Global weak solutions in a self-consistent chemotaxis-fluid system with prescribed signal concentration on the boundary

In this article, we investigate an incompressible chemotaxis-Stokes system with nonlinear diffusion and rotational flux $ \begin{equation*} \begin{cases} n_t+{\bf u}\cdot\nabla n =\Delta n^m - \nabla \cdot (n S(x,n,c)\cdot\nabla c)+\nabla \cdot(n\nabla \phi),\quad &x\in \Omega,\, t>0,\\ c_t +{\bf u}\cdot\nabla c=\Delta c -nc,\quad &x\in \Omega,\, t>0,\\ {\bf u}_t+\nabla P = \Delta {\bf u}-n\nabla \phi+nS(x,n,c)\cdot\nabla c,\quad\nabla\cdot{\bf u}=0,\quad &x\in \Omega,\, t>0 \end{cases} \end{equation*} $ in a bounded domain $ \Omega \!\!\subset\!\! \mathbb{R}^3 $ with smooth boundary $ \partial \Omega $. The corresponding boundary conditions satisfy$ \begin{equation*} (\nabla n^m-nS(x, n, c)\cdot \nabla c+n\nabla \phi)\cdot \nu = 0, \, c = c_{*}(x, t) , \, {\bf u} = {\bf 0}, \quad x\in\partial\Omega, \, t>0, \end{equation*} $with $ m>1 $ and a given nonnegative function $ c_{*}(x, t)\in C^{2, 1}(\overline{\Omega}\times(0, \infty)) $. The chemotatic sensitivity $ S $ is a given tensor-valued function fulfilling $ |S(x, n, c)| < S_0(c) $ for all $ \overline{\Omega}\times[0, \infty)^2 $ with $ S_0(c) $ nondecreasing on $ [0, \infty) $. It is shown that the corresponding initial-boundary problem possesses at least one global bounded weak solution when $ m>\frac{7}{6} $. In the homogeneous Dirichlet signal boundary condition (i.e., $ c_*(x, t)\equiv 0 $) case, we further prove that the solutions will stabilize to the mass-preserving spatial equilibrium $ (\overline{n}_0, 0, {\bf 0}) $, where $ \overline{n}_0: = \frac{1}{|\Omega|} \int_\Omega n_0(x)dx $.