Applied analysis links pure mathematics and scientific computing to engineering and natural sciences. Modeling in the applied sciences typically leads to systems of nonlinear partial differential equations or variational problems. The nonlinearities reflect complex phenomena which result in oscillations, concentrations, or singularities in the mathematical model. Most problems involve the interaction of processes on different time and length scales. Resolution on all scales is usually impossible, which requires deriving hierarchies of models to describe the problem on the respective scales. Since experiments can now be carried out on the atomic scale, the passage from discrete to continuum models is an important topic. A general challenge is to further develop techniques, such as variational convergence and homogenization, for applications where the relevant scales are not known a priori. Berlin applied analysis comprises of groups at FU, HU, and WIAS. The very active cooperation is established through several joint research activities, the seminar on Nonlinear Dynamical Systems (FU and WIAS) and the traditional Langenbach Seminar on nonlinear PDEs (WIAS and HU).

Several research projects of the applied analysis groups in Berlin are motivated by phase transformation phenomena (Melcher, Mielke, Sprekels). For example, temperature driven martensitic phase transformations in solids are the basis of the shape-memory effect in alloys. Techniques from calculus of variations are relevant in the study of local or global minimizers of certain energies. Often small parameters are involved, representing a specific regime of interest for the corresponding applications. The framework of Γ-convergence allows one to study the corresponding family of minimization problems and to identify the limit problem. Since certain limit procedures lead to the study of lower-dimensional singular structures, there is a strong link to differential geometry.

Solid-liquid phase transitions occur in metal soldering or in crystals used in wafer fabrication. The early stage of such a phase transformation requires a microscopic description on the atomic scale. Here is a strong link to stochastic analysis, where the corresponding particle models are studied. The second stage, growth, is often modeled by nonlinear parabolic systems for the respective fields (temperature, concentration, potentials), whereas the late stage, coarsening, is best described as a free boundary problem. Homogenization and averaging techniques are used to extract reduced models for the statistics of large systems.

Asymptotic analysis also plays an important role in the context of thin liquid films (Münch, Wagner). In this case of separation of length scales, simplified models can be derived from the basic equations of fluid mechanics. Corresponding reduced theories result in degenerate higher-order equations for which a general mathematical theory is still lacking. Multiple scale asymptotics has recently opened highly innovative routes of theoretical research in meteorology and climate science (Klein). It provides a unifying framework for the large variety of simplified models known in theoretical meteorology, and it paves the way for the systematic investigation and robust numerical simulation of the ubiquitous scale interactions in the atmosphere and the oceans.

Nonlinear dynamical systems (Fiedler, Mielke, Recke) are studied in Berlin, for example, in the context of nonlinear processes in opto-electronic devices or in spatio-temporal patterns in nonlinear excitable systems, such as the Belousov– Zhabotinsky reaction. The theory of global bifurcations and center manifold theory are the key analytical tools for the understanding of these processes. Due to the inherent multiple scales singular perturbation theory is also relevant.

All of the research described above requires at some level efficient simulations of the different mathematical models. The corresponding link to numerical analysis, optimization and scientific computing is also reflected in the RTG (HU and WIAS) Analysis, Numerics and Optimization of Multiphase Problems and in the Application Areas C and D of MATHEON. As pointed out, there are also strong links to differential geometry, established also through the SFB Space, Time, Matter (FU and HU), and in particular to probability. The combination of analytical and stochastic tools to understand complex phenomena is currently a very active research area, which is also emphasized by the recently installed DFG Research Group Analysis and Stochastics in Complex Physical Systems (HU, TU, and Leipzig).