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
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 involves
First, 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.
- M. Becker, Th. Frenzel, Th. Niedermayer, S. Reichelt, A. Mielke, and M. Bär: Local control of globally competing patterns in coupled Swift-Hohenberg equations, Chaos 28(4), 043121, 2018.
- P. Gurevich and S. Reichelt: Pulses in FitzHugh-Nagumo systems with rapidly oscillating coefficients, Multiscale Model. Simul. 16(2), 833-856, 2018.
- S. Reichelt: Corrector estimates for a class of imperfect transmission problems, Asymptot. Anal., 105, 3-26, 2017.
- A. Mielke: Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics, in Pattern of Dynamics, Eds. P. Gurevich, J. Hell, B. Sanstede, A. Scheel, Springer Proc. in Math. & Stat. Vol. 205, 149-171, 2017.
- A. Muntean and S. Reichelt: Corrector estimates for a thermo-diffusion model with weak thermal coupling, Multiscale Model. Simul. 16(2), 807-832, 2018.
- S. Reichelt: Error estimates for elliptic equations with not-exactly periodic coefficients, Adv. Math. Sci. Appl., 25, 117-131, 2016.
- S. Reichelt: Two-scale Homogenisation of Systems of Nonlinear Parabolic Equations, PhD Thesis, Humboldt-Univerität zu Berlin, 2015.
- A. Mielke: On evolutionary Γ-convergence for gradient systems, in: Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing, Heidelberg, 187-249, 2016.
- A. Mielke: Deriving effective models for multiscale systems via evolutionary Γ-convergence, in: Control of Self-Organizing Nonlinear Systems, Understanding Complex Systems, Springer, 235-251, 2016.
- W. Dreyer, R. Huth, A. Mielke, J. Rehberg, and M. Winkler: Global existence for a nonlocal and nonlinear Fokker-Planck equation, ZAMP Zeitschrift für Angewandte Mathematik und Physik, 66: 293-315, 2015.
- M. Liero and S. Reichelt: Homogenization of Cahn-Hilliard-type equations via evolutionary Γ-convergence, NoDEA Nonlinear Differential Equations Appl. 25, no. 1, Art. 6, 31 pp., 2018.
- S. Reichelt: Error estimates for nonlinear reaction-diffusion systems involving different length scales, MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), vol. 727 of Journal of Physics: Conference Series, IOP Publishing, 012013/1--012013/15, 2016.
- A. Mielke: Deriving amplitude equations via Γ-convergence, Discr. Cont. Dynam. Systems Ser. A, 35(6), 2679-2700, 2015.
- A. Mielke, S. Reichelt, and M. Thomas: Two-scale homogenization of nonlinear reaction-diffusion systems with small diffusion, Netw. Heterog. Media, 9(2): 353-382, 2014.
- S. Yanchuk, L. Lücken, M. Wolfrum, and A. Mielke: Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discrete Contin. Dyn. Syst., 35(1): 537-553, 2015.
- A. Mielke and E. Rohan: Homogenization of elastic waves in fluid-saturated porous media using the Biot model, Math. Models Meth. Appl. Sci., 23(5): 873-916, 2013.
- A. Mielke, J. Haskovec, and P.A. Markowich: On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27: 897-928, 2015.
- A. Mielke: Multiscale gradient systems and their amplitude equations, Oberwolfach Report 9: 3588-3591, 2012.
- M. Heida, A. Mielke, Ch. Kraus, and M. Thomas: Effective models for interfaces with many scales, SCCS Days, Ketzin, October 10-12, 2016.
- M. Becker, Th. Frenzel, Th. Niedermayer, S. Reichelt, A. Mielke, and M. Bär: Competing patterns in anti-symmetrically coupled Swift-hohenberg equations, International Conference on Control of Complex Systems and Networks, Heringsdorf, September 4-8, 2016.
- S. Reichelt: Homogenization of Cahn-Hilliard equations, Conference "Patterns of Dynamics", FU Berlin, July 25-27, 2016.
- S. Reichelt: Homogenization of degenerated reaction-diffusion equations, Doktorandenforum der Leibniz-Gemeinschaft Sektion D, Berlin, June 6-7, 2013.
- M. Thomas: Coupling of reaction diffusion processes with thermomechanics using GENERIC (joint work with A. Mielke), Winter School "Calculus of Variations in Physics and Materials Science", Julius-Maximilians-Universität Würzburg, January 9-13, 2012.
- S. Reichelt: Pulses in FitzHugh-Nagumo systems with rapidly oscillating coefficients, 89th Annual Meeting of GAMM, Section "Applied Analysis", München, March, 19-23, 2018.
- S. Reichelt: Traveling waves in FitzHugh-Nagumo systems with rapidly oscillating coefficients, SFB 910 Workshop, Wittenberg, August, 29-31, 2017.
- S. Reichelt: Corrector estimates for elliptic and parabolic equations with periodic coefficients,
Analysis Kolloquium, FAU Erlangen-Nürnberg, May 18, 2017, and Analysis Seminar, Universität Augsburg, May 23, 2017.
- A. Mielke: On self-induced oscillations for friction reduction with applications to walking, Conference "Dynamical Systems and Geometric Mechanics",
June 12-14, 2017, Technische Universität München, Zentrum für Mathematik, June 13, 2017.
- A. Mielke: Uniform exponential decay for energy-reaction-diffusion systems, Analysis Seminar, University of Pavia, Department of Mathematics, Italien,
March 21, 2017.
- S. Reichelt: Corrector estimates for a class of imperfect transmission problems, 88th Annual Meeting of GAMM, Weimar, March 6-10, 2017.
- A. Mielke: Exponential decay into thermodynamical equilibrium for reaction-diffusion systems with detailed balance, Conference "Patterns of Dynamics", FU Berlin, July 25-27, 2016.
- S. Reichelt: Homogenization of Cahn-Hilliard equations via evolutionary Γ-convergence,
11th AIMS Conference on Dynamical Systems, Differential Equations, and Applications, Orlando, USA, July 1-5, 2016.
- S. Reichelt: Konvergenz und Homogenisierung, 21. Tag der Mathematik, FU Berlin, April 30, 2016.
- S. Reichelt: Error estimates for elliptic and parabolic equations, Karlstads Universitet, Karlstad, Schweden, April 13, 2016.
- S. Reichelt: Homogenization of Cahn-Hilliard equations via evolutionary Γ-convergence, GAMM-DMV Jahrestagung, Young Researcher's Minisymposium YR1, TU Braunschweig, March 7-11, 2016.
- S. Reichelt: On periodic homogenization, 20th Harz-Seminar, Hahnenklee-Goslar, February 21-23, 2016.
- S. Reichelt: Homogenization of Cahn-Hilliard equations, SFB 910 Workshop, Wittenberg, September 14-19, 2015.
- S. Reichelt: Achilles und die Schildkröte, Lange Nacht der Wissenschaften, Berlin, June 13, 2015.
- A. Mielke: Homogenizing the Penrose-Fife system via evolutionary Γ-convergence, INdAM Workshop ``Special Materials in Complex Systems - SMaCS 2015'', Rome, May 18-20, 2015.
- S. Reichelt: Two-scale homogenization and error estimates for nonlinear reaction-diffusion systems with slow diffusion, CASA Colloquium, TU Eindhoven, March 11, 2015.
- S. Reichelt: Two-scale homogenization and error estimates for nonlinear reaction-diffusion systems, MATHEON Multiscale Seminar, TU Berlin, December 3, 2014.
- A. Mielke: Homogenization of parabolic gradient systems via evolutionary Γ-convergence, Second Workshop of the GAMM Activity Group on ``Analysis of Partial Differential Equations", Universität Stuttgart, September 29 - October 1, 2014.
- A. Mielke: Multiscale modeling and evolutionary Gamma-convergence for gradient flows, BMS-WIAS Summer School ``Applied Analysis for Materials'', TU Berlin, August 25 - September 5, 2014.
- S. Reichelt: Two-scale homogenization of nonlinear reaction-diffusion systems involving different diffusion length scales, AIMS Conference on Dynamical Systems and Differential Equations, Madrid, June 7-11, 2014
- S. Reichelt: Effective model for a reaction-diffusion system in strongly heterogenous media, MURPHYS-HSFS, WIAS Berlin, April 7-11, 2014.
- A. Mielke: Evolutionary Gamma convergence and amplitude equations, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Friedrich-Alexander Universität Erlangen-Nürnberg, March 10-14, 2014.
- S. Reichelt: Two-scale homogenization of nonlinear reaction-difusion problems with small diffusion, 13th GAMM seminar on microstructures, Bochum, January 17-18, 2014.
- S. Reichelt: Two-scale homogenization of nonlinear reaction-difusion problems with small diffusion, BMS Intensive Course on Evolution Equations and Applications, Berlin, November 27-29, 2013.
- A. Mielke: Introduction to evolutionary Gamma convergence for gradient systems, School "Multi-scale and Multi-field Representations of Condensed Matter Behavior", Centro di Ricerca Matematica "Ennio De Giorgi", Pisa, Italy, November 25-29, 2013.
- M. Thomas: Local versus energetic solutions in rate-independent brittle delamination, Diffuse interface models -- DIMO 2013, Levico (Italy), September 10-13, 2013.
- A. Mielke: Deriving the Ginzburg-Landau equation as amplitude equation via evolutionary Gamma convergence, ERC Workshop on Variational Views on Mechanics and Materials, University of Pavia, June 24-26, 2013.
- A. Mielke: Gradient structures and uniform global decay for reaction-diffusion systems, Mathematisches Kolloquium, Universität Bielefeld, April 25, 2013.
- S. Reichelt: Introduction to homogenization, Seminar, FU Berlin, April 11, 2013.
- A. Mielke: Evolutionary Gamma convergence and amplitude equations, Matheon Multiscale Seminar, TU Berlin, April 8, 2013.
- A. Mielke: Multiscale gradient systems and their amplitude equations, Dynamics of Patterns, Oberwolfach, December 17-21, 2012.
- A. Mielke: Linearized elastoplasticity is the evolutionary Γ-limit of finite elastoplasticity, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), TU Darmstadt, March 26-30, 2012.
- M. Thomas: Delamination in visco-elastic materials with thermal effects, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), TU Darmstadt, March 26-30, 2012.
- M. Thomas: Thermomechanical modeling via energy and entropy using GENERIC (joint work with A. Mielke),
MFO-Workshop "Mechanics of Materials", Oberwolfach, March 18-24, 2012.
- A. Mielke: Gamma convergence and evolution, International Conference "Trends in Mathematical Analysis", Politecnico di Milano, March 1-3, 2012.
- M. Thomas: Thermomechanical modeling via energy and entropy (joint work with A. Mielke), University of Pavia, February 14, 2012.
- S. Reichelt: Homogenization in reaction-diffusion problems, SFB 910 Symposium, TU Berlin, November 25, 2011.
- S. Reichelt: Homogenization in reaction-diffusion problems, SFB 910 Workshop, Wittenberg, August 31 - September 2, 2011.
Symposia and Workshops
Last modified: June 4, 2018 CET Sina Reichelt