This paper deals with the singular chemotaxis-Navier-Stokes system with indirect signal consumption $n_{t}+u\cdot\nabla n=\Delta n-\chi\nabla\cdot(\frac{n}{v}\nabla v)$; $v_{t}+u\cdot\nabla v=\Delta v-vw$; $w_{t}+u\cdot\nabla w=\Delta w-w+n$; $u_t+(u\cdot\nabla)u=\Delta u-\nabla P+n\nabla\Phi$; $\nabla\cdot u=0$, $x\in \Omega$, $t>0$ in a bounded and smooth domain $\Omega\subset\mathbb{R}^2$ with no-flux/no-flux/no-flux/no-slip boundary conditions, where $\Phi\in W^{2,\infty}(\Omega)$. A recent literature [Dai F, Liu B. J Differential Equations, 2023, 369: 115--155] has proved that for all reasonably regular initial data, the associated initial-boundary value problem possesses a global classical solution, but qualitative information on the behavior of solution has never been touched so far. In stark contrast to the positive effect of indirect signal consumption mechanism on the global solvability of system, the analysis of asymptotic behavior of solution to the system with indirect signal consumption is essentially complicated than that with direct signal consumption because the favorable coupled structure between cells and signal is broken down by the indirect signal consumption mechanism. The present study shows that the global classical solution exponentially stabilizes toward the corresponding spatially homogeneous equilibria under a smallness condition on the initial cell mass. In comparison to the previously known result concerning the uniform convergence of solution to the system with direct signal consumption, our result inter alia provides a more in-depth understanding on the asymptotic behavior of solution.
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