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 = \Delta n - \chi \nabla \cdot (\frac{n}{c} \nabla c), \\ c_t + u \cdot \nabla c = \Delta c - nf(c), \\ u_t + (u \cdot \nabla) u = \Delta u + \nabla P + n \nabla \phi, \quad \nabla \cdot u = 0, \end{cases} \end{align*} $in smooth bounded domains $ \Omega \subset \mathbb R^N $, $ N \in \mathbb N $, for given $ f \ge 0 $, $ \phi $ 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 $ \chi $ nor of the initial data.