Approach to self-similarity in Smoluchowski's coagulation equations
Govind Menon and Robert L. Pego
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Submission date: 11. Jul. 2003
published as: Approach to self-similarity in Smoluchowski's coagulation equations.
In: Communications on pure and applied mathematics, 57 (2004) 9, p. 1197-1232
DOI number (of the published article): 10.1002/cpa.3048
published as: Schouten tensor equations in conformal geometry with prescribed boundary metric.
In: Electronic journal of differential equations, 2005 (2005) 81, p. 1-17
Keywords and phrases: dynamic scaling, regular variation, agglomeration, coalescence, self-preserving spectra, heavy tails, mittag-leffler function, \levy\ flights
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We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels K(x,y)=2,x+y and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K=2 the size distribution is Mittag-Leffler, and for K=x+y and K=xy it is a power-law rescaling of a maximally skewed -stable distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.