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Lecture

Methods of Multiscale Analysis

The lecture is devoted to mathematical methods of multiscale and asymptotic analysis with application to differential equations. The methods are motivated by problems of model derivation and reduction. Real systems can be modeled at various levels of resolution. Effective behavior of the system depends on parameters and, ideally, by changing these parameters we should be able to move smoothly between microscopic, mesoscopic and macroscopic level depending on whether we are interested in the accuracy of the description and disregard the cost, or conversely. The goal of this lecture is to provide a systematic way of deriving effective equations, the coefficients of which encapsulate relevant information from the original regime. We aim to investigate convergence of solutions of the original equations in the micro regime to solutions of the equations in the macro regime when the identified small parameter converges to zero. The program of the lecture includes:
1. Singular perturbation methods:
(i) Introduction to regular and singular perturbation:
(ii) Tikhonov Theorem with application to ODEs (quasi-stationary approximation):
(iii) Singular perturbation methods for reaction-diffusion equations (shadow systems, SLEP method):
(iii) Renormalization Group method applied to ODE and PDE problems of singular perturbation (quasi-stationary approximation; shadow limit, etc):
2. Methods of homogenization:
(i) Heuristic approach and two-scale expansions. Examples of diffusion and filtration in porous medium.:
(ii) Techniques to study convergence of multi-scale expansions: two-scale convergence, Tartar's energy method, Gamma-convergence, periodic unfolding. Applications to reactive flows in multiscale systems, including heterogeneous surface reactions.

Lecture (2 h)	Tutorials (2 h)
Prof. Dr. Anna Marciniak-Czochra	Jan-Erik Busse
Time: Thursdays 9 - 11	Time: Mondays 11-13
Place: INF 294, HS -111	Place: INF 294, HS -111

Prerequisites:
Functional Analysis. Helpful previous knowledge: Basic theory of ODEs and PDEs.

Grading policy:
Solution of exercises and a final exam in written or oral form. Details will be given by the lecturer at the beginning of the course.

Scripts:
There are four scripts in a password-protected file.

Suggested Literature
[1] J.Banasiak and M.Lachowicz, Methods of small parameter in mathematical biology. Birkh�user, Boston 2014.
[2] L. Y. Chen, N .Goldenfeld, Y. Oono, Renormalization group and singular perturbations : Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E 54, (1996), 376-394.
[3] U. Hornung, Homogenization and Porous Media, Springer 1997.
[4] A. Marciniak-Czochra and A.Mikelic, Multiscale Methods with Applications in Sciences, chapters from the book in preparation, 2014.
[5] I.Takagi, Mathematical Analysis of Biological Pattern Formation - Singular Perturbation Methods, Script, 2011.
[6] V.V. Zhikov, S.M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, 1994.