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Hölder estimates of weak solutions to chemotaxis systems of fast diffusion type

M. Marras, F. Ragnedda, S. Vernier-Piro, V. Vespri

TL;DR

This work analyzes a chemotaxis system with fast (singular) diffusion $\Delta u^{m}$ for $0<m<1$ in a parabolic–parabolic setting and nonlinear chemotactic drift. The authors develop a self-contained regularity theory based on a refined De Giorgi–Di Benedetto iteration adapted to the coupled diffusion–drift structure, combining energy estimates, De Giorgi lemmas, and local logarithmic estimates to prove local Hölder continuity of bounded weak solutions for $m$ in the admissible range. The main contribution is establishing local Hölder regularity and clarifying how singular diffusion regularizes aggregation, with precise dependence on data and model exponents. The results extend porous-medium-type regularity theory to nonlinear chemotaxis systems, providing a rigorous foundation for the fine qualitative behavior of solutions and informing the balance between diffusion and chemotactic aggregation in fast-diffusion regimes.

Abstract

We study a quasilinear chemotaxis system of singular type, where the diffusion operator is given by $Δu^m$ with $0<m<1$, corresponding to the fast diffusion regime, and where the chemotactic drift is nonlinear. Since Hölder 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.

Hölder estimates of weak solutions to chemotaxis systems of fast diffusion type

TL;DR

This work analyzes a chemotaxis system with fast (singular) diffusion for in a parabolic–parabolic setting and nonlinear chemotactic drift. The authors develop a self-contained regularity theory based on a refined De Giorgi–Di Benedetto iteration adapted to the coupled diffusion–drift structure, combining energy estimates, De Giorgi lemmas, and local logarithmic estimates to prove local Hölder continuity of bounded weak solutions for in the admissible range. The main contribution is establishing local Hölder regularity and clarifying how singular diffusion regularizes aggregation, with precise dependence on data and model exponents. The results extend porous-medium-type regularity theory to nonlinear chemotaxis systems, providing a rigorous foundation for the fine qualitative behavior of solutions and informing the balance between diffusion and chemotactic aggregation in fast-diffusion regimes.

Abstract

We study a quasilinear chemotaxis system of singular type, where the diffusion operator is given by with , corresponding to the fast diffusion regime, and where the chemotactic drift is nonlinear. Since Hölder 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.
Paper Structure (16 sections, 14 theorems, 146 equations)

This paper contains 16 sections, 14 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Energy Estimates for $k>u$
  4. A De Giorgi Type Lemma for $k > u$
  5. Energy Estimates for $u > k$
  6. A De Giorgi-Type Lemma for $u > k$
  7. Local Logarithmic Estimates for $u > k$
  8. Estimate of $I_{1}$.
  9. Estimate of $I_{2}$.
  10. Combination of terms.
  11. Conclusion.
  12. Hölder Continuity
  13. Preliminaries for the Proof of Theorem \ref{['thm:holder']}
  14. First Alternative.
  15. Second Alternative.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $u$ be a locally bounded weak solution of the first equation in 1.1. Then $u$ is locally Hölder continuous in $\mathbb{R}^N \times (0,T]$. More precisely, there exist constants $\gamma>0$ and $\alpha \in (0,1)$, depending only on the data, such that for every compact set ${\mathcal{K}} \subset \ for all $(x_1,t_1), (x_2,t_2) \in {\mathcal{K}} \times (0,T]$.

Theorems & Definitions (26)

  • Theorem 1.1: Hölder regularity
  • Lemma 2.1: Embedding Lemma
  • Lemma 2.2: De Giorgi's Lemma
  • Lemma 2.3: Fast Geometric Convergence
  • Lemma 2.4: Heat Estimate
  • Remark 2.1
  • Lemma 3.1: Local Energy Estimates for $k>u$
  • Remark 3.1: On the proof strategy
  • proof : Proof of Lemma \ref{['lem:energy']}
  • Lemma 4.1: De Giorgi type lemma for $k > u$
  • ...and 16 more