Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $ \quad n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nχ(c)\nabla c), $ $ \quad c_t+u\cdot\nabla c=Δc-nf(c), $ $ \quad u_t+κ(u\cdot\nabla)u=Δu+\nabla P+n\nablaΦ, $ $ \quad \nabla\cdot u=0, $ in a bounded convex domain $Ω\subset\mathbb{R}^3$. It is proved that if $m\geq\frac{2}{3}$, $κ\in\mathbb{R}$, $0