Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients

José A. Cañizo, Stéphane Mischlerj, Clément Mouhot

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)

Abstract

We show that solutions to Smoluchowski's equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the L2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case. © 2009 Society for Industrial and Applied Mathematics.
Original languageEnglish
Pages (from-to)2283-2314
JournalSIAM Journal on Mathematical Analysis
Volume41
DOIs
Publication statusPublished - 1 Jan 2009

Keywords

  • Coagulation equation
  • Constant coagulation kernel
  • Explicit
  • Self-similar variables
  • Smoluchowski's equation
  • Spectral gap

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