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MiS Preprint

Approach to self-similarity in Smoluchowski's coagulation equations

Govind Menon and Robert L. Pego


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 $\alpha$-stable Levy 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.

Jul 11, 2003
Jul 11, 2003
dynamic scaling, regular variation, agglomeration, coalescence, self-preserving spectra, heavy tails, mittag-leffler function, levy flights

Related publications

2004 Repository Open Access
Govind Menon and Robert L. Pego

Approach to self-similarity in Smoluchowski's coagulation equations

In: Communications on pure and applied mathematics, 57 (2004) 9, pp. 1197-1232
2005 Journal Open Access
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Schouten tensor equations in conformal geometry with prescribed boundary metric

In: Electronic journal of differential equations, 2005 (2005) 81, pp. 1-17