\alpha + \beta\) are necessarily equal, while when \(\alpha < 0\), it is proved that two profiles with the same moments of order \(\alpha\) and \(\beta\) and with the same asymptotic behaviour at \(y=0\) are equal. The proofs are based on a distributional representation of the coagulation operator along with estimates of its regularity by using the theory of the derivatives of fractional order.

Reviewer: Vittorio Romano (Catania)

### MSC:

82C21 | Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics |

45K05 | Integro-partial differential equations |

82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |

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\textit{J. A. Cañizo} and \textit{S. Mischler}, Rev. Mat. Iberoam. 27, No. 3, 803--839 (2011; Zbl 1242.82031)

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