Partial Hölder Regularity for a Class of Cross-Diffusion Systems with Entropy Structure

We obtain partial $C^{0,\alpha }$-regularity for bounded solutions of a certain class of cross-diffusion systems, which are strongly coupled, degenerate quasilinear parabolic systems. Under slightly more restrictive assumptions, we obtain partial $C^{1,\alpha }$-regularity. The cross-diffusion systems that we consider have a formal gradient flow structure, in the sense that they are formally identical to the gradient flow of a convex entropy functional. The main novel tool that we use is a "glued entropy density", which allows us to emulate the classical theory of partial Hölder regularity for nonlinear parabolic systems. We are, in particular, able to obtain partial $C^{1,\alpha }$-regularity for solutions of the Maxwell-Stefan system, as well as partial $C^{1,\alpha }$-regularity for bounded solutions of the Shigesada-Kawasaki-Teramoto model.

This talk is based on joint work with Marcel Braukhoff and Nicola Zamponi.