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

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $\mathrm{\Omega }$: \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 $r,\mu ,\alpha ,\beta ,\chi$ are positive parameters, $k,\eta \in (0,1)$, $\phi \in W^{2,\mathrm{\infty }}(\mathrm{\Omega })$, and $f\in C^1(\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty }))\cap L^{\mathrm{\infty }}(\mathrm{\Omega }\times (0,\mathrm{\infty }))$. 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 $C^0(\bar{\mathrm{\Omega }})\times W^{1,\mathrm{\infty }}(\mathrm{\Omega })\times C^0(\bar{\mathrm{\Omega }};{\mathbb{R}}^2)$.