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

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, $k, η\in (0,1)$, $φ\in W<sup>{2,\infty}(Ω)$, 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 $$C^0(\barΩ) \times W^{1, \infty}(Ω) \times C^0(\barΩ; \mathbb{R}<sup>2)$$.