Global existence and boundedness of classical solutions in chemotaxis-(Navier-)Stokes system with singular sensitivity and self-consistent term

This paper addresses the global existence and boundedness of classical solutions to the Neumann-Neumann-Dirichlet value problem for the chemotaxis system, as described by (*)$\left(\begin{array}{cc}{n}_t+u\cdot \nabla n=\Delta n-\chi \nabla \cdot \left(\frac{n}{1+c}\nabla c\right)+\nabla \cdot (n\nabla \varphi ),& x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-n^{\alpha }c,& x\in \Omega ,t>0,\\ u_t+\nabla P=\Delta u-n\nabla \varphi +\frac{n}{1+c}\nabla c,& x\in \Omega ,t>0,\\ \nabla \cdot u=0,& x\in \Omega ,t>0\\ \frac{\partial n}{\partial \nu }=\frac{\partial c}{\partial \nu }=0,u=0,& x\in \partial \Omega ,t>0\\ n(x,0)=n_0\left(x\right),c(x,0)=c_0\left(x\right),u(x,0)=u_0\left(x\right),& x\in \Omega \end{array}\right)$in a smoothly bounded domain $\Omega \subset R^N(N=2,3)$, where $\alpha >0$ and the gravitational potential function $\varphi \in W^{1,\infty }\left(\Omega \right)$. It is confirmed that for any selection of $\chi$ satisfying $0<\chi \le \frac{\sqrt{k^2{\delta }_1^2{ln}^2(1+{\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)})+k{(k-1)}^2({\pi }^2+{ln}^2(1+{\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)}))}-k{\delta }_{1}ln(1+{\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)})}{k(k-1)ln(1+{\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)})}$with $k\to \frac{N}{2}$ and ${\delta }_1\to k-1$, then the corresponding problem (*) admits a globally and uniformly bounded classical solution via an approach to introduce a trigonometric-type weight function.