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Fallstudien der Mathematischen Modellbildung, part 3: Asymptotic methods for perturbation problems

The goal of this lecture is to introduce some fundamental notions and techniques used in the asymptotic analysis of perturbation problems. Such problems are called singular if they undergo a change in their mathematical structure as the perturbation parameter ε tends to zero. A solution of the reduced problem (ε = 0) coincides with the limit solution of the full problem as ε --> 0 only if the perturbation is regular. It is the subject of asymptotic analysis to find approximate solutions of the full problem that are valid uniformly for 0 < ε <= ε0, even if the perturbation is singular. Singular perturbation problems usually arise at the most critical (and interesting) regimes of physical modeling - their analysis and ultimate resolution has often lead to major advances in a specific field of science. In the first part of this course we focus on some basic principles and examples in the context of ordinary differential equations: we introduce the principle of dominant balance and discuss boundary layers, the WKB method, the method of (variational) averaging and the method of multiple scales. The guiding-center approximation of plasma physics is considered as a generic example of nonlinear perturbation theory. In the second part we extend our analysis to partial differential equations and present Prandtl's boundary layer for the Navier-Stokes equation. Moreover, we elaborate on macroscopic limits of kinetic equations in the strongly collisional regime, leading to fluid models of reduced dimensionality.

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The lectures will be complemented with an exercise class where the notions developed in the lecture will be applied to concrete examples.

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