Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation
José A. Cañizo, Stéphane Mischler · Revista Matemática Iberoamericana · 2011
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.