Boundedness and stabilization of a three-dimensional parabolic-elliptic Keller-Segel-Stokes system

This paper is concerned with the volume-filling effect on global solvability and stabilization in a parabolic-elliptic Keller-Segel-Stokes systems

$\begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(n)\nabla c),\quad x\in \Omega, t>0,\\ u\cdot\nabla c = \Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P = \Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} \;\;\;\;\;\;\;\;\;\;\;\;(KSF)$

with no-flux boundary conditions for $ n $ and $ c $ as well as no-slip boundary condition for $ u $ in a bounded domain $ \Omega \subseteq \mathbb{R}^3 $ with smooth boundary. Here the nonnegative function $ S\in C^2(\bar{\Omega}) $ denotes the chemotactic sensitivity which fulfills

$ |S(n)|\leq C_S(1 + n)^{-\alpha} \; \; \; \; \text{for all}\; \; n\geq0 $

with some $ C_S > 0 $ and $ \alpha> 0 $. Imposing no restriction on the size of the initial data, by seeking some new functionals and using the bootstrap arguments on the system, we establish the existence and boundedness of global classical solutions to parabolic-elliptic Keller-Segel-Stokes system under the assumption $ \alpha> \frac{1}{2} $. On the basis of this, we further prove that if the chemotactic coefficient $ C_S $ is appropriately small, the obtained solutions are shown to approach the spatially homogeneous steady state $ (\bar{n}_0, \bar{n}_0, 0) $ in the large time limit, where $ \bar{n}_0 = \frac{1}{|\Omega|}\int_{\Omega}n_0 $, provided that merely $ n_0\not \equiv0 $ on $ \Omega $.