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

Abstract

In this contribution 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. Furthermore, we assume that the cross-diffusion systems are not volume-filling. The main novel tool that we introduce in this contribution is a "glued entropy density," which allows us to emulate the classical theory of partial Hölder regularity for nonlinear parabolic systems by Giaquinta and Struwe within this new setting. To demonstrate the applicability of our results, we give two examples of well-studied cross-diffusion systems that satisfy our assumptions --one of which is the two component Shigesada-Kawasaki-Teramoto (SKT) model for population dynamics.

arXiv commentMerged into new submission, "Partial Holder Regularity for Solutions of a Class of Cross-Diffusion Systems with Entropy Structure", with substantially broader scope. This submission only included non volume-filling models, such as the Shigesada-Kawasaki-Teramoto model, whereas the new preprint can also handle volume-filling models, such as the Maxwell-Stefan model