The chemotaxis-Stokes system with flux limitation and nonlinear production \(\begin{aligned} \left\{ \begin{array}{ll} n_t=\Delta n- \nabla \cdot (n F(|\nabla c|^2)\nabla c)-u\cdot \nabla n,& (x,t)\in \Omega \times (0,T),\\ c_t=\Delta c-c+g (n)-u\cdot \nabla c,& (x,t)\in \Omega \times (0,T),\\ u_t=\Delta u+\nabla P+n\nabla \phi , \ \ \nabla \cdot u=0 ,& (x,t)\in \Omega \times (0,T) \end{array}\right. \end{aligned}\) is considered in a smoothly bounded domain \(\Omega \subset \mathbb {R}^2\) associated with Neumann conditions for n, c and Dirichlet boundary condition for u, where \(\phi \in W^{2,\infty }(\Omega )\) , \(F(s)\in C^2([0,\infty ))\) and \(g(s)\in C^1([0,\infty ))\) satisfy \(|F(s)|\le K_F (1+s)^{-\frac{\alpha }{2}}\) and \(0\le g(s)\le K_g s^{\beta }+K_g\) for all \(s\ge 0\) with \(K_F, K_g, \beta >0\) and \(\alpha \in \mathbb {R}\) . We show that the system admits a global bounded classical solution if \(\alpha >1-\frac{1}{(2\beta -1)_+}\) . Furthermore, in the critical case that \(\alpha =1-\frac{1}{(2\beta -1)_+}\) with \(\beta >\frac{1}{2}\) , we obtain global bounded solutions if the total mass of cells is small.
