Global solvability to a two competing species chemotaxis Navier-Stokes system with singular sensitivity

In this paper, we deal with a two-species chemotaxis Navier-Stokes system with singular sensitivity under the homogeneous Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^2 $ with smooth boundary. For any appropriately regular initial data, we prove that the system has a global classical solution when $ \chi: = \max\{\chi_1,\chi_2\}<1 $. Moreover, we showed the global boundedness of the solution for sufficiently large $ b_1 $ and $ b_2 $.