Abstract
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
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
. 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
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.