Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension

Motivated by the study of bacteria’s response to environmental conditions, we consider the doubly degenerate nutrient taxis system \begin{align*} \begin{cases} u_t=\nabla\cdot(uv\nabla u)-χ\nabla\cdot(u^αv\nabla v)+\ell uv,\ v_t=Δv-uv, \end{cases} \end{align*} subjected to no-flux boundary conditions and smooth initial data, where $α\in\mathbb{R}$ is the bacterial response parameter. Global solvability of weak solutions to this taxis system is highly challenging due to not only the doubly nonlinear diffusion and its degeneracy but also the strong chemotactic effect, where the latter is strong at the large species density if $α$ is close to $2$. Recent findings on the global weak solvability for the considered system are summarised as follows \begin{itemize} \item In [M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021] for $α=2$, $N=1$; \item In [M. Winkler, \textit{J. Differ. Equ.}, 2024] for $1\leα\le 2$, $N=2$ with initial data of small size if $α=2$; \item In [Z. Zhang and Y. Li, \textit{arXiv:2405.20637}, 2024] for $α=2$, $N=2$; and \item In [G. Li, \textit{J. Differ. Equ.}, 2022] for $\frac{7}{6}<α<\frac{13}{9}$, $N=3$. \end{itemize} Our work aims to provide a picture of global weak solvability for $0\leα<2$ in the physically dimensional setting $N=3$. As suggested by the analysis, it is divided into three separable cases, including (i) $0\leα\le 1$: Weak chemotaxis effect; (ii) $1<α\le 3/2$: Moderate chemotaxis effect; and (iii) $3/2<α<2$: Strong chemotaxis effect.

💡 Research Summary

This paper presents a comprehensive analysis establishing the global existence of weak solutions to a “doubly degenerate nutrient taxis system” in three physical dimensions (N=3), for the full range of the bacterial response parameter α where 0 ≤ α < 2. The system models the biased movement of bacteria (density u) towards a nutrient (concentration v), featuring nonlinear diffusion and chemotaxis fluxes that both degenerate when u or v are small, making the analysis highly challenging.

The primary achievement is proving global solvability without any smallness assumption on the initial data, which had been a major open problem for most α values in 3D. The authors strategically divide the parameter range into three regimes based on the strength of the chemotactic effect and devise distinct, sophisticated proof strategies for each:

1. Weak Chemotaxis (0 ≤ α ≤ 1): The proof leverages a rediscovered “competitive structure” of the fluxes, allowing a direct bootstrap argument. The key energy identity (9) provides sufficient regularity to quickly establish L^p bounds for u for any p, leading to global existence.

2. Moderate Chemotaxis (1 < α ≤ 3/2): The analysis combines the known logarithmic energy structure (5) with L^p-energy estimates. The core innovation here is a two-phase bootstrap argument. The authors construct sequences that establish a mutual feedback loop between the boundedness of ∫∫ u^m v and the boundedness of ∫ u^p. This iterative process gradually improves the solution’s regularity until arbitrary L^p bounds for u are secured.

3. Strong Chemotaxis (3/2 < α < 2): This is the most technically intricate and novel part. The authors start from very low regularity using estimates for negative exponent energies. They then introduce a new mixed energy functional of the form ∫ u^p v^q (Lemma 5.4) to carefully control cross-terms. The masterstroke is the design of a cyclic bootstrap loop based on three inter-dependent implications (S1, S2, S3). This loop creates a virtuous cycle where control over the chemotactic term ∫∫ u^(q+2) v |∇v|^2, control over the L^p norm of u, and control over a “half-weighted” chemotactic term ∫∫ u^(q+2) √v |∇v|^2 mutually reinforce and enhance each other. A critical ingredient in step (S2) is the simultaneous use of both the boundedness of ∫ u^(3-α) and ∫ |∇v|^4 / v^3 from the logarithmic energy structure—an aspect unused in prior work. This loop is iterated to finally achieve the necessary high regularity for global existence.

Throughout the paper, the authors refine and adapt functional inequalities to the three-dimensional setting, overcoming the dimensional limitations of tools used in earlier studies for 1D and 2D. The work nearly completes the global existence theory for this biologically relevant model in 3D, leaving only the borderline case α = 2 open for future investigation. The methodologies developed, particularly the cyclic bootstrap for strong chemotaxis, provide a powerful new framework for analyzing complex degenerate chemotaxis systems.