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 \[ \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c) &\\ & c_t + u\cdot \nabla c &= Δc - c + n &\\ & u_t + κ(u\cdot \nabla) u &= Δu + \nabla P + n\nabla Φ& \end{split}\end{cases} \] 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 \[ 0<χ<\sqrt{\frac{2}{N}}. \]