Chemotaxis-Fluid Systems

Updated 7 January 2026

Chemotaxis-fluid systems are PDE models that couple biological cell movement in response to chemical gradients with fluid mechanics to capture complex interactions.
They integrate nonlinear features like chemotactic sensitivity, logistic growth, and fluid forcing, employing techniques such as enhanced dissipation and entropy methods.
Analyses focus on global well-posedness, stability, and numerical simulation, with applications from microbial plume formation to microfluidic experiments.

Chemotaxis-fluid systems are partial differential equation (PDE) models describing the interaction between self-propelled biological populations (e.g., bacteria, sperm, or microbial ecologies) exhibiting chemotactic movement in response to chemical gradients and the dynamics of the surrounding incompressible fluid. These systems generalize the classical Patlak–Keller–Segel (PKS) chemotaxis equations by coupling them with the (Navier-)Stokes equations, typically via density-dependent buoyancy, chemotactic feedback on the fluid, or additional nonlinearities involving cross-diffusion and nonlinear growth sources. They appear at the nexus of mathematical biology, nonlinear PDE theory, fluid mechanics, and numerical analysis due to their rich pattern formation, nonlinear stability/bifurcation structure, and deep analytic challenges.

1. Canonical PDE Models and Structural Variants

The prototypical chemotaxis–Navier–Stokes system in a smooth bounded domain $\Omega\subset\mathbb{R}^d$ ( $d=2,3$ ) involves the coupled evolution

$\begin{cases} n_t + u\cdot\nabla n = \Delta n - \nabla\cdot(\chi(c)n\nabla c) + \Psi_{\text{growth}}(n, c), \ c_t + u\cdot\nabla c = \Delta c - n\,f(c), \ u_t + (u\cdot\nabla)u = \nu\Delta u + \nabla P + \Gamma(n, c, x, t), \ \nabla\cdot u = 0, \end{cases}$

where $n$ is the microorganism density, $c$ is the chemoattractant (e.g., oxygen), $u$ is fluid velocity, and $P$ is pressure.

Major structural choices include:

Chemotactic sensitivity function $\chi(c)$ (scalar/tensor, possibly volume-filling or with saturation/logarithmic form) (Fuest, 2022, Wang et al., 2024, Ma et al., 21 Sep 2025).
Consumption law $f(c)$ , allowing linear, sublinear, superlinear, or even degenerate uptake (Fuest, 2022).
Source terms $\Psi_{\text{growth}}$ (e.g., logistic growth $+\kappa n - \mu n^2$ ), modeling birth/death (Lankeit, 2016).
Fluid forcing $\Gamma$ (e.g., $n\nabla\phi$ with external potential $\phi$ , or active force $n\nabla c$ ) (Carrillo et al., 2023, 2207.13494).
Nonlinear/degenerate diffusion of $n$ (porous media, signal-dependent, volume-filling) (Black, 2017, Ma et al., 21 Sep 2025, Wang et al., 2024).
Domain geometry and boundary conditions: no-flux for $n,c$ , no-slip or Navier-type for $u$ , variable free-surface, or complex sessile drop geometries (Wang et al., 2022, Jiang et al., 2014).

Selected representative models from the literature:

Reference	Main features	Nonlinearity
(Lankeit, 2016)	3D, logistic source, oxygen consumption, Navier–Stokes	$\Psi = +\kappa n - \mu n^2$
(Fuest, 2022)	Logarithmic sensitivity, slow consumption, $N=2$ and $N\ge3$ (fluid-free)	$\chi(c)=\text{const}/c$ , $f(c)$ superlinear decay
(Carrillo et al., 2023)	Self-consistent force: cells affect and are affected by the fluid and signals	$-\nabla\cdot(n\nabla\phi)$ , $\chi(c) n\nabla c$ fluid forcing
(Black, 2017, Ma et al., 21 Sep 2025)	Degenerate/signal-dependent diffusion	$\Delta(n\phi(c))$ with $\phi(c)\to0$ as $c\to0$
(Wang et al., 2024)	Volume-filling sensitivity, supercritical regime, enhanced dissipation	$\chi(n)=n(n+1)$

2. Global Well-Posedness, Regularity, and Blowup Criteria

Existence theory critically depends on space dimension, nonlinear structure, and the magnitude/smallness of key parameters (e.g., cell mass, initial $c_0$ ). Principal results include:

3D with logistic damping: For arbitrary data, the strong quadratic decay $-\mu n^2$ controls classical chemotactic blowup mechanisms. Uniform-in-time $L^p$ estimates can be closed, yielding the existence of global weak solutions and eventual smoothness. Solutions converge in $C^1$ to $(n_*,c_*,u_*) = (\kappa/\mu, 0, 0)$ as $t\to\infty$ (Lankeit, 2016).
Degenerate/Volume-filling diffusion ( $m>4/3$ , or more strongly $m>5/3$ in 3D): non-classical “very weak” solutions exist globally even with full fluid coupling. The porous medium exponent threshold $m>4/3$ mirrors the sharp threshold for Keller–Segel in 3D without fluid (Black, 2017).
Logarithmic sensitivity with slow consumption: Global generalized solutions exist for $N=2$ (fluid coupling) and $N\geq3$ (no fluid) without any smallness assumptions, provided the oxygen consumption decays fast enough near $c=0$ . Eventual smoothing and convergence to spatially uniform state occur in 2D (Fuest, 2022).
Dominant chemotactic blowup (without stabilizing mechanisms): Classical PKS in 2D and 3D admits finite-time blowup above critical mass ( $8\pi$ in 2D). Fluid coupling generically raises the threshold; sufficiently strong flow or enhanced dissipation (e.g., shear-driven mixing) can suppress blowup for arbitrarily large cell mass, provided the shear or viscosity is large enough (2207.13494, Wang et al., 2024).
Two-dimensional systems: For Navier–Stokes or Stokes, global classical solutions exist for any initial data (subcritical mass), converging to the uniform state for $t\to\infty$ even on general nonconvex domains (Jiang et al., 2014, Kong et al., 2023).
Measure-valued data: For 2D models, global solutions exist starting from Dirac, filament, or general Radon measures for cell densities and fluid vorticity, provided the initial signal $c_0$ is sufficiently small. This covers extremely singular initial configurations (Ferreira et al., 2023).

The analytic toolkit includes entropy–energy methods, bootstrapping via Moser–Alikakos iteration, Lyapunov/quasi-energy functionals, blowup-preventing cross-diffusion and logistic damping, and spectral gap or enhanced dissipation techniques in the presence of strong background flows.

3. Linear Stability, Pattern Formation, and Nonlinear Dynamics

Chemotaxis-fluid systems exhibit a variety of nonlinear instabilities, patterning, and bioconvection phenomena:

Convection-coupled chemotaxis: At suitable taxis Rayleigh number $Ra_\tau$ (ratio of buoyant to viscous and diffusive relaxation rates), one observes the onset of hydrodynamic instability and plume formation analogous to Rayleigh–Bénard convection. Chemotactic terms can destabilize or stabilize the system depending on their relative strength ( $S\,H$ product, where $S$ is chemotaxis sensitivity and $H$ is oxygen consumption per cell) (Deleuze et al., 2014).
Linear stability analysis: Critical $Ra_{\tau, c}(S\,H)$ curves are U-shaped: minimal threshold at $S\,H\sim O(1)$ ; diffusion-dominated $(S\,H\to0)$ and chemotaxis-dominated $(S\,H\to\infty)$ regimes both suppress instability. Strong chemotaxis homogenizes cell concentration, while diffusion suppresses perturbations.
Plume and aggregation patterns: In bioconvective regimes, structured high-density bacterial “stack” layers form below oxygenated surfaces, then break into descending plumes with exponential amplitude growth and specific intrinsic wavelengths. Late-time plume positions may remain unpredictable except in degenerate initial geometries (Deleuze et al., 2014, Wang et al., 2022).
2D bounded-mass aggregation: Below critical mass, solutions remain globally bounded and relax to uniformity; at critical mass, “spot” equilibrium solutions localize at domain boundaries, constructed rigorously via inner–outer gluing methods with Stokes operator analysis (Kong et al., 2023).

4. Analytical and Numerical Techniques

A sophisticated array of PDE, functional, and numerical methods has been developed for chemotaxis-fluid systems:

Entropy, Lyapunov, and Quasi-Energy Methods: Weighted entropy/dissipation inequalities for $\int n\ln n$ , $\int |\nabla c|^2$ , $\int |u|^2$ , and specialized functionals (e.g., lower-bound Lyapunov for logistic sources) control cross-diffusion and align with weak/very weak solvability (Lankeit, 2016, Fuest, 2022).
Enhanced Dissipation: Shear flows (e.g., Couette) create mixing that amplifies viscous dissipation, introducing a spectral gap and exponential energy decay of nonzero modes in the co-moving frame. This removes the classical chemotactic blowup threshold for any mass (2207.13494, Wang et al., 2024).
Carleman Inequalities and Controllability: Advanced Carleman estimates for the coupled chemotaxis and Stokes components enable local exact controllability results—solutions can be steered to prescribed final states by distributed controls, leveraging observability duality (Chaves-Silva et al., 2016).
Numerical Schemes: High-resolution, positivity-preserving finite volume/element and projection methods, including hybrid strategies for complex geometries (diffuse-domain) (Wang et al., 2022), splitting and semi-implicit time advancement (Duarte-Rodríguez et al., 2019, Li, 7 Jun 2025), and SUPG stabilization for convection (Deleuze et al., 2014), allow quantitative simulation of patterning, aggregation, and intricate bioconvective flows with rigorous error bounds.
Boundary and Domain Effects: Weak solution theory extends to nonconvex and general bounded domains via handling of curvature-induced boundary terms (Mizoguchi–Souplet lemma), enabling entropy–energy estimates and global existence results without convexity preconditions (Jiang et al., 2014).

5. Nonlinear Features: Degeneracy, Tensor Sensitivity, and Volume-Filling

Chemotaxis-fluid models display rich nonlinearities, including:

Degenerate Diffusion: Signal-dependent or porous medium-type diffusion ( $\Delta (n\phi(v))$ , $\phi(0)=0$ ) dramatically alters well-posedness and smoothing. The cross-diffusive term $\phi'(v)\,n\,\nabla v$ can dominate when random motility vanishes on signal depletion, leading to mathematically delicate L $\log$ L entropy frameworks (Ma et al., 21 Sep 2025).
Tensor-Valued Sensitivity: Matrix-valued $S(x,n,c)$ introduces rotational chemotaxis, breaking the classical entropy structure, and demands smallness assumptions on the initial chemical to secure uniform bounds and stabilization (Cao, 2016).
Volume-Filling: Supercritical volume-filling sensitivity ( $\chi(n)=n(n+1)$ ) suppresses aggregation-induced blowup only under sufficiently strong background flow (enhanced dissipation) or mass subcriticality (Wang et al., 2024).
Measure-valued Initial Data: Recent works construct global mild solutions for arbitrary nonnegative Radon measures as initial densities and vorticities, enabling analysis of singular “peak” or “filament” aggregations (Ferreira et al., 2023).

6. Applications, Experimental Validation, and Open Directions

Chemotaxis-fluid systems underpin a wide array of phenomena:

Coral Spawning, Microbial Plume Formation, Sperm/Ovum Dynamics: Two-population and reaction–advection–diffusion–chemotaxis–fluid models capture fertilization and aggregation in water column and marine environments. Global existence, decay rate, and positivity preservation are resolved for both empirically-motivated and abstracted forms (Chae et al., 2019).
Microfluidic Experiments: State-of-the-art devices quantify bacterial drift velocity vs. chemical gradient, confirming predicted nonlinear $\chi(c)$ responses and highlighting surface-quenching of chemotactic drift at boundaries (hydrodynamic reorientation near solid interfaces) (Gargasson et al., 12 Nov 2025).
Numerical Pattern Simulation and Geometry Effects: Machine-verified code demonstrates robustness of plume dynamics and solution regularity across sessile drop and capsule geometries, as well as surface uptake and gravity-induced flows (Wang et al., 2022).
Open Problems and Limitations:
- Extension to 3D full Navier–Stokes with singular/supercritical nonlinearity remains largely open, barring special structures.
- Volume-filling and nonlocal effects (e.g., non-instantaneous signal response, Ostwald ripening, aggregation with transport delay) present ongoing analytic and computational challenges.
- Realistic boundary and biochemical coupling (chemotaxis with partial signal reflection, nontrivial surface rheology) remain only partially captured in available theory and numerics.

References reflect main results cited:

(Lankeit, 2016): global weak solution, boundedness, convergence for 3D with logistic source.
(Fuest, 2022): global generalized solutions for logarithmic sensitivity, eventual regularity.
(Jiang et al., 2014): classical solutions without convexity, global weak solutions in 3D Stokes.
(Ma et al., 21 Sep 2025): 2D degenerate, signal-dependent diffusion (global regularity, convergence).
(Kong et al., 2023): global existence and spot equilibrium for 2D PKS–NS in bounded domains.
(Wang et al., 2024, 2207.13494): enhanced dissipation, blowup suppression by shear.
(Ferreira et al., 2023): global solutions from singular measures in 2D.
(Li, 7 Jun 2025, Duarte-Rodríguez et al., 2019, Wang et al., 2022): finite element, finite difference, and diffuse-domain numerical methods.
(Chaves-Silva et al., 2016): Carleman inequalities and controllability of chemotaxis–Navier–Stokes.
(Gargasson et al., 12 Nov 2025): microfluidics, experimental chemotaxis–fluid coupling.
(Black, 2017): very weak/weak solutions for nonlinear/degenerate diffusion.
(Cao, 2016): tensor-valued (rotational) chemotactic flux.