Partial H\"{o}lder Regularity for Bounded Solutions of a Class of Cross-Diffusion Systems with Entropy Structure

TL;DR

This paper establishes partial Hölder regularity for bounded solutions of certain strongly coupled, degenerate quasilinear cross-diffusion systems with an entropy structure, introducing a new 'glued entropy density' technique.

Contribution

It introduces a novel 'glued entropy density' method to prove partial regularity for cross-diffusion systems with entropy structure, including the SKT model.

Findings

Partial $C^{0,eta}$-regularity for bounded solutions.

Partial $C^{1,eta}$-regularity under additional assumptions.

Applicability demonstrated on well-known cross-diffusion models.

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 H\"{o}lder 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.