## Collaborative Research Center 910:

Control of self-organizing nonlinear systems: Theoretical methods and concepts of application

Subproject A5: Pattern formation in systems with multiple scales

Project Head: Prof. Dr. Alexander Mielke Investigators: Dr. Sina Reichelt, Dr. Marita Thomas Funding period I: January 1, 2011 - Dezember 31, 2014 Funding period II: January 1, 2015 - Dezember 31, 2018

### Description

Pattern formation in nonlinear partial differential equations depends on nontrivial interactions between different internal length scales of the system, the nonlinearities and the size and geometry of the underlying domain. The challenge is to understand how effects on small spatial scales generate effective pattern formation on large spatial scales. With a view to well-chosen model problems reflecting the focus application of the CRC 910, we will investigate the mathematical foundations of the derivation of effective models for pattern formation in multi-scale systems. Controls for the effective models will be used in order to construct controls for the original system.

Mathematical systems with effects on microscopic length scales may arise from biological problems (e.g. neuron models), chemical problems (e.g. micro-emulsions) or engineering (e.g. composite materials). Models depending on scales of different order challenge numerical and analytical treatment. This expresses a need for effective models, which are defined on macroscopic scales, only, but which approximate the properties of the original microscopic systems well. We aim to derive such effectice models by means of multi-scale convergence and classical results from homogenization theory.

The aim of this project is to investigate the effective influence of microstructure on the formation of macroscopic pattern. This involvesFirst, general analytical methods will be developed for abstract reaction-diffusion systems. They will be used later to investigate specific systems related to the application area of the CRC 910.

- the derivation of effective macroscopic equations for pattern forming systems,
- the study of bifurcations in the effective equations and
- the control of pattern in the effective and the original equations by using spatially localized controls.

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### Symposia and Workshops

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Last modified: November 7, 2018 CET Sina Reichelt