Derivation and analysis of cross-diffusion systems

Many multi-species applications from physics, chemistry, and biology can be modeled by systems of partial differential equations which describe the evolution of the particle density, temperature, velocity etc. Examples are ultracold atoms in optical lattices, multicomponent gas mixtures like air, and chemical reaction kinetics mixing reactants. The aim is to understand the behavior of the partial differential equations far from equilibrium, like pattern formation and equilibration properties. The difficulty is the multi-species aspect which leads to coupled systems with sometimes unexpected properties. In recent years, significant progress in the mathematical analysis of such systems has been made, partially based on so-called entropy methods. Physically, the entropy is a measure of molecular disorder or energy dispersal; mathematically, it may be employed as an analytical tool in many partial differential equations. The discovery of a mathematical entropy is usually a delicate task. In this project, we aim to construct entropies by investigating more fundamental models which possess clearly identifiable entropies (or free energies). The novelty is the systematic approach to construct or identify entropies in partial differential equations. The project tasks include the rigorous derivation and analysis of energy-transport equations for ultracold atoms in optical lattices and extended Maxwell-Stefan systems for gas mixtures, as well as fast-reaction limits in chemical reaction networks.

Many applications from physics, chemistry, and biology can be modeled by nonlinear partial differential equations. Examples include charged-particle transport, chemical reaction kinetics, and population growth. The particles of these systems evolve according to reaction, diffusion, and cross-diffusion (which means that they are influcenced by the diffusion of other particle species). The aim of the project is to derive simplified partial differential systems from more fundamental models and to explore the structure of the original and the simplified models. When described by thermodynamic principles, these models possess an entropy, which is a measure of the disorder in a (physical) system. The entropy can be exploited as a tool for the mathematical analysis of the partial differential equations. The long-term perspective is to identify the entropy structure in any cross-diffusion system that is derived from fundamental equations. In this project, we have explored the preservation of this structure in a number of situations, including energy-transport equations in optical lattices, Maxwell-Stefan systems for gas mixtures, and reaction-diffusion systems from chemical kinetics and biofilm growth. We have been able to derive some models in a mathematical rigorous way and to identify the associated, preserved entropy structures.