This work is concerned with the global solvability analysis of the Keller-Segel-Stokes system in two-dimensional space \(\begin{aligned} \left\{ \begin{aligned}&n_t=\Delta (\gamma (c)n)+\mu n(1-n^{\sigma })-\textbf{u}\cdot \nabla n, & x\in \Omega , t>0,\\&c_t=\Delta c-c+n^{1-\alpha }c^{\alpha }-\textbf{u}\cdot \nabla c, & x\in \Omega , t>0,\\&\textbf{u}_t=\Delta \textbf{u}-\nabla P+n\nabla \phi , & x\in \Omega , t>0,\\&\nabla \cdot \textbf{u}=0, & x\in \Omega , t>0,\\ \end{aligned}\right. \end{aligned}\) where \(\Omega \subset \mathbb {R}^2\) denotes a bounded domain with smooth boundary, and the system parameters satisfy \(\sigma >0\) , and \(\alpha \in (0,1)\) . The function \(\gamma \) belongs to \( C^3([0,\infty ))\) with \(\gamma (s)>0\) for all \(s\ge 0\) , as well as the gravitational potential \(\phi \in W^{2,\infty }(\Omega )\) . Under homogeneous Neumann boundary conditions, we establish the existence of globally defined classical solutions for this chemotaxis-fluid coupled system.
