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

In bounded smooth domains $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\in \{2,3\}$, 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 $$\chi <\{\begin{array}{ll}\mathrm{\infty },& N=2,\\ \frac{5}{3},& N=3.\end{array}$$ These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.