A Keller-Segel-fluid system with singular sensitivity: Generalized solutions

In bounded smooth domains $\Omega\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} 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 &= \Delta u + \nabla P + n\nabla \phi, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if $$\chi<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases}$$ These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.