Area of Research
Partial Differential Equations; Mathematical Modelling; Thermodynamical Principles
Biography
Dr Alexander Mielke studied mathematics with a minor subject in mechanics in Stuttgart, where he also finished his PhD and Habilitation. After professorships in Hannover and Stuttgart he moved to Berlin in 2004, where he became the Head of Research Group on Partial Differential Equations at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) and a professor at the Humboldt-Universität zu Berlin. Since his retirement in 2023, he is an Honorary Member of WIAS.
His scientific interest lies in the mathematical modelling and the analysis of phenomena in continuum systems, typically described by partial differential equations. The earlier research focused on mathematical problems in fluid dynamics, drawing on ideas from dynamical systems theory. A later focus included continuum mechanical models for hysteresis in shape-memory alloys and elastoplastic materials. More recently, he studies reaction-diffusion processes in chemistry and semiconductor physics.
As a central guiding theme, his research is driven by the fundamental question of how general principles of thermodynamics can be transformed into modern mathematical concepts, providing new tools for understanding Nature. This topic was also central to the research within the ERC Advanced Grant “Analysis of Multiscale Systems Driven by Functionals” which ran from 2011 to 2017.
Citizenship
Germany
ViCAS Project
The topics I want to investigate can be summarised by the title
VARIATIONAL PRINCIPLES AND MATHEMATICAL METHODS FOR CLOSED AND OPEN DISSIPATIVE SYSTEMS IN THE SCIENCES
Dissipation is a concept fundamental to many areas in the sciences: it tends to reduce the complexity of states, often generating convergence to steady states. This applies especially to closed systems where a total free energy or a negative entropy is decreasing with time. However, the situation changes when such systems are connected to the environment, leading to open systems that are able to exchange mass or energy with the environment. This includes classical physical systems with suitable forcing, as well as biological models and systems with active particles. The goal is the development of mathematical structures and variational principles for the modelling and analysis of such systems by highlighting the roles of dissipation and inflow/outflow of energy or mass. Variational principles are especially useful for studying qualitative properties of solutions and, more importantly, for deriving effective models in multiscale problems. ViCAS' interdisciplinary approach will be ideal for stimulating mathematical developments beyond current boundaries.
Recent Publications
- Mielke A, Peletier MA & Zimmer J (2025). “Deriving a GENERIC system from a Hamiltonian system.” Archive Rational Mechanics Analysis, 249(62) 1-71. doi.org/10.1007/s00205-025-02119-7.
- Mielke A (2023). “Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems.” Journal of Mathematical Physics 64(12)1-10. doi.org/10.1063/5.0149910.
- Liero M, Mielke A & Savaré G (2018). “Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures.” Inventiones mathematicae, 211(3) 969-1117. doi.org/10.1007/s00222-017-0759-8.