We study the following chemotaxis-Navier–Stokes system with general sensitivity and nonlinear production ⋆ \(\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nf(n)\nabla c), \\&c_t+u\cdot \nabla c=\Delta c-c+g(n),\\&u_t+(u\cdot \nabla )u+\nabla P=\Delta u+n \nabla \phi , ~ \nabla \cdot u=0 \end{aligned} \right. \end{aligned}\) in a bounded domain \(\Omega \subset \mathbb {R}^2\) , where the chemotaxis sensitivity function \(f\in C^2([0,\infty ))\) satisfies that \(|f(s)|\le K_f(1+s)^{-\alpha }\) for all \(s\ge 0\) with \(K_f>0\) and \(\alpha \in \mathbb {R}\) , and the signal production function \(g\in C^1([0,\infty ))\) is such that \(0\le g(s)\le K_g s(1+s)^{\beta -1}\) for all \(s\ge 0\) with \(K_g,\beta >0\) . It is shown in this paper that for all reasonably regular initial data, the corresponding initial-boundary value problem of ( \(\star \) ) possesses a unique globally bounded classical solution if \(\alpha >\frac{1}{2}[(2\beta -1)_+-1]\) for \(0<\beta <1\) , or if \(\alpha >\beta -1\) for \(\beta \ge 1\) . Our work is one of the few explorations involving chemotaxis-fluid models with nonlinear production mechanisms and greatly extends the global solvability result obtained in Black (Nonlinear Anal Real World Appl 31:593–609, 2016) only for the chemotaxis-Stokes variant of ( \(\star \) ) with \(\alpha =0\) and the sublinear signal production of \(\beta \in (0,1)\) .
