Global boundedness of classical solutions to a Keller-Segel-Navier-Stokes system involving saturated sensitivity and indirect signal production in two dimensions

This paper is concerned with the following Keller–Segel–Navier–Stokes system with indirect signal production and tensor-valued sensitivity:

$$\{\begin{array}{ll}{n}_t+u\cdot \mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }\cdot (nS(x,n,v,w)\mathrm{\nabla }v),& x\in \mathrm{\Omega },t>0,\\ v_t+u\cdot \mathrm{\nabla }v=\mathrm{\Delta }v-v+w,& x\in \mathrm{\Omega },t>0,\\ w_t+u\cdot \mathrm{\nabla }w=\mathrm{\Delta }w-w+n,& x\in \mathrm{\Omega },t>0,\\ u_t+\kappa (u\cdot \mathrm{\nabla })u+\mathrm{\nabla }P=\mathrm{\Delta }u+n\mathrm{\nabla }\varphi ,& x\in \mathrm{\Omega },t>0,\\ \mathrm{\nabla }\cdot u=0,& x\in \mathrm{\Omega },t>0,\end{array}(♡)$$

in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with smooth boundary, where $\kappa \in \mathbb{R}$, $\varphi \in W^{2,\mathrm{\infty }}(\mathrm{\Omega })$, and $S$ is a given function with values in ${\mathbb{R}}^{2\times 2}$ which satisfies $|S(x,v,w,u)|\le C_S(n+1{)}^{-\alpha }$ with $C_S>0$. If $\alpha >0$, then for any sufficiently smooth initial data, there exists a globally classical solution which is bounded for the corresponding initial-boundary value problem of system (♡).