Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion

This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nχ(c)∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nf(c),x∈Ω,t>0,ut+(u⋅∇)u=Δu+∇P+n∇Φ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0under homogeneous boundary conditions of Neumann type for n and c, and of Dirichlet type for u in a bounded convex domain Ω⊂R3 with smooth boundary. Here, Φ∈W1,∞(Ω), 0<χ∈C2([0,∞)) and 0≤f∈C1([0,∞)) with f(0)=0. It is proved that if p>3215 and under appropriate structural assumptions on f and χ, for all sufficiently smooth initial data (n0,c0,u0) the model possesses at least one global weak solution.