Multicomponent systems with incomplete diffusion

Nature contains mainly of mixtures of multiple species or particles. Examples are gas mixtures like air, ion transport through cell membranes, and segregation of animal populations. Previous scientific research was mainly focused on mathematical models for single species, since multicomponent systems show a very complex behavior which is difficult to predict or to control. Recent progress in the mathematics of multicomponent systems makes it possible to consider mathematical models for multiple species. The idea is to combine the newest techniques from mathematical fluid dynamics with thermodynamic principles and so-called entropy methods. The focus of this project is to investigate mathematical models with diffusion, which expresses the erratic movement of animals or bacteria, for instance. In applications, many systems do not show complete diffusion, but there are also transport phenomena without any diffusion in that direction. We call these systems having incomplete diffusion. Standard mathematical methods cannot be used for such systems, and the aim of the project is to develop new techniques which are able to deal with systems with incomplete diffusion.

The project was concerned with diffusion phenomena in multi-species systems like gas mixtures, ion transport in biological channels, and segregation in animal populations. These phenomena can be described by diffusive and viscous partial differential equations. A key feature is that diffusion is incomplete in these models, which means that parts of the equations are of diffusive (parabolic) type, while other parts are of transport (hyperbolic) type. Consequently, the solutions do not enjoy the usual regularity properties of parabolic equations. Thus, novel mathematical techniques need to be devised to compensate the loss of parabolicity. - In this project, we have analyzed two model classes. The first class describes heat-conducting fluid mixtures with Maxwell-Stefan diffusion. By advancing convex free energy techniques and exploiting thermodynamic consistency, the existence of a weak solution to the stationary Navier-Stokes-Fourier system was proved. The incomplete diffusion was compensated by the volume-filling constraint. This condition also appeared in other models like volume-filling Nernst-Planck equations for ion transport. We have proved the global existence of weak solutions and a weak-strong uniqueness property. Furthermore, the understanding of thermodynamic principles helped us to develop a mathematical theory for time-dependent non-isothermal Maxwell-Stefan equations. - The second model class consists of mass conservation equations and velocities that are given by the nonlinear pressure gradient according to Darcy's law. They were suggested by Busenberg and Travis to model segregating populations. We have proved, under rather general conditions, the global existence of measure-valued solutions and a weak-strong uniqueness property. The established hyperbolic-parabolic normal form yields a better understanding of these models. Surprisingly, the Busenberg-Travis equations possess two mathematical entropies (usually, there is only one), namely the Boltzmann entropy, well known in thermodynamic, and the Rao entropy, describing the diversity of the species. A fluid approximation of these equations allowed us to explain this fact: The Boltzmann entropy comes from the thermodynamic fluid entropy, while the Rao entropy can be associated to the kinetic and potential energy. The approximation was made mathematically rigorous by including regularizing Korteweg terms. - The main outcome of the project is an improved understanding how thermodynamic properties and mathematical techniques for diffusive and fluid models are intertwined. The incomplete diffusion was successfully compensated by physical volume-filling constraints or mathematical hyperbolic structure properties.