Global boundedness in a 3D quasilinear Keller-Segel-Stokes system with nonlinear sensitivity and indirect signal production

Abstract

This paper is concerned with the Keller-Segel-Stokes system{nt+u⋅∇n=Δnm−∇⋅(n(1+n)−α∇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 m>0 and α∈R. It is shown that for all suitably regular initial data an associated initial-boundary value problem (⁎) possesses at least one global boundedness weak solution provided m+α>109. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if m+2α>2 and m>34 in [24] or m>43 and α=0 in [61] or m=1 and α>13 in [51]. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the three-dimensional Keller-Segel-Stokes system.