Publication 06-CNA-08
**The Scaling Attractor and Ultimate Dynamics in Smoluchowski's Coagulation
Equations **

Govind Menon

Division of Applied Mathematics

Brown University

Providence, RI 02912

menon@dam.brown.edu

and

Robert L. Pego

Department of Mathematical Sciences

Carnegie Mellon Univeristy

Pittsburgh, PA 15213

rpego@cmu.edu

**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 , 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).

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