Ansgar Jüngel: Multi-species systems in biology: cross-diffusion and hidden gradient-flow structure

Abstract:
Nature is dominated by systems composed of many individuals with a collective behavior. Examples include wildlife populations, biological cell dynamics, and tumor growth. There is a fast growing interest in multi-species systems both in theoretical biology and applied mathematics, but because of their enormous complexity, the scientific understanding is still very poor. On a macroscopic level, such systems may be modeled by systems of partial differential equations with cross diffusion, which reveals surprising effects such as uphill diffusion and diffusion-induced instabilities, seemingly contradicting our intuition on diffusion.

Major difficulties of the mathematical analysis of the cross-diffusion equations are their highly nonlinear structure and the lack of positive definiteness of the diffusion matrix. In this talk, a method inspired from non-equilibrium thermodynamics is proposed, which allows for a mathematical theory of a large class of such systems. The idea is to exploit the hidden formal gradient-flow structure by introducing so-called entropy variables. The analysis in these variables leads to global existence results, L^\infty bounds, and large-time asymptotics results. We apply the technique to some systems modeling populations and tumor growth.