Global solutions for chemotaxis-fluid systems with singular chemotactic sensitivity

This paper concerns the chemotaxis-Navier-Stokes system \begin{document}$\begin{equation} \begin{cases} n_t+u\cdot {\nabla } n = \Delta - {{\nabla }\cdot}(n\chi(c) c),\\ c_t+u\cdot c -nf(c),\\ u_t+\kappa (u\cdot }) u + P +n }\Phi, \quad }\cdot} 0 \end{cases} \end{equation} \;\;\;\;\;\;\;\;\;\;\;\;(\star)$ \end{document} in a smoothly convex bounded domain under Neumann/Neumann/Dirichlet boundary conditions. The existing literature reveals that $ (\star) possesses global solution some mild assumptions on \chi and f [25,28]. It is shown this even though admits singularity (e.g. \chi(c) c^{-\alpha} for \alpha>0 $), appropriate use of quantity involving guarantees existence to any choice suitably regular (nonnegative) initial data.