Summary: In a bounded planar domain \(\varOmega\) with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system \[ \begin{aligned}\begin{cases}n_t+\nabla\cdot(nu)=\Delta n-\nabla\cdot(n\nabla c),& x\in\varOmega, t> 0,\\ 0=\Delta c-c+n, & x\in\varOmega,t> 0, \end{cases}\end{aligned} \] is considered, where \(u\) is a given sufficiently smooth velocity field on \(\overline{\varOmega}\times[0,\infty)\) that is tangential on \(\partial \varOmega\) but not necessarily solenoidal.
It is firstly shown that for any choice of \(n_0\in C^0(\overline{\varOmega})\) with \(\int_{\varOmega}n_0< 4\pi\), this problem admits a global classical solution with \(n(\cdot ,0)=n_0\), and that this solution is even bounded whenever \(u\) is bounded and \(\int_{\varOmega}n_0<2\pi\). Secondly, it is seen that for each \(m> 4\pi\) one can find a classical solution with \(\int_{\varOmega}n(\cdot,0)=m\) which blows up in finite time, provided that \(\varOmega\) satisfies a technical assumption requiring \(\partial\varOmega\) to contain a line segment.
In particular, this indicates that the value \(4\pi\) of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.