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{nt+u⋅∇n=Δnm−∇⋅(nχ(c)∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nf(c),x∈Ω,t>0,ut+κ(u⋅∇)u=Δu+∇P+n∇Φ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, in a bounded convex domain Ω⊂R3Ω⊂R3. Here κ∈Rκ∈R, Φ∈W1,∞(Ω)Φ∈W1,∞(Ω), 0<χ∈C2([0,∞))0<χ∈C2([0,∞)) and 0≤f∈C1([0,∞))0≤f∈C1([0,∞)) with f(0)=0f(0)=0. It is proved that under appropriate structural assumptions on f and χ  , for any choice of m≥23 and all sufficiently smooth initial data (n0,c0,u0)(n0,c0,u0) the model possesses at least one global weak solution.