Global boundedness and absorbing sets in two-dimensional chemotaxis-Navier-Stokes systems with weakly singular sensitivity and a sub-logistic source

TL;DR

This work analyzes a two-dimensional chemotaxis–Navier–Stokes system with weakly singular sensitivity $-\chi \nabla\cdot\left( \frac{n}{c^k} \nabla c\right)$ $(k\in(0,1))$ and a sub-logistic damping term $-\frac{\mu n^2}{\log^\eta(n+e)}$ $(\eta\in(0,1))$ in a bounded domain, under homogeneous Neumann boundaries for $n,c$ and no-slip for the fluid. The authors develop a robust suite of a priori estimates, including an $L\log L$ bound for $n$, $L^p$ bounds for $n,c,u$, and control of the coupled chemotaxis–fluid interactions, to establish global-in-time, bounded classical solutions, as well as an absorbing set in $C^0(\bar{\Omega})\times W^{1,\infty}(\Omega)\times C^{0,\theta}(\bar{\Omega})$. The analysis leverages energy functionals, Moser–Trudinger-type bounds, Neumann heat semigroup smoothing, and the Stokes operator to manage the singular sensitivity and fluid coupling in 2D. These results contribute to the theory of chemotaxis-fluid systems by showing that sub-logistic damping suffices to prevent blow-up in the presence of weakly singular sensitivity and fluid advection.

Abstract

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $Ω$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= Δn - χ\nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n - \frac{μn^2}{\log^η(n+e)}, c_t + u \cdot \nabla c &= Δc - αc + βn, u_t + u \cdot \nabla u &= Δu - \nabla P + n \nabla φ+ f, \nabla \cdot u &= 0, \end{cases} \end{equation*} where $r, μ, α, β, χ$ are positive parameters, $k, η\in (0,1)$, $φ\in W^{2,\infty}(Ω)$, and $f \in C^1\left(\barΩ\times [0, \infty)\right) \cap L^\infty\left(Ω\times (0, \infty)\right)$. We show that, under suitable conditions on the initial data and with no-flux/no-flux/Dirichlet boundary conditions, this system admits a globally bounded classical solution. Furthermore, the system possesses an absorbing set in the topology of $C^0(\barΩ) \times W^{1, \infty}(Ω) \times C^0(\barΩ; \mathbb{R}^2)$.

Theorems & Definitions (48)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof