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.