In this paper, we consider the chemotaxis-fluid model with singular sensitivity and indirect signal production \(\begin{aligned} \left\{ \begin{array}{lll} n_{t}+u\cdot \nabla n=\Delta n-\chi \nabla \cdot (\frac{n}{c}\nabla c)+rn-\mu n^{2},& \quad x\in \Omega ,t>0,\\ c_{t}+u\cdot \nabla c=\Delta c-c+v,& \quad x\in \Omega ,t>0,\\ v_{t}+u\cdot \nabla v=\Delta v-v+n,& \quad x\in \Omega ,t>0,\\ u_{t}+k(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \phi , ~~\nabla \cdot u=0, & \quad x\in \Omega ,t>0,\\ \end{array} \right. \end{aligned}\) which is considered under non-flux boundary conditions for n,c and v, along with the non-slip boundary condition for u within a bounded domain \(\Omega \subset \mathbb {R}^{N}(N = 2,3)\) possessing a smooth boundary. Here, \(\phi \in W^{1,\infty }\) , \(k\in \{0,1\}\) and \(\chi \) , \(\mu \) , \(r>0\) are given constants. Through the application of heat semigroup theory combined with coupled nonlinear estimation techniques, we rigorously establish the existence of a global classical bounded solution for the system with \(k=1,N=2\) and \(k=0,N=3\) , contingent upon the chemotaxis sensitivity parameter \(\chi \) fulfilling the criterion that \(\begin{aligned} 0<\chi <2(r+1)+\sqrt{3r^{2}+8r+4}, r>0. \end{aligned}\) In the special case where \(u = 0\) , Xing et al. (Z. Angew. Math. Phys. 72:105, 2021) established the existence of unique global solutions for the mentioned system in two dimensions. The present work extends these results by proving the global existence and boundedness of classical solutions to the system with the special case of \(u = 0\) in two-dimensional and three-dimensional spaces.
