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

We consider Smoluchowski&#39;s equation with a homogeneous kernel of the form $a(x,y) = x^αy ^β+ x^βy^α$ with $-1 < α\leq β< 1$ and $λ:= α+ β\in (-1,1)$. 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 $α< 0$. 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 a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.