Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation

This paper is concerned with the following Keller-Segel-Navier-Stokes system(⋆){nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, where Ω⊂R3 is a bounded domain with smooth boundary ∂Ω,κ∈R and S denotes a given tensor-valued function fulfilling|S(x,n,c)|≤CS(1+n)α with some CS>0 and α>0. As the case κ=0 has been considered in [25], it is shown in the present paper that the corresponding initial-boundary problem with κ≠0 admits at least one global weak solution if α≥37.