Global existence and regularity in a three-dimensional Keller-Segel-Navier-Stokes system with indirect signal production

We consider a degenerate quasilinear Keller-Segel-Navier-Stokes system with indirect signal production$\begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n^m-\nabla\cdot( nS(x,n,v)\nabla v),\quad x\in \Omega, t>0,\\ { }{ v_{ t}+u\cdot\nabla v = \Delta v-v+w},\quad x\in \Omega, t>0,\\ { }{w_{t}+u\cdot\nabla w = \Delta w-w+n},\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P = \Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} \quad\quad\quad (KSNF)$in a bounded domain $ \Omega\subseteq \mathbb{R}^3 $ with a smooth boundary, where $ m>0,\kappa\in \mathbb{R} $ is a given constant, $ \phi\in W^{2,\infty}(\Omega) $. Here, $ S(x,n, v) $ is a given parameter matrix on $ \Omega\times[0,\infty)^2 $ whose Frobenius norm satisfies $ |S(x,n,v)|\leq C_S(1+n)^{-\alpha} $ with $ C_S>0 $ and $ \alpha\geq0 $. It is shown that whenever $ m +\alpha> \frac{5}{4} $, for any nonnegative initial data which is sufficiently smooth, the corresponding Neumann-Neumann-Neumann-Dirichlet initial-boundary problem possesses at least one globally weak solution, and that this solution is uniformly bounded with respect to the norm in $ L^1(\Omega)\times L^2(\Omega)\times L^2(\Omega)\times L^2(\Omega; \mathbb{R}^3) $. This work improves previous results of several other authors (see Remark 1.1).