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Approach to self‐similarity in Smoluchowski's coagulation equations
Author(s) -
Me Govind,
Pego Robert L.
Publication year - 2004
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3048
Subject(s) - mathematics , lemma (botany) , smoluchowski coagulation equation , laplace transform , algebraic number , self similarity , scaling , distribution (mathematics) , infinite divisibility , heavy tailed distribution , probability distribution , similarity (geometry) , statistical physics , pure mathematics , mathematical analysis , statistics , physics , ecology , geometry , poaceae , artificial intelligence , computer science , image (mathematics) , biology
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 α‐stable Lévy 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. © 2003 Wiley Periodicals, Inc.