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

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} \displaystyle n_t=Δn-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in Ω\times (0,T),\\ c_t=Δc-nc-u\cdot\nabla c , &(x,t)\inΩ\times (0,T),\\ u_t=Δu-κ(u\cdot\nabla)u+\nabla P+n\nablaϕ, &(x,t)\inΩ\times (0,T),\\ \nabla\cdot u=0,&(x,t)\inΩ\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 $Ω\subset\mathbb{R}^N$ ($N=2,3$), $κ=0,1$. 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 $\|c_0\|_{L^\infty(Ω)}$ and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization.