An optimal result for global existence and boundedness in a three-dimensional Keller-Segel(-Navier)-Stokes system (involving a tensor-valued sensitivity with saturation)

The coupled quasilinear Keller-Segel-Navier-Stokes system $$\{\begin{array}{l}{n}_t+u\cdot \mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }\cdot (nS(x,n,c)\mathrm{\nabla }c),x\in \mathrm{\Omega },t>0,c_t+u\cdot \mathrm{\nabla }c=\mathrm{\Delta }c-c+n,x\in \mathrm{\Omega },t>0,u_t+\kappa (u\cdot \mathrm{\nabla })u+\mathrm{\nabla }P=\mathrm{\Delta }u+n\mathrm{\nabla }\varphi ,x\in \mathrm{\Omega },t>0,\mathrm{\nabla }\cdot u=0,x\in \mathrm{\Omega },t>0\end{array}\text{eqno}(KSNF)$$ is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ with smooth boundary, where $\kappa\in $\mathbb{R}$$isgivenconstant,$\phi\in W^{1,\infty}(\Omega),m>0$,$|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$andtheparameter$\alpha\geq0$. %For any small $\mu >0$, If $\alpha >\frac{1}{3}$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution. This result improves the result of Wang (Math. Models Methods Appl. Sci., 27(14):2745–2780, 2017), where the global {\bf very weak} solution for system $(KSNF)$ is obtained. Moreover, if $\kappa =0$ and $S(x,n,c)=C_S(1+n{)}^{-\alpha }$, then the system $(KSF)$ exists at least one global classical solution which is bounded in $\mathrm{\Omega }\times (0,\mathrm{\infty })$. These results significantly improve or extend previous results of several authors. In comparison to the result for the corresponding fluid-free system, the {\bf optimal condition} on the parameter $\alpha$ for global (weak) existence and boundedness is obtained. Our proofs rely on Maximal Sobolev regularity techniques and on a variant of the natural gradient-like energy functional.