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This research paper establishes the Hölder continuity of bounded weak solutions to a class of degenerate drift-diffusion equations, extending previous results to cover a wider range of nonlinear diffusion processes and boundary conditions relevant to chemotaxis models.

Abstract

**Bibliographic Information:**Black, Tobias. "Refining Hölder regularity theory in degenerate drift-diffusion equations."*arXiv preprint arXiv:2410.03307*(2024).**Research Objective:**This paper aims to prove the Hölder continuity of bounded weak solutions to a class of degenerate parabolic equations with specific structural conditions on the diffusion and convection terms, motivated by limitations in existing regularity theory for chemotaxis models.**Methodology:**The author employs a novel approach based on transforming the original equation into a form resembling Stefan problems and adapting the classical energy estimate techniques developed by DiBenedetto and Friedman. This involves analyzing the transformed equation's sub and superlevel sets and establishing decay estimates for the oscillation of solutions over shrinking nested cylinders.**Key Findings:**The paper successfully establishes the local Hölder continuity of bounded weak solutions to the considered class of degenerate drift-diffusion equations under specific assumptions on the diffusion function Φ and the convection terms. It also extends the regularity results to the associated initial-boundary value problem with no-flux boundary conditions.**Main Conclusions:**The results presented in this paper significantly contribute to the regularity theory of degenerate parabolic equations by providing a more flexible framework that can handle a wider array of nonlinear diffusion processes, including those relevant to chemotaxis models. The findings have implications for understanding the qualitative behavior of solutions to such equations, particularly in the context of chemotaxis systems.**Significance:**This research addresses a gap in the existing literature on Hölder regularity for degenerate parabolic equations, particularly concerning the interplay between degenerate diffusion and convection terms. The results have direct applications in the mathematical modeling of chemotaxis and other biological phenomena governed by similar equations.**Limitations and Future Research:**The study focuses on a specific class of degenerate drift-diffusion equations with particular structural assumptions. Further research could explore extending the results to more general forms of degenerate parabolic equations or investigating the regularity of solutions under weaker assumptions on the diffusion and convection terms.

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by Tobias Black at **arxiv.org** 10-07-2024

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Beyond the evident benefit of gaining insight into the regularity properties of solutions, establishing Hölder continuity for the solution $w$ (and under suitable assumptions on $\Phi$ also for $u$) can be particularly advantageous for the analysis of chemotaxis models in the following ways:
Improved compactness properties: Hölder continuity implies strong compactness of families of solutions in appropriate function spaces. This compactness is crucial when studying the convergence of approximate solutions obtained via regularization or discretization methods, ultimately aiding in proving the existence of solutions to the chemotaxis system.
Facilitating bootstrapping arguments: Hölder continuity of the solution can be used as a starting point for bootstrapping arguments. In such arguments, the established regularity is leveraged to show higher regularity for the solution by treating the equation in a hierarchical manner. This can lead to solutions belonging to a more regular class, potentially even classical solutions.
Analyzing long-term behavior: Understanding the regularity of solutions is often a prerequisite for investigating the long-term behavior of solutions to chemotaxis systems. For instance, one might be interested in whether solutions converge to a steady state, exhibit periodic behavior, or display more complex dynamics.
Justifying qualitative analysis techniques: Many qualitative analysis techniques, such as studying the evolution of level sets or applying comparison principles, rely on certain regularity assumptions on the solutions. Hölder continuity can serve as a justification for employing these techniques, leading to insights into properties like finite-time blow-up, formation of singularities, or asymptotic behavior.

The structural assumptions on the diffusion function $\Phi$, particularly the condition $s\Phi''(s) \leq C_\Phi \Phi'(s)$, play a crucial role in controlling the degenerate diffusion and ensuring Hölder continuity. Relaxing these assumptions while still guaranteeing Hölder continuity is a delicate issue, and any potential relaxation would likely come at the cost of additional restrictions on other aspects of the equation or the solutions themselves.
Here are some potential avenues for relaxation and their implications:
Weakening the growth condition on Φ'': Instead of requiring a global bound on $s\Phi''(s)/\Phi'(s)$, one could explore conditions that allow for more rapid growth of $\Phi''$ near zero, potentially involving additional powers of $s$. This might necessitate a refined analysis of the degenerate terms in the energy estimates and could lead to a smaller Hölder exponent or restrictions on the admissible range of the solution.
Allowing for non-convex Φ: While convexity of $\Phi$ simplifies the analysis significantly, it might be possible to relax this assumption to a certain extent. For instance, one could consider functions that are "almost convex" in a suitable sense. This would likely require a more intricate analysis of the level sets and could potentially lead to weaker regularity results, such as continuity but not Hölder continuity.
Imposing additional conditions on the solutions: Another possibility is to compensate for weaker assumptions on $\Phi$ by imposing additional conditions on the solutions themselves. For example, one could require a certain degree of positivity or boundedness of the gradient. This approach would restrict the admissible class of solutions but could potentially allow for a wider range of diffusion functions.

The regularity properties of solutions to degenerate drift-diffusion equations, such as those studied in the paper, can have profound implications for the emergence of pattern formation in biological systems. Here are some potential connections:
Suppression of blow-up and formation of aggregates: In chemotaxis models, degenerate diffusion can counteract the aggregation-promoting effects of the chemotactic sensitivity, potentially preventing finite-time blow-up of solutions. Hölder continuity of solutions, as established in the paper, provides further evidence for the regularizing effect of degenerate diffusion, suggesting the formation of bounded aggregates rather than singular structures.
Influence on pattern wavelength and regularity: The specific form of the diffusion function $\Phi$ can influence the characteristic length scales and regularity of emerging patterns. For instance, stronger degeneracy near zero (corresponding to slower diffusion at low densities) might lead to more spread-out patterns with smoother boundaries, while weaker degeneracy could result in more concentrated aggregates with potentially sharper interfaces.
Connection to traveling waves and other coherent structures: The regularity properties of solutions can impact the existence and stability of traveling wave solutions, which are often observed in chemotaxis systems and play a crucial role in pattern formation. Hölder continuity can facilitate the analysis of these coherent structures, providing insights into their speed, shape, and persistence.
Impact on the long-term dynamics of patterns: The regularity of solutions can influence the long-term evolution of patterns. For instance, smoother solutions might lead to more stable patterns that persist over time, while less regular solutions could exhibit more complex dynamics, such as pattern coarsening, merging, or even chaotic behavior.
Overall, the regularity properties of solutions to degenerate drift-diffusion equations are intricately linked to the emergence and characteristics of pattern formation in biological systems. Understanding these connections is crucial for unraveling the complex interplay between diffusion, chemotaxis, and other factors that govern the spatial organization of cells and organisms.

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