Control of Self-Organizing Nonlinear Systems by Eckehard Schöll Sabine H. L. Klapp & Philipp Hövel

Author:Eckehard Schöll, Sabine H. L. Klapp & Philipp Hövel
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

As a major assumption of the abstract result in Theorem 7 one needs the liminf estimate if in .

In Sect. 12.5 we discuss a few possible applications of the general results. We first consider the classical question of homogenization of reaction-diffusion systems as a didactical example. There we treat the diffusion part as a gradient part associated with the convex and quadratic Dirichlet energy. Because of the semilinear structure, all the nonlinear reaction terms can be treated as non-gradient perturbations. We are able to apply the -convex theory and refer to [19] for a comparison of the strengths and weaknesses of the two different approaches discussed on the basis of the homogenization of a Cahn–Hilliard-type problem.

In Sect. 12.5.2 we reconsider the theory developed in [22] for pure gradient systems. There it was shown that the Ginzburg–Landau equations can be understood as the evolutionary -limit of the suitable scaled Swift-Hohenberg equation. We discuss the usage of perturbed gradient systems to analyze a coupled system of Swift-Hohenberg equations introduced in [33].

Finally we speculate concerning the usage of evolutionary -convergence to derive a nonlinear reaction-diffusion system from a single Fokker–Planck-type master equation of diffusion in physical space as well as along a chemical reaction path. This follows the spirit of [2, 20, 29, 30], where chemical reaction is understood as a limit of diffusion.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.