H\"older estimates of weak solutions to chemotaxis systems of fast diffusion type

TL;DR

This paper proves H"older continuity for weak solutions to a chemotaxis system with fast diffusion, using a refined De Giorgi iteration adapted to the coupled nonlinear structure.

Contribution

It extends regularity results to singular chemotaxis systems with nonlinear diffusion, revealing a regularizing mechanism similar to porous medium equations.

Findings

Established H"older continuity for solutions in the fast diffusion chemotaxis model.

Developed a refined De Giorgi iteration scheme for coupled nonlinear systems.

Demonstrated regularization effects due to the interplay of singular diffusion and chemotactic aggregation.

Abstract

We study a quasilinear chemotaxis system of singular type, where the diffusion operator is given by $\Delta u^m$ with $0<m<1$, corresponding to the fast diffusion regime, and where the chemotactic drift is nonlinear. Since H\"older continuity constitutes the optimal regularity class for weak solutions to the porous medium equation, we establish analogous regularity results for bounded solutions of parabolic--parabolic chemotaxis systems in this setting. The proof is based on a refined De Giorgi--Di Benedetto iteration scheme adapted to the coupled structure of the system. These results advance the understanding of the fine regularity properties of chemotaxis models with nonlinear diffusion, and demonstrate that the interplay between singular diffusion and aggregation exhibits a regularizing mechanism consistent with the porous medium paradigm.