2D Chemotaxis-Fluid Model Dynamics

• The model is defined by coupled PDEs for bacterial density, chemoattractant concentration, and fluid velocity, integrating chemotaxis with Navier–Stokes dynamics.
• Analysis shows global classical solutions exist under conditions like small initial mass or presence of logistic damping, ensuring stabilization to equilibrium.
• Degenerate diffusion in low-signal regions introduces analytical challenges that require entropy-based methods and delicate handling of nonlinear cross-diffusion effects.

A two-dimensional chemotaxis-fluid model describes the coupled evolution of cell (bacterial) density, chemoattractant (signal) concentration, and incompressible fluid velocity, accounting for mechanisms such as chemotaxis, nonlinear cross-diffusion, nonlinear or degenerate mobilities, and explicit coupling with the Navier–Stokes equations. This class of models is pivotal in mathematical biology and fluid dynamics as it connects pattern formation, aggregation, and stabilization phenomena in populations of motile organisms to the fluid flows they both generate and inhabit. The rigorous treatment of such systems elucidates under what structural and parameter regimes solutions exist globally, remain bounded, and converge to equilibrium, as well as the mechanisms driving blowup or loss of regularity.

1. Mathematical Structure of the Two-Dimensional Chemotaxis-Fluid System

The prototypical two-dimensional chemotaxis-fluid model considered in recent literature is formulated as the following system, posed over $\Omega \subseteq \mathbb{R}^2$:

$$\begin{aligned} & n_t + u \cdot \nabla n = \Delta (n \phi(v)) + \mu n(1-n), \ & v_t + u \cdot \nabla v = \Delta v - n v, \ & u_t + \kappa (u \cdot \nabla) u = \Delta u + n \nabla\Phi - \nabla P, \quad \nabla \cdot u = 0, \end{aligned}$$

where:

• $n = n(x, t)$: bacterial density,
• $v = v(x, t)$: chemoattractant or signaling molecule concentration,
• $u = u(x, t)$: fluid velocity,
• $P = P(x, t)$: pressure,
• $\mu \geq 0$: logistic damping parameter,
• $\phi \in C^1[0, \infty) \cap C^3(0, \infty)$: motility function with $\phi(0) = 0, \phi|_{(0, \infty)} > 0, \phi'(0)>0$,
• $\kappa$ parameterizes inertial effects in Navier–Stokes,
• $\Phi$: prescribed gravitational or external potential, with suitable initial and no-slip, no-flux (or Navier/Neumann) boundary conditions (Ma et al., 21 Sep 2025).

The degenerate diffusion nature comes from the $v$-dependence in $\phi(v)$: diffusion vanishes as $v \to 0$, substantially changing regularity and long-term behavior compared to classical, linear-diffusive chemotaxis models.

2. Existence of Global Classical Solutions and Parameter Regimes

(A) Degenerate Diffusion, No Logistic Source ($\mu=0$):

• For reasonably regular initial data and a sufficiently small initial mass of bacteria ($\int_\Omega n_0 < \delta(K)$, with $\|v_0\|_{L^\infty} \leq K$), there exists a unique global classical (smooth) solution.
• The smallness condition on $\int_\Omega n_0$ is essential: when $\phi(0)=0$, the degenerate parabolicity fails in low-signal regions, which can propagate loss of regularity unless the mass—and thus the capacity for aggregation-driven instabilities—is limited.
• The proof employs a quasi-entropy functional of the form $$\mathcal{F}(t) = \int_\Omega n \log n + \int_\Omega (|\nabla v|^2/v) + C\int_\Omega |u|^2$$, leading to global-in-time a priori bounds on all components (Ma et al., 21 Sep 2025).

(B) Degenerate Diffusion with Logistic Source ($\mu>0$):

• The logistic damping $\mu n(1-n)$ regularizes the equation, providing decay at high densities and preventing blowup even for arbitrarily large initial bacterial mass.
• For all reasonably regular initial data there exists a unique global bounded classical solution.
• Moreover, under a smallness assumption on the initial signal mass ($\int_\Omega v_0 < \delta(K)$), the solution converges globally in time:

$$n(x, t) \to 1 \text{ in } L^2(\Omega),\quad v(x, t) \to 0~\text{(uniformly and in } W^{1, \infty}),\quad u(x, t) \to 0~\text{in } W^{1,2}.$$

• This convergence is proved via uniform-in-time $L^p$ and higher norm estimates, exploiting the dissipative properties of the logistic term (Ma et al., 21 Sep 2025).

3. Role of Degenerate Diffusion and Comparison to Classical Chemotaxis-Fluid Models

The degenerate diffusion $\phi(v)$ (with $\phi(0)=0$, $\phi'(0) > 0$) weakens or eliminates the regularizing effect of diffusion in low-signal regions, which is a fundamentally different regime from classical chemotaxis-fluid systems with linear cell diffusion. This gives rise to several phenomena:

• Analytical Challenges: Standard parabolic regularity may not be directly available or may fail in the vanishing-signal regime. This necessitates the use of tailored entropy functionals combining $n\log n$ and signal gradient terms.
• Critical Mass Phenomena: Boundedness and global existence often only hold under mass (or signal-mass) constraints, similarly to critical mass behavior in the minimal Keller–Segel system, but with new subtleties due to nonlinear and degenerate structure.
• Asymptotic Behavior: In the presence of logistic damping and small initial signal, the system stabilizes to $(n,v,u) = (1,0,0)$, i.e., full population at carrying capacity, signal extinction, and fluid at rest (Ma et al., 21 Sep 2025).

4. Analytical Framework and Main Techniques

The mathematical analysis of this model is built on the following components:

• Quasi-Entropy Functionals: Appropriate combinations of $n\log n$, $\|\nabla v\|^2/v$, and kinetic energy yield differential inequalities used to derive global $L^p$-in-time and $W^{1,\infty}$ spatial bounds.
• Small Mass Condition: For $\mu=0$, this is indispensable to ensure the quasi-entropy dissipates fast enough to preclude singularity formation and maintain regularity.
• Parabolic Regularity Theory: Despite the degeneracy for small $v$, estimates are bootstrapped from the entropic structure and continuity arguments; separation from the vanishing set of $v$ can be maintained via signal consumption damping or bounds on the signal's initial mass.
• Navier–Stokes Coupling: The incompressible fluid equation introduces additional technical challenges: nonlinear transport of both $n$ and $v$, direct coupling through the buoyancy term $n\nabla\Phi$, and regularity loss from the convective nonlinearity. Uniform $L^2$ and $W^{1,2}$ bounds for $u$ are obtained via energy estimates and properties of the Stokes operator (Ma et al., 21 Sep 2025).

5. Broader Context and Applications

These results generalize previous global existence and stabilization theorems from "fluid-free" degenerate chemotaxis-consumption models to cases where the population is embedded in a dynamically evolving incompressible fluid. This extension is crucial for accurately modeling motile bacterial populations in aquatic environments, for example, in the paper of oxygentaxis, pattern formation in biofilms, and aggregation under nutrient depletion.

Potential applications and implications include:

• Biological Pattern Formation: Predicting under what environmental and biochemical conditions bacterial (or more generally, microbial) populations will form persistent aggregations, remain dispersed, or relax to homogeneity in the presence of flow fields.
• Fluid Influence on Chemotactic Instabilities: Quantifying how advection and shear flows alter critical mass thresholds, stabilize spatial inhomogeneities, or influence the formation and dissipation of chemotactic patterns.
• Modeling with Indirect Signal Dependence: Understanding the interplay between cell motility laws that directly depend on environmental signals and regularizing (e.g., logistic) population feedbacks under hydrodynamic coupling.
• Mathematical Fluid Mechanics: Extending techniques of entropy methods, quasi-linear parabolic theory, and Navier–Stokes regularity to settings involving strong nonlinearities and degenerate cross-diffusion effects.

6. Relation to Prior and Related Work

• If $\phi(v) \equiv 1$ (linear diffusion), classical results guarantee global bounded solutions in 2D without smallness constraints (Chae et al., 2011, Heihoff, 2019).
• Degenerate signal-dependent diffusion ($\phi(0) = 0$) in the absence of fluid ($u \equiv 0$) was analyzed previously in the chemotaxis literature, and the present work recovers and extends such results to the coupled fluid environment.
• The role of logistic damping in regularizing chemotaxis or chemotaxis-fluid systems is consistent with a variety of models in which overgrowth instabilities are suppressed by population feedback.
• The results complement recent advances on blowup criteria, critical mass, stabilization by shear flows, nontrivial pattern dynamics, and numerical analysis for chemotaxis-fluid systems with and without structural degeneracy (Kong et al., 2023, Carrillo et al., 2023, Biler et al., 2014).

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In summary, the two-dimensional chemotaxis-fluid model with signal-dependent degenerate diffusion exhibits global existence, boundedness, and stabilization of classical solutions under biologically relevant smallness or damping conditions. These findings extend fluid-free and nondegenerate theories, providing a rigorous mathematical framework for understanding complex population–signal–fluid interactions in structured environments (Ma et al., 21 Sep 2025).