A new result for global solvability of a Keller-Segel-Navier-Stokes system with nonlinear diffusion and matrix-valued sensitivities in three dimensions

In this paper, we are concerned with the Keller-Segel-Navier-Stokes system with nonlinear diffusion and matrix-valued sensitivities as(KSNF)$\{\begin{array}{c}{n}_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot \left(nS\right(x,n,c\left)\nabla c\right),x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+n,x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+n\nabla \varphi ,x\in \Omega ,t>0,\\ \nabla \cdot u=0,x\in \Omega ,t>0,\end{array}$ where $\Omega \subseteq R^3$ is a bounded domain with smooth boundary, $m>0,\kappa \in R$ are two given constants, $\varphi \in W^{2,\infty }\left(\Omega \right)$ is a given function, and the chemotactic sensitivity S is a given matrix-valued function on $\Omega \times {[0,\infty )}^2$ satisfying$\left|S\right(x,n,c\left)\right|\le C_S{(1+n)}^{-\alpha }\text{with}{C}_S>0\text{and}\alpha \ge 0.$ With suitable regular nonnegative initial data, we establish the global solvability of a very weak solution to the no-flux-Dirichlet boundary value problem for ($KSNF$) provided that$m+\alpha >\frac{4}{3}.$ Comparing with the known results for the fluid-free setting of $\left(KS\right)$, the condition appears to be optimal. Moreover, if $m>\max \{\frac{5}{3}-2\alpha ,\frac{4}{3}-\alpha \}$, we establish the existence of at least one global weak solution in the standard sense.