ERC

European Research Council Advanced Grant
within the 7th Framework Programme


Analysis of Multiscale Systems Driven by Functionals

Project Logo

Grant Agreement Number 267802; April 2011 - March 2017



Researchers

Project Head: Prof. Dr. Alexander Mielke

Investigators: Karoline Disser, Florian Eichenauer, Thomas Frenzel, Markus Mittnenzweig, Hagen Neidhardt

Associated Researchers: Annegret Glitzky, Joachim Rehberg, Marita Thomas

Former Members: S. Heinz, C. Kreisbeck, S. Neukamm, P. Racec, N. Rotundo, L. Wilhelm, S. Yanchuk


Recent Event

ERC Workshop Modeling Materials and Fluids using Variational Methods, WIAS Berlin, February 22 - 26, 2016
jointly with ERC Group 1.



Description

Many complex phenomena in the sciences are described by nonlinear partial differential equations, the solutions of which exhibit oscillations and concentration effects on multiple temporal or spatial scales. To understand the interplay of effects on different scales, it is central to determine those quantities on the microscale that are needed for the correct description of the macroscopic evolution. Our aim is to develop a mathematical framework for modeling and analyzing systems with multiple scales. In particular, we want to derive new effective equations on the macroscale that fully take into account the effects on the microscale. This will include Hamiltonian dynamics as well as different types of dissipation like gradient flows or rate-independent dynamics. The choice of models will be guided by specific applications in
  • material modeling (e.g., thermoplasticity, pattern formation, rate-independent models, delamination, micro-structure evolution) and
  • optoelectronics (drift-diffusion equations, chemical reactions, coupling to quantum mechanics).
The research will address mathematically fundamental issues like existence and stability of solutions but will be mainly devoted to the modeling of multiscale phenomena in evolution systems. We will focus on systems with geometric structures, where the dynamics is driven by functionals. Thus, we can go much beyond the classical theory of homogenization and singular perturbations. The novel features of our approach to multiscale problems are
  • the combination of different dynamical effects in one framework,
  • the use of geometric and metric structures for partial differential equations,
  • the exploitation of Gamma-convergence for evolution systems driven by functionals.




Publications




Last modified: September 20, 2016 Thomas Frenzel