Uniqueness and regularity of scaling profiles for Smoluchowski’s coagulation equation

We consider Smoluchowski’s equation with a homogeneous kernel of the form a(x, y) = xy + yx with −1 < α ≤ β < 1 and λ := 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 a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.