Boundedness in a chemotaxis-fluid system involving a gradient-dependent flux limitation and indirect signal production mechanism

Abstract

This paper is concerned with the Keller-Segel-Stokes system(⁎){nt+u⋅∇n=Δn−∇⋅(nf(|∇v|2)∇v),vt+u⋅∇v=Δv−v+w,wt+u⋅∇w=Δw−w+n,ut=Δu+∇P+n∇ϕ,∇⋅u=0, under no-flux/no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded domains Ω⊂R3, with given suitably regular functions f and ϕ, as well as f satisfies f(|∇v|2)≤Kf(1+|∇v|2)−α2. It is shown that for all suitably regular initial data an associated initial-boundary value problem (⁎) possesses a globally defined bounded classical solution provided α>14. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if α>12 in [50]. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the three-dimensional Keller-Segel-Stokes system.