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Chemotaxis(-fluid) systems with logarithmic sensitivity and slow consumption: global generalized solutions and eventual smoothness

We consider the system \begin{align*} \begin{cases} n_t + u \cdot \nabla n = Δn - χ\nabla \cdot (\frac{n}{c} \nabla c), \\ c_t + u \cdot \nabla c = Δc - nf(c), \\ u_t + (u \cdot \nabla) u = Δu + \nabla P + n \nabla ϕ, \quad \nabla \cdot u = 0, \end{cases} \end{align*} in smooth bounded domains $Ω\subset \mathbb R^N$, $N \in \mathbb N$, for given $f \ge 0$, $ϕ$ and complemented with initial and homogeneous Neumann--Neumann--Dirichlet boundary conditions, which models aerobic bacteria in a fluid drop. We assume $f(0) = 0$ and $f'(0) = 0$, that is, that $f$ decays slower than linearly near $0$, and construct global generalized solutions provided that either $N=2$ or $N > 2$ and no fluid is present. If additionally $N=2$, we next prove that this solution eventually becomes smooth and stabilizes in the large-time limit. We emphasize that these results require smallness neither of $χ$ nor of the initial data.