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.

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 FG1) 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.

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.

The German journal "Spektrum der Wissenschaft" has a standing comparable to the "Scientific American" in the USA. It has published recently the article ”Energie effizienter speichern" (”Storing energy more efficiently") by Martin Heida and Manuel Landstorfer (RG 7), in which the significance of efficient storage systems with regard to the upcoming energy transition is discussed. The research focuses on understanding the aging mechanisms of batteries to develop better energy storage options. Mathematical models and computer simulations are employed to understand complex processes within batteries and predict their performance evolution. The goal is to produce high-performance and sustainable batteries that ensure reliable power supply for electromobility and renewable energy systems.

The German Research Foundation has agreed to fund the project ”Analysis of energy-variational solutions for hyperbolic conservation laws" in the scope of the Priority Program "Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness (CoScaRa)" (SPP 2410), which was proposed by Thomas Eiter together with Robert Lasarzik (RG 4). The project aims at extending the analytic investigation of a novel solution concept for a general class of conservation laws, in particular, in connection with the introduction of suitable selection criteria for ruling out unphysical solutions.

Several coworkers of RG 1 gave talks at the Annual GAMM Meeting (Dresden, 29 May -- 2 June 2023): Thomas Eiter on ”Energy-variational solutions for a class of hyperbolic conservation laws”, Annegret Glitzky on ”An effective bulk-surface thermistor model for large-area organic light-emitting diodes”, Martin Heida on ”Perspectives for homogenization on randomly perforated domains”, Matthias Liero on ”Balanced-viscosity solutions for a Penrose--Fife phasefield model with friction”, Michael Kniely on ”On a thermodynamically consistent electro-energy-reaction-diffusion system”, Leon Schütz on ”An existence theory for solitary waves on a ferrofluid jet”, and Willem van Oosterhout on ”Analysis of poro-visco-elastic solids at finite strains”.

Michael O'Donovan successfully defended his PhD thesis entitled “Theory of carrier transport in III-Nitride based heterostructures” on 23 March 2023. Congratulations!

Successful start of the collaborative project UV Lasers: From Modeling and Simulation to Technology - on 12 January 2023, 25 people participated in the kickoff meeting to discuss the challenges of realizing UVC laser diodes and approaches to overcome them.

We are delighted by the news that Alexander Mielke, Head of RG 1 and Professor at Humboldt-Universität zu Berlin, has been awarded the “MATH+ Distinguished Fellowship”. The “MATH+ Distinguished Fellowship” is to recognise outstanding achievement within the mathematical sciences. Congratulations!

Anieza Maltsi successfully defended her PhD thesis entitled “A mathematical study of the Darwin--Howie--Whelan equations for Transmission Electron Microscopy” on 11 November 2022. Congratulations!

From 12 to 14 September 2022, the workshop “MathBio22 - - Mathematical Models for Biological Multiscale Models" took place in hybrid mode. It was jointly organized by Barbara Wagner (RG 7), Alfonso Caiazzo (RG 3), Dirk Peschka (WG 1), and Matthias Liero (RG 1). In 23 talks and four posters, mathematical models and methods were discussed that bridge the gap in scale and complexity between microscopic descriptions of biological and biophysical systems with corresponding PDE descriptions.

Matthias Liero successfully completed his habilitation at the Humbold-Universitat zu Berlin. His habilitation thesis “Mathematical Analysis of Charge and Heat Flow in Organic Semiconductor Devices", defended on 4 July 2022, discusses mathematical models describing the electrothermal behavior of organic semiconductor devices. Congratulations!

Thomas Eiter was chosen to give this year's Junior Richard-von-Mises-Lecture at the Humboldt-Universität zu Berlin. In his talk on June 17, 2022 with the title ”On time-periodic viscous flow around a moving body", he presented his newest results on the fluid flow around a rotating obstacle. His analytical investigations are mainly based on suitable estimates for the associated resolvent problem, which strongly depend on whether or not an additional translation is present.

On 20 and 21 June 2022, the Young Researchers' Forum on Mathematical Fluid Mechanics took place, which was organized by Thomas Eiter together with Ryosuke Nakasato and Keiiche Watanabe (both from Waseda University, Tokyo). In particular to enhance the exchange between early-career scientists, besides plenary talks and discussion rounds, there were short talks where PhD students could present their research topics.

SPHInX-Tutorial 2022: from 21 March to 11 April 2022, Oliver Marquardt successfully hosted a weekly tutorial on the calculation of electronic properties of semiconductor heterostructures using the SPHInX software library, which is in part maintained by WIAS Berlin. The tutorial took place in hybrid mode, was funded by means of the cluster of excellence MATH+ (project IN-7), and was attended by about 20 members of both the Berlin scientific community and by international partners.

Several coworkers of RG 1 hosted minisymposia at the online conference “SIAM PDE 2022”, 14 -- 18 March 2022: Thomas Eiter with Keiichi Watanabe (Waseda University, Japan) on “Recent Developments in the Mathematical Analysis of Viscous Fluids (MS13)”; Alexander Mielke with Hong-Ming Yin (Washington State University, USA) and William Fitzgibbon (College of Technology, USA) on “Nonlinear Parabolic Equations and Systems (MS15)”; Artur Stephan with Oliver Tse (Eindhoven University of Technology, Netherlands) on “Variational Evolution: Analysis and Multi-Scale Aspects (MS45)”; Martin Heida on “Homogenization of Random Singular Structures (MS49)”; Matthias Liero with Dirk Peschka and Marita Thomas (both WG 1, WIAS) on “Energy-Based Mathematical Methods and Thermodynamics (MS80)”.

Welcome! Michael Kniely has been a visiting scientist within RG 1 since 1 March 2022. His research stay is funded by an Erwin Schrödinger Fellowship of the Austrian Science Fund (FWF). In the next two years, Michael Kniely will conduct research on electro-energy-reaction-diffusion systems together with members of RG 1.

The Leibniz Association will fund the joint project “UV Lasers - From Modeling and Simulation to Technology (UVSimTech)” for three years within the Leibniz Competition. The consortium is coordinated by Thomas Koprucki at WIAS and includes besides WIAS also the FBH Berlin, the IKZ Berlin, the TU Berlin und the FAU Erlangen.

The interdisciplinary workshop ”Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (AMaSiS 2021)” took place from 6 -- 9 September 2021, and attracted more than 100 participants from 17 countries (Germany, France, USA, China, Italy, UK, Austria, ...). This online-workshop was dedicated to the mathematical modeling of semiconductors and electrochemical systems. Due to inherent similarities between both disciplines, AMaSiS 2021 explored synergies in mathematical modeling, analysis, numerics, and simulation techniques. The conference brought together experts from applied mathematics, physics, engineering, chemistry, and material science. AMaSiS 2021 covered the topics electronic structure theory, non-equilibrium thermodynamics and transport theories, mathematical upscaling from quantum mechanics and particle systems to continuum scale, numerical methods, as well as special semiconductor devices, and electrochemical systems.

Artur Stephan successfully defended his PhD thesis entitled ”Coarse-graining for gradient systems and Markov processes” on 11 May 2021. Congratulations!

Closing conference "Structures in Evolution: Theory and Applications" (23 -- 25 February 2021) online within the Thematic Einstein Semester, winter term 2020/21: A programme that included renowned speakers such as Pamela Cook (U Delaware), Günther Grün (U Erlangen-Nürnberg), Paul Kotyczka (TU München), Josef Málek (Charles University Prague), Marcel Oliver (U Bremen), Jacquelien Scherpen (U Groningen), Guido Schneider (U Stuttgart), and Ulisse Stefanelli (U Wien) was offered. The full thematic range of application, scale transitions, numerical methods, and mathematical analysis was covered by the talks, which were complemented by discussions in breakout rooms. The conference had 79 registered participants from 10 countries.

"MA4M: Mathematical Analysis for Mechanics" (23 -- 25 November 2020) online within the Thematic Einstein Semester, winter term 2020/21: This workshop was organised by Matthias Liero, Alexander Mielke, and Marita Thomas and realised in close cooperation with the DFG Priority Programme SPP 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials”. The focus was on the derivation of effective models in continuum mechanics for problems with different scales, on mechanical modelling and mathematical analysis for complex materials, and on variational formulations and relaxation methods. Despite the limitations of the online format, the opportunity to discuss and to compare methods and approaches in the above-mentioned fields was given by virtual discussion boards and extensive breaks between talks. Thereby, exchange between junior and senior scientists was ensured. There were about 120 participants and 19 invited talks, eight of which were given by junior researchers.

"Student Compact Course on Variational Methods and Fluids" online (12 -- 23 October 2020) and "Kick-off Conference" online (26 -- 30 October 2020) within the Thematic Einstein Semester, winter term 2020/21: A broad programme of online lectures on gradient structures, Hamiltonian systems, GENERIC and port-Hamiltonian structures, evolutionary Gamma-convergence, large deviations theory, structure-preserving numerical methods, and much more was offered to about 100 participants of the ”Student Compact Course”. Interested PhD-students had the opportunity to give short presentations of their work and to further explore the TES topics. Subsequently, a ”Kick-off Conference” with 25 invited talks and about 150 participants took place.