Singular sensitivity in a Keller-Segel-fluid system

Abstract: In bounded smooth domains $\Omega\subset\mathbb{R}^N$, $N\in{2,3}$, considering the chemotaxis--fluid system [ \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= \Delta n - \chi \nabla \cdot(\frac{n}{c}\nabla c) &\ & c_t + u\cdot \nabla c &= \Delta c - c + n &\ & u_t + \kappa (u\cdot \nabla) u &= \Delta u + \nabla P + n\nabla \Phi & \end{split}\end{cases} ] with singular sensitivity, we prove global existence of classical solutions for given $\Phi\in C^2(\bar{\Omega})$, for $\kappa=0$ (Stokes-fluid) if $N=3$ and $\kappa\in{0,1}$ (Stokes- or Navier--Stokes fluid) if $N=2$ and under the condition that [ 0<\chi<\sqrt{\frac{2}{N}}. ]