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 \(\begin{aligned} \left\{ \begin{array}{l} n_t+\textbf{u}\cdot \nabla n=\Delta n-\nabla \cdot (n\mathcal {S}(n)\nabla v),\quad x\in \Omega , t>0,\\ v_t+\textbf{u}\cdot \nabla v=\Delta v-v+w,\quad x\in \Omega , t>0,\\ w_t+\textbf{u}\cdot \nabla w=\Delta w-w+n,\quad x\in \Omega , t>0,\\ \textbf{u}_t+\kappa (\textbf{u}\cdot \nabla )\textbf{u}+\nabla P=\Delta \textbf{u}+n\nabla \phi , \quad x\in \Omega , t>0,\\ \nabla \cdot \textbf{u}=0,\quad x\in \Omega , t>0,\\ \end{array}\right. \qquad \qquad (*) \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{n}_t+\mathbf{u}·\mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }·(n\mathcal{S}(n)\mathrm{\nabla }v),x\in \mathrm{\Omega },t>0,\hfill \\ v_t+\mathbf{u}·\mathrm{\nabla }v=\mathrm{\Delta }v-v+w,x\in \mathrm{\Omega },t>0,\hfill \\ w_t+\mathbf{u}·\mathrm{\nabla }w=\mathrm{\Delta }w-w+n,x\in \mathrm{\Omega },t>0,\hfill \\ {\mathbf{u}}_t+\kappa (\mathbf{u}·\mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }P=\mathrm{\Delta }\mathbf{u}+n\mathrm{\nabla }\varphi ,x\in \mathrm{\Omega },t>0,\hfill \\ \mathrm{\nabla }·\mathbf{u}=0,x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)(\ast )\end{array}$$ which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid, where \(\kappa \in \mathbb {R}\) $\kappa \in \mathbb{R}$ , the gravitational potential \(\phi \in W^{2,\infty }(\Omega )\) $\varphi \in W^{2,\infty }(\mathrm{\Omega })$ , and \(\mathcal {S}(n)\) $\mathcal{S}(n)$ satisfies \(\begin{aligned} |\mathcal {S}(n)|\le C_\mathcal {S}(1+n)^{-\alpha } \quad \text{ for } \text{ all }~~ n\ge 0,~~C_\mathcal {S}>0~~\text{ and }~~\alpha >-1. \end{aligned}\) $$\begin{array}{c}\hfill |\mathcal{S}(n)|\le C_{\mathcal{S}}{(1+n)}^{-\alpha }\text{for}\text{all}n\ge 0,C_{\mathcal{S}}>0\text{and}\alpha >-1.\end{array}$$ Under the boundary conditions \(\begin{aligned} (\nabla n-n\mathcal {S}(n)\nabla v)\cdot \nu =\partial _\nu v=\partial _\nu w=0, \textbf{u}=0, \quad x\in \partial \Omega , t>0, \end{aligned}\) $$\begin{array}{c}\hfill (\mathrm{\nabla }n-n\mathcal{S}(n)\mathrm{\nabla }v)·\nu ={\partial }_{\nu }v={\partial }_{\nu }w=0,\mathbf{u}=0,x\in \partial \mathrm{\Omega },t>0,\end{array}$$ it is shown in this paper that suitable regularity assumptions on the initial data entail the following: (i) If \(\alpha >-1\) $\alpha >-1$ and \(\kappa =0\) $\kappa =0$ , then the simplified chemotaxis-Stokes system possesses a unique global classical solution which is bounded. (ii) If \(\alpha \ge 0\) $\alpha \ge 0$ and \(\kappa \in \mathbb {R}\) $\kappa \in \mathbb{R}$ , then the full chemotaxis-Navier–Stokes system admits a unique global classical solution.