An optimal result for global classical and bounded solutions in a two-dimensional Keller-Segel-Navier-Stokes system with sensitivity

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0, \end{array}\right.$$ which describes chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells, where $\kappa\in \mathbb{R},\phi\in W^{2,\infty}(\Omega)$ and $S$ is a given function with values in $\mathbb{R}^{2\times2}$ which fulfills $$|S(x,n,c)| \leq C_S (1 + n)^{-\alpha}$$ with some $C _S > 0$ and $\alpha \geq 0.$ If $\alpha>0$ and $\Omega\subseteq \mathbb{R}^2$ is a {\bf bounded} domain with smooth boundary, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a global classical solution which is bounded on $\Omega\times(0,\infty)$. This extends a recent result by Wang-Winkler-Xiang (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. XVIII, (2018), 2036--2145) which asserts global existence of bounded solutions under the constraint $\Omega\subseteq \mathbb{R}^2$ is a bounded {\bf convex domain} with smooth boundary. Moreover, we shall improve the result of Wang-Xiang (J. Diff. Eqns., 259(2015), 7578--7609), who proved the possibility of global and bounded, in the case that ${\bf\kappa\equiv0}$ and $\alpha>0$. In comparison to the result for the corresponding fluid-free system, the {\bf optimal condition} on the parameter $\alpha$ for both {\bf global existence} and {\bf boundedness} are obtained.