**Pattern formation, energy landscapes, and scaling laws**

**Head: **

Felix Otto

**Contact: **Email*Phone:*

+49 (0) 341 - 9959 - 950

**Address:**

Inselstr. 22

04103 Leipzig

**Administrative Assistant:**

Katja Heid

Email, *Phone/Fax:*

+49 (0) 341 - 9959

- 951

- 658

# Overview over our work

Below, we very briefly introduce some of the problems that we have investigated. For details, please visit the related links or contact the researchers working on the topic.

## Micromagnetics

The magnetization of a ferromagnetic sample is influenced by external magnetic fields, by the magnetic field it creates itself, material anisotropies and the quantum mechanical spin exchange effect.

Together, these lead to various interesting patterns, for example in magnetic thin films. Amongst other questions, we have investigated the coarsening of the concertina pattern and the transition from symmetric to asymmetric domain walls. Read more >>>

## Coarsening

Binary mixtures far from thermodynamic equilibrium tend to separate their two components; droplets spread on a thin liquid substrate tend to merge. Typically, such systems develop a characteristic length scale that reflects the current state of ordering and which grows during the evolution.

Despite the possibly very complicated pattern, the rate at which the coarsening proceeds follows simple growth laws. We have approached some of these coarsening rates rigorously. Read more >>>

## Stochastic Homogenization

Many heterogeneous media are described by partial differential equations with random coefficients varying on small scales. On macroscopic scales such media often show a less complex, deterministic behavior. The corrector problem from stochastic homogenization provides an equation that relates the microscopic random description of the medium to its macroscopic behavior. Since, in practice, the corrector problem has to be approximated, a quantitative analysis is needed to estimate the approximation error.

Motivated by this, we have developed quantitative methods for elliptic equations with random (possibly correlated) coefficients. In particular, we obtained an optimal error estimate for the approximation of the homogenized coefficients by periodization. Read more >>>

## Hydrodynamic Limit

The hydrodynamic limit associates to a simple microscopic model, that is exposed to thermal noise, a macroscopic continuous evolution. This macroscopic evolution turns out to be deterministic like in most physical models. In this way, the hydrodynamic limit helps to answer the question how macroscopic phenomena originate from microscopic rules.

A prototype for a real world application would be understanding how the interaction of spins leads to the continuous description of a magnet. Because such a real world application is out of reach yet, we mainly consider toy models from statistical mechanics to gain insight and develop tools needed for more realistic and hence more complicated models. Read more >>>

## Viscous Thin Films

We investigate the time evolution of thin viscous films on a flat solid. In particular we are interested in properties of mathematical models for the triple junction between liquid, solid, and the surrounding gas. The choice of proper boundary conditions at the junction for the underlying partial differential equation is decisive for the macroscopic behavior of the thin film. Read more >>>

## Rayleigh-Bénard Convection

Rayleigh-Bénard convection models a fluid enclosed between two parallel isothermal plates held at fixed temperatures. If the temperature of the bottom plate is set to be higher than the top one and the distance between the plates is large enough then strong buoyancy forces will trigger bulk motion. The initially periodic motion will eventually break down, becoming turbulent and chaotic. It is of interest for applications to quantify the enhancement of heat transport due to convection in the regime of turbulences. Read more >>>