Global existence of weak solutions for 2D chemotaxis-Navier-Stokes system with fractional diffusion

In this paper, we consider the following 2D incompressible chemotaxis-Navier-Stokes equations with the fractional diffusion { ∂ t n + u · ∇ n − ∆ n = − ∇ · ( n ∇ c )+ n ( 1 − n )( n − a ) , ∂ t c + u · ∇ c − ∆ c = − cn , ∂ t u + u · ∇ u + ∧ 2 α u + ∇ P = − n ∇ ϕ , where ∧ : = ( − ∆ ) 1 2 and α ∈ 1 2 , 1 . By establishing the new priori estimates, we get the global well-posedness for the above system with large initial data.