Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

The coupled quasilinear Keller-Segel-Navier-Stokes system(KSNF){nt+u⋅∇n=Δnm−∇⋅(n∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+κ(u⋅∇)u+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 is considered under Neumann boundary conditions for n and c and no-slip boundary conditions for u in three-dimensional bounded domains Ω⊆R3 with smooth boundary, where m>0, κ∈R are given constants, ϕ∈W1,∞(Ω). If m>2, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (KSNF) possesses a globally defined weak solution.