This paper deals with the two-competing-species chemotaxis–(Navier)–Stokes system with indirect signal consumption, as given by \(\begin{aligned} \left\{ \begin{array}{llll} \left( n_1 \right) _t+\textbf{u}\cdot \nabla n_1=d_1\Delta n_1-\chi _1\nabla \cdot \left( n_1\nabla c \right) +\mu _1n_1\left( 1-n_1-a_1n_2 \right) ,& in~~\Omega \times \left( 0,\infty \right) , \\ \left( n_2 \right) _t+\textbf{u}\cdot \nabla n_2=d_2\Delta n_2-\chi _2\nabla \cdot \left( n_2\nabla c \right) +\mu _2n_2\left( 1-a_2n_1-n_2 \right) ,& in~~\Omega \times \left( 0,\infty \right) , \\ c_t+\textbf{u}\cdot \nabla c=d_3\Delta c-\alpha _1cv,& in~~\Omega \times \left( 0,\infty \right) , \\ v_t+\textbf{u}\cdot \nabla v=d_4\Delta v-\alpha _2v+\alpha _3n_1+\alpha _4n_2,& in~~\Omega \times \left( 0,\infty \right) , \\ \textbf{u}_t+\kappa \left( \textbf{u}\cdot \nabla \right) \textbf{u}=\Delta \textbf{u}+\nabla P+\left( \beta _1n_1+\beta _2n_2 \right) \nabla \phi ,& in~~\Omega \times \left( 0,\infty \right) , \\ \nabla \cdot \textbf{u}=0, & in~~\Omega \times \left( 0,\infty \right) , \end{array} \right. \end{aligned}\) under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^N\) , \(N=2,3\) , with the nonnegative initial data \(\left( n_{1,0}, n_{2,0}, c_0, v_0, \textbf{u}_0 \right) \) , and the parameters \(d_i, \alpha _i\left( i = 1, 2, 3, 4 \right) \) and \(\chi _j, \mu _j, a_j, \beta _j\left( j = 1, 2 \right) \) are positive. It is proved that, in the two-dimensional case with \(\kappa =1\) , the system possesses a globally bounded classical solution based on the standard heat semigroup argument. Furthermore, by leveraging the maximal Sobolev regularity, we demonstrate the existence of a globally bounded classical solution in three dimensions for \(\kappa =0\) , provided that there exists a positive constant \(\gamma \) such that \(\frac{\max \{\chi _1,\chi _2\}}{\min \{\mu _1,\mu _2\}}<\gamma \) . Additionally, through the utilization of energy functionals and comparison arguments, we reveal that the globally bounded solution converges to distinct constant steady states, contingent upon the values of \(a_1\) and \(a_2\) , and the explicit convergence rates for these global solutions are provided.
