Partial Hölder Regularity for a Class of Cross-Diffusion Systems with Entropy Structure (Raithel, Wien)
Partial Hölder Regularity for a Class of Cross-Diffusion Systems with Entropy Structure
Speaker: Dr. Claudia Raithel
Affiliation: Vienna University of Technology, Austria
Abstract: 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 H\”{o}lder regularity for nonlinear parabolic systems. To demonstrate the applicability of our results and motivate our techniques, we consider the two component Shigesada-Kawasaki-Teramoto (SKT) model for population dynamics. This talk is based on a joint work with Marcel Braukhoff and Nicola Zamponi.