Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation

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.