This paper deals with the following system \(\begin{aligned} \left\{ \begin{array}{llll} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(n)\nabla v),& x\in \Omega ,t>0,\\ v_t+u\cdot \nabla v=\Delta v-v+w, & x\in \Omega ,t>0,\\ w_t+u\cdot \nabla w=\Delta w-w+n, & x\in \Omega ,t>0,\\ u_t+(u\cdot \nabla ) u=\Delta u+\nabla P+n\nabla \phi ,\quad \nabla \cdot u=0, & x\in \Omega ,t>0\\ \end{array} \right. \end{aligned}\) in a smoothly bounded domain \(\Omega \subset \mathbb {R}^2\) , where \(\phi \in W^{2,\infty }(\Omega )\) and \(S\in C^{2}([0,\infty ))\) satisfies \(|S(n)|\le C_S(1+n)^{-\alpha }\) with \(\alpha \in \mathbb {R}\) and \(C_S>0\) . When \(\alpha \ge 0\) , the global boundedness (or existence) of classical solutions was observed in Dai–Liu (J Differ Equ 314:201–250, 2022), Gao (Electron Res Arch 31:1710–1736, 2023) and Zheng–Liu (Z Angew Math Phys 75(5):179, 2024). Now, it is shown that for all suitably regular initial data, the associated initial-boundary value problem possesses a global classical solution provided \(-\frac{1}{2}<\alpha <0\) .
