Boundedness in a chemotaxis-fluid system involving a saturated sensitivity and indirect signal production mechanism

Abstract

The Keller-Segel-Stokes type system with indirect signal production, as given by{nt+u⋅∇n=Δn−∇⋅(nS(n)∇v),x∈Ω,t>0,vt+u⋅∇v=Δv−v+w,x∈Ω,t>0,wt+u⋅∇w=Δw−w+n,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,∇⋅u=0,x∈Ω,t>0, is considered in a bounded domain Ω⊂RN, N∈{2,3}, with smooth boundary, where ϕ∈C2(Ω‾) and S∈C2([0,∞)).Under the assumption that there exist CS>0 and α≥0 such that|S(n)|≤CS(n+1)−αfor all n≥0, it is shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined bounded classical solution in the case N=2. In the case of N=3, the conclusion is also true provided α>19.