A note to the large-time behavior of a 3D chemotaxis-Navier-Stokes system with porous medium slow diffusion

In this note, we consider the large time behavior of the following chemotaxis-Navier-Stokes system$ \begin{equation*} \label{AH4} \left\{ \begin{split} n_t + {\bf u}\cdot\nabla n & = \Delta n^m - \nabla \cdot\big(n\nabla c\big), & \qquad x\in\Omega ,\, t>0,\\ c_t + {\bf u}\cdot\nabla c & = \Delta c - nc, &\qquad x\in\Omega ,\, t>0,\\ {\bf u}_t + {\bf u}\cdot\nabla {\bf u} & = \Delta{\bf u} +\nabla P + n\nabla \phi, &\qquad x\in\Omega ,\, t>0,\\ \nabla \cdot{\bf u} & = 0, &\qquad x\in\Omega ,\, t>0 \end{split} \right. \end{equation*} $with $ m>1 $ in spatially three-dimensional setting. The global weak solution $ (n, c, {\bf u}) $ to the no-flux/no-flux/no-slip initial-boundary value problem has been constructed by Zhang and Li (J. Differential Equations, 2015). Here, we will show that such a weak solution will stabilize to the constant equilibrium $ (\overline{{n}_0}, 0, {\bf 0}) $ with $ \overline{{n}_0} = \frac{1}{|\Omega |}\int_\Omega n_0 $ as $ t\rightarrow \infty $.