Nonlinear Sciences – Adaptation and Self-Organizing Systems
Scientific paper
2006-09-11
Nonlinear Sciences
Adaptation and Self-Organizing Systems
Scientific paper
10.1007/s00332-007-9007-5
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 $K(x,y)=2$, $x+y$ and $xy$. 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 L\'{e}vy-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).
Menon Govind
Pego Robert L.
No associations
LandOfFree
The scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-25666