### Abstract

We consider Smoluchowski's equation with a homogeneous kernel of the form a (x, y) = xα yβ+withxα yβ -1 < α ≤ β < 1 and λ := a α+ β ∈ (-1,1). We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at y = 0 in the case α < 0. We also give some partial uniqueness results for self-similar profiles: in the case α = 0 we prove that two profiles with the same mass and moment of order λ are necessarily equal, while in the case α < 0 we prove that two profiles with the same moments of order α and β, and which are asymptotic at y = 0, are equal. Our methods include α new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

Original language | English |
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Pages (from-to) | 803-839 |

Journal | Revista Matematica Iberoamericana |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jan 2011 |

### Keywords

- Asymptotic behavior
- Coagulation
- Regularity
- Self-similarity
- Uniqueness

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## Cite this

Cañizo, J. A., & Mischler, S. (2011). Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski's coagulation equation.

*Revista Matematica Iberoamericana*,*27*(3), 803-839. https://doi.org/10.4171/RMI/653