Convergence to Equilibrium and Fast Reaction Limit for Coagulation-Fragmentation type Systems
by
Klemens Fellner
University of Vienna
Coauthors: Jose A. Carrillo, Laurent Desvillettes

The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations in which polymers diffuse in space with a non-degenerate size-dependent coefficient. Both the break-up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels. We demonstrate that the entropy-entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension, but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non-degenerate diffusion in dimension one. We finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial. Up to our knowledge, this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation-fragmentation models.

Secondly, in a work in progress, we consider the inhomogeneous Aizenman-Bak model rescaled for fast reactions which tends formally to a nonlinear diffusion equation for the mass density (with the total mass being the conserved quantity of the model). In making the limit rigorous, we exploit the entropy dissipation as well as the properties of the limiting equation. So far, to prove convergence, we have to assume a uniform lower and upper bound on the number density of the polymer distribution.

Date received: June 25, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavg-69.