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We consider the spatially $3$-D version of the followingKeller-Segel-Navier-Stokes system with rotational flux$$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Deltan-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nablac=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nablaP=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in\Omega, t>0 \end{array}\right.\qquad(*)$$ under no-flux boundary conditions ina bounded domain $\Omega\subseteq \mathbb{R}^{3}$ with smooth boundary, where$\phi\in W^{2,\infty} (\Omega)$ and $\kappa\in \mathbb{R}$ represent theprescribed gravitational potential and the strength of nonlinear fluidconvection, respectively. Here the matrix-valued function $S(x,n,c)\inC^2(\bar{\Omega}\times[0,\infty)^2 ;\mathbb{R}^{3\times 3})$ denotes therotational effect which satisfies $|S(x,n,c)|\leq C_S(1 + n)^{-\alpha}$ withsome $C_S > 0$ and $\alpha\geq 0$. In this paper, by seeking some newfunctionals and using the bootstrap arguments on system $(*)$, we establish theexistence of global weak solutions to system $(*)$ for arbitrarily largeinitial data under the assumption $\alpha\geq1$. Moreover, under an explicitcondition on the size of $C_S$ relative to $C_N$, we can secondly prove that infact any such {\bf weak} solution $(n,c,u)$ becomes smooth ultimately, and thatit approaches the unique spatially homogeneous steady state$(\bar{n}_0,\bar{n}_0,0)$, where $\bar{n}_0=\frac{1}{|\Omega|}\int_{\Omega}n_0$and $C_N$ is the best Poincar\'{e} constant. To the best of our knowledge,there are the first results on asymptotic behavior of the system.

Global existence, smooth and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux