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 {n_t+u⋅∇n=Δn−∇⋅(nS(x,n,c)∇c),x∈Ω, t>0,c_t+u⋅∇c=Δc−c+n,x∈Ω, t>0,u_t+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 C_S>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.