Blow-up prevention by indirect signal production mechanism in a two-dimensional Keller–Segel–(Navier–)Stokes system

This paper deals with an initial-boundary value problem in two-dimensional smoothly bounded domains for the system

$$\begin{aligned} \left\{ \begin{array}{l} n_t+\textbf{u}\cdot \nabla n=\Delta n-\nabla \cdot (n\mathcal {S}(n)\nabla v),\quad x\in \Omega , t>0,\\ v_t+\textbf{u}\cdot \nabla v=\Delta v-v+w,\quad x\in \Omega , t>0,\\ w_t+\textbf{u}\cdot \nabla w=\Delta w-w+n,\quad x\in \Omega , t>0,\\ \textbf{u}_t+\kappa (\textbf{u}\cdot \nabla )\textbf{u}+\nabla P=\Delta \textbf{u}+n\nabla \phi , \quad x\in \Omega , t>0,\\ \nabla \cdot \textbf{u}=0,\quad x\in \Omega , t>0,\\ \end{array}\right. \qquad \qquad (*) \end{aligned}$$

which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid, where \(\kappa \in \mathbb {R}\), the gravitational potential \(\phi \in W^{2,\infty }(\Omega )\), and \(\mathcal {S}(n)\) satisfies

$$\begin{aligned} |\mathcal {S}(n)|\le C_\mathcal {S}(1+n)^{-\alpha } \quad \text{ for } \text{ all }~~ n\ge 0,~~C_\mathcal {S}>0~~\text{ and }~~\alpha >-1. \end{aligned}$$

Under the boundary conditions

$$\begin{aligned} (\nabla n-n\mathcal {S}(n)\nabla v)\cdot \nu =\partial _\nu v=\partial _\nu w=0, \textbf{u}=0, \quad x\in \partial \Omega , t>0, \end{aligned}$$

it is shown in this paper that suitable regularity assumptions on the initial data entail the following: (i) If \(\alpha >-1\) and \(\kappa =0\), then the simplified chemotaxis-Stokes system possesses a unique global classical solution which is bounded. (ii) If \(\alpha \ge 0\) and \(\kappa \in \mathbb {R}\), then the full chemotaxis-Navier–Stokes system admits a unique global classical solution.