Partial Differential Equations

Many fundamental processes in nature and technology can be described by partial differential equations. The research group is working on the analytical theory of such equations (existence, uniqueness, qualitative behaviour) and on the development and implementation of algorithms for their numerical solution. The algorithms are used for the numerical simulation in industrial applications. The functionality of modern materials, for instance, relies on the complex interplay of effects on several length and time scales as well as on different physical effects, such as mechanics, thermodynamics, optics, and electromagnetism. The main topics of research are mathematical models of carrier transport in semiconductors and optoelectronic devices and reaction-diffusion equations for the transport of dopants in solids. Furthermore, nonlinear material models for linearized and nonlinear elasticity and plasticity as well as for systems with internal variables are under study. In this context, we develop in particular methods for abstract evolutionary equations, e.g. gradient systems, and for multiscale problems.

From October 9 to 13, the conference "Variational and Geometric Structures for Evolution" took place in Levico Terme (Italy). It was jointly organized by Marita Thomas (RG 1), Dorothee Knees (U Kassel), Riccarda Rossi (U Brescia), and Giuseppe Savaré (U Milano). The scientific focus was on recent developments in the identification and analytic exploration of variational and geometric structures for evolutionary problems. Katharina Hopf and Artur Stephan from RG 1 contributed talks to the programme. The conference also marked the 65th birthday of Alexander Mielke, former head of RG 1, and honoured his achievements in this field.

The workshop "Energetic Methods for Multi-Component Reactive Mixtures - Modelling, Stability, and Asymptotic Analysis" took place at WIAS from September 13 to 15, 2023. It was organized by Katharina Hopf and Michael Kniely (both RG 1) in cooperation with Ansgar Jüngel (TU Wien). The 22 talks and three poster presentations covered a variety of aspects in the modelling and analysis of reaction-diffusion systems and fluid models, including viscoelastic fluids and multiphase flow. Special emphasis was put on energy and entropy methods, which are known as a versatile tool in the context of complex systems subject to temperature effects, electrostatic forces, and compressibility.

On September 6, 2023, Alexander Mielke's 65th birthday was celebrated with a scientific colloquium. The event started with welcome addresses by the institute's director and close colleagues. The scientific part of the event featured presentations by renowned speakers such as Rupert Klein (FU Berlin), Felix Otto (MPI Leipzig), and Riccarda Rossi (U Brescia).

Several coworkers of RG 1 hosted minisymposia at the "10th International Congress on Industrial and Applied Mathematics (ICIAM 2023)", Tokyo,
20 -- 25 August 2023; Thomas Eiter with Florian Oschmann (Czech Academy of Sciences) on "Limit behaviour and asymptotic properties in fluid mechanics", and Matthias Liero and Artur Stephan with Yuan Gao (Purdue University) on "Variational methods for multi-scale dynamics. Additionally, Thomas Eiter, Dirk Peschka, and Artur Stephan gave talks at the conference.