Small-density solutions to Keller-Segel-Navier-Stokes system with rapidly decaying diffusivities

In the present work, chemotaxis-Naiver-Stokes system with rapidly decaying diffusivities \begin{document}$ \begin{eqnarray*} { \bf{(KSNS)} }\ \ \left\{ \begin{array}{llll} n_{t}+u\cdot\nabla n = \nabla\cdot(D(n)\nabla n)-\nabla\cdot(S(n)\nabla c), &&x\in\Omega, \, t>0, \\ c_{t}+u\cdot\nabla c \Delta - c+ n, u_{t}+\kappa(u\cdot\nabla)u u +\nabla P+ n\nabla\Phi, \nabla\cdot 0, t>0\ \end{array} \right. \end{eqnarray*} $\end{document} is considered in bounded domain $ \Omega\subseteq \mathbb{R}^d ($ d\in\{2, 3\} $) smooth boundary. Let D, S\in C^2([0, \infty)) be such that S(0) 0 and D(s)\geq\eta>0\ \text{in}\ [0, R]\ \text{with}\ R>0. Here, we aim at proving for all K>0 there exist \epsilon_1(K)\in(0, \frac{R}{2}) \epsilon_2(K)>0 if initial data 0<n_0, c_0\in W^{1, \infty}(\Omega)\times \infty}(\Omega) as well u_0\in (W^{1, \infty}(\Omega))^d satisfy \|n_0\|_{L^\infty(\Omega)}\leq \epsilon_1(K), \|\nabla c_0\|_{L^\infty(\Omega)}\leq K, \|u_0\|_{L^d(\Omega)}\leq \epsilon_2(K), then {\bf (KSNS)} admits a global small-density solution. As technical key point, use Moser-type iterative arguments, which transfer time-independent L^1 $-bound of density L^\infty to density, control upper-bound component $. our result, smallness condition velocity u_0 imposed L^d( \Omega) $, known critical space coupled Navier-Stokes equation.