Collective Behavior: Macroscopic versus Kinetic Descriptions

Analysis of cross-diffusion models for multi-species systems: How entropy can help

Ansgar Jüngel

Vienna University of Technology


Some systems of collective behavior for multiple species can be described in the continuum limit by cross-diffusion systems, derived e.g. from lattice models. Examples are coming from population dynamics, cell biology, and gas dynamics. A common feature of these strongly coupled parabolic differential equations is that the diffusion matrix is often neither symmetric nor positive definite, which makes the mathematical analysis very challenging.
In this talk, we explain that for certain cross-diffusion systems, these difficulties can be overcome by exploiting the so-called entropy structure. This means that there exists a transformation of variables (called entropy variables) such that the transformed diffusion matrix becomes positive definite, and there exists a Lyapunov functional (called entropy) which enables suitable a priori estimates. Although the maximum principle generally does not hold for systems, we show that the entropy concept helps to prove lower and upper bounds for the solutions to systems with volume-filling effects (boundedness-by-entropy principle).
We detail this theory for several examples coming from tumor-growth modeling, population dynamics, and multicomponent gas dynamics. The existence of global weak solutions and their long-time behavior is investigated and some numerical examples are presented.