Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

Abstract: The coupled chemotaxis fluid system \begin{equation} \left{ \begin{array}{llc} \displaystyle n_t=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in \Omega\times (0,T),\ c_t=\Delta c-nc-u\cdot\nabla c , &(x,t)\in\Omega\times (0,T),\ u_t=\Delta u-\kappa(u\cdot\nabla)u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\ \nabla\cdot u=0,&(x,t)\in\Omega\times (0,T), \end{array} \right.(\star) \end{equation} is considered under the no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $\Omega\subset\mathbb{R}^N$ ($N=2,3$), $\kappa=0,1$. We assume that $S(x,n,c)$ is a matrix-valued sensitivity under a mild assumption such that $|S(x,n,c)|<S_0(c_0)$ with some non-decreasing function $S_0\in C^2((0,\infty))$. It contrasts the related scalar sensitivity case that $(\star)$ does not possess the natural {\em gradient-like} functional structure. Associated estimates based on the natural functional seem no longer available. In the present work, a global classical solution is constructed under a smallness assumption on $|c_0|_{L^\infty(\Omega)}$ and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization.