Analysis – nonlinearity, memory effects and randomness

In the field of analysis, we introduce Professor Petra Wittbold’s and PD Dr Aleksandra Zimmerman’s work on nonlinear evolution equations. Evolution equations are used to describe the temporal development of systems. Complex issues in biology, medicine, physics, engineering, economics and the social sciences require nonlinear models for adequate description.

Memory effects and random effects are an important factor in describing the development of such processes, as they influence the present and the future of humans and materials alike. Take filtration processes as an example: if a liquid flowing through a porous medium contains particles that interact with that medium (e.g., by settling in its pores), its flow behaviour will change with time. This is a typical memory effect. Random parameters are introduced into the models in order to account for the heterogeneity of the medium, interactions on various scales and the inaccuracy of measured data. Over the past two years, there has been a lot of research into whether nonlinear evolution equations with memory and random effects are well posed. Many projects were carried out in collaboration with mathematical institutes in Brazil and France. Within the scope of her own externally funded project, Dr Aleksandra Zimmerman has worked with an interdisciplinary research group from the Laboratoire de Mécanique et d’Acoustique (LMA) of Aix-Marseille University on modelling the interfaces of composite materials in consideration of stochastic effects.

Professor Paola Pozzi’s research group focuses on geometric aspects of evolutionary processes. In particular, its members were able to produce new results in the field of curvature flows. These flows describe changes in surfaces occurring due to physical processes, which are often described in terms of the minimisation of surface tensions. They can also be used to obtain classification results on the shape of geometrical structures in cases where it can be proven that the limits  of the flows have special geometrical properties. A collaborative project between mathematicians from Australia and Japan was able to obtain new existence results for such flows. In the years ahead, the researchers will work on questions on the long-term evolution of networks under various flows, working with research groups in Ulm and Taiwan within the scope of the DFG project on the flow of elastic networks.