JournalsrmiVol. 27, No. 3pp. 803–839

Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation

  • José A. Cañizo

    Universitat Autònoma de Barcelona, Bellaterra, Spain
  • Stéphane Mischler

    Université de Paris-Dauphine, Paris, France
Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation cover
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Abstract

We consider Smoluchowski's equation with a homogeneous kernel of the form a(x,y)=xαyβ+xβyαa(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha with 1<αβ<1-1 < \alpha \leq \beta < 1 and λ:=α+β(1,1)\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=0y = 0 in the case α<0\alpha < 0. We also give some partial uniqueness results for self-similar profiles: in the case α=0\alpha = 0 we prove that two profiles with the same mass and moment of order λ\lambda are necessarily equal, while in the case α<0\alpha < 0 we prove that two profiles with the same moments of order α\alpha and β\beta, and which are asymptotic at y=0y = 0, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

Cite this article

José A. Cañizo, Stéphane Mischler, Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 803–839

DOI 10.4171/RMI/653