Analysis of diffusion models for multi-species systems

Many systems in chemistry and biology which consist of several species are modeled by cross-diffusion equations. Examples are diffusion models in population dynamics, reaction-diffusion models in chemistry, ion transport through membranes, and Maxwell- Stefan systems for multicomponent fluids. Cross-diffusion systems consist of nonlinear parabolic equations in divergence form whose diffusion matrix is nondiagonal. Because of the strong coupling, many tools of the theory of partial differential equations, like maximum principles and regularity theory, cannot be applied to such systems. While Ladyzenskaya, Solonnikov, and Ural`ceva and later Amann have developed a local existence theory for quasilinear systems with elliptic differential operators, a global theory for weak solutions is still missing.. The first aim of this project is to develop a global well-posedness theory for certain cross-diffusion systems, strengthening recent results obtained by the PI. The second aim is to prove some qualitative properties of the solutions, like boundedness, and long-time behavior of weak solutions. The innovative features of the analysis are the systematic use of entropy-dissipation methods, the boundedness-by-entropy principle, a refined Bakry-Emery approach, and hypocoercivity concepts.

Many applications in physics, biology, and chemistry include systems for multiple species or components. At the macroscopic level, these systems can be described by special par- tial differential equations, so-called cross-diffusion systems. These equations allow for the prediction of the dynamical behavior of particle densities or chemical concentrations of the multi-component systems. The aim of this project was to study cross-diffusion equations from applications analytically and numerically. The following applications were analyzed in detail for this study: a model for the trans- port of ions through biological channels and nanopores; a population model describing the segregation of species; a model to describe the diffusion and chemical reactions in fluid mixtures; and kinetic Boltzmann equations which model the transport and collisions of particles. The main results of this project are the existence of global-in-time solutions, the exponential decay of the solutions towards the equilibrium state and numerical simulations using finite- volume and finite-element methods.