Global weak solutions in a parabolic–elliptic–elliptic Keller–Segel–Stokes system with nonlinear diffusion and indirect signal production

In this paper, we consider the parabolic–elliptic–elliptic Keller–Segel–Stokes system with nonlinear diffusion and indirect signal production nt+u⋅∇n=Δnm−∇⋅(nS(n)∇v);u⋅∇v=Δv−v+w;u⋅∇w=Δw−w+n;ut=Δu−∇P+n∇ϕ;∇⋅u=0 in a bounded domain Ω⊂R3 with smooth boundary, where ϕ ∈ W2,∞(Ω) and m > 0, and the chemotactic sensitivity function S∈C2(0,∞) satisfies 0 ≤ S(ξ) ≤ CS(ξ + 1)−α for all ξ ≥ 0 with some CS > 0 and α∈R. It is proved that under the structural assumption m+α>1 and proper regularity assumptions on the initial data, the associated initial-boundary value problem possesses at least one global bounded weak solution. Our result not only accommodates both porous-medium-type diffusion and fast (singular) diffusion, but also covers saturated sensitivity (α > 0) and stronger chemotactic sensitivity (α ≤ 0). To the best of our knowledge, this is the first result on the existence of global bounded weak solutions in a parabolic–elliptic–elliptic Keller–Segel–Stokes system of type (⋆) and covers the result of linear diffusion under the condition that α > 0.