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

TL;DR

This paper proves the global boundedness and existence of absorbing sets for solutions to a complex two-dimensional chemotaxis-Navier-Stokes system with weakly singular sensitivity and a sub-logistic source, under certain conditions.

Contribution

It establishes the global boundedness and absorbing set existence for solutions to a novel chemotaxis-fluid system with weakly singular sensitivity and sub-logistic source.

Findings

Existence of globally bounded classical solutions.

Presence of an absorbing set in specified function spaces.

Conditions under which solutions remain bounded over time.

Abstract

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $\Omega$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= \Delta n - \chi \nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n - \frac{\mu n^2}{\log^\eta(n+e)}, c_t + u \cdot \nabla c &= \Delta c - \alpha c + \beta n, u_t + u \cdot \nabla u &= \Delta u - \nabla P + n \nabla \phi + f, \nabla \cdot u &= 0, \end{cases} \end{equation*} where $r, \mu, \alpha, \beta, \chi$ are positive parameters, $k, \eta \in (0,1)$, $\phi \in W^{2,\infty}(\Omega)$, and $f \in C^1\left(\bar{\Omega}\times [0, \infty)\right) \cap L^\infty\left(\Omega \times (0, \infty)\right)$. 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 $C^0(\bar{\Omega}) \times W^{1, \infty}(\Omega) \times C^0(\bar{\Omega}; \mathbb{R}^2)$.