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Publication 06-CNA-08
The Scaling Attractor and Ultimate Dynamics in Smoluchowski's Coagulation Equations Govind Menon and Robert L. Pego Abstract: We describe a basic framework for studying dynamic
scaling that has roots in dynamical systems and probability theory. Within
this framework, we study Smoluchowski's coagulation equation for the three
simplest rate kernels Get the paper in its entirety as Back to CNA Publications 2006 |
, 
and 
. In another work, we
classified all self-similar solutions and all universality classes (domains of
attraction) for scaling limits under weak convergence (Comm. Pure Appl. Math
57 (2004) 1197-1232). Here we add to this a complete description of the set
of all limit points of solutions modulo scaling (the scaling attractor) and
the dynamics on this limit set (the ultimate dynamics). The main tool is
Bertoin's Levy-Khintchine representation formula for eternal solutions of
Smoluchowski's equation (Adv. Appl. Prob. 12 (2002) 547-64). This
representation linearizes the dynamics on the scaling attractor, revealing
these dynamics to be conjugate to a continuous dilation, and chaotic in a
classical sense. Furthermore, our study of scaling limits explains how
Smoluchowski dynamics ``compactifies'' in a natural way that accounts for
clusters of zero and infinite size (dust and gel).