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

Stéphane Mischler (1) , José Cañizo (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 UAB - Departament de Matemàtiques [Barcelona]

We consider Smoluchowski's equation with a homogeneous kernel of the form a(x, y) = xαyβ + yβxα with −1 < α ≤ β ≤ 1 and λ := α+β ∈ [0, 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.

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Mischler, José Cañizo. Uniqueness and regularity of scaling profiles for Smoluchowski's coagulation equation. Revista Matemática Iberoamericana, 2011, 27 (3), pp.803-839. ⟨hal-00726385⟩