Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation

Details Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation Regularity, asymptotic behavior and partial uniqueness for Smoluchowski's coagulation equation

2008-03-10

arxiv.org/abs/0803.1462v2

Revista Matem\'atica Iberoamericana, Vol. 27, No. 3, pp. 803-839 (2011)

Physics

Mathematical Physics

Scientific paper

We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with $-1 < \alpha \leq \beta < 1$ and $\lambda := \alpha + \beta \in (-1,1)$. 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 $\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\alpha = 0$ we prove that two profiles with the same mass and moment of order $\lambda$ are necessarily equal, while in the case $\alpha < 0$ we prove that two profiles with the same moments of order $\alpha$ and $\beta$, and which are asymptotic at $y=0$, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

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