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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.

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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References:

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