Boundedness and decay property in a three-dimensional Keller–Segel–Stokes system involving tensor-valued sensitivity with saturation

In this paper, we consider the following Keller–Segel–Stokes system{nt+u⋅∇n=∇⋅(D(n)∇n)−∇⋅(nS(x,n,c)∇c)+ξn−μn2,ct+u⋅∇c=Δc−c+n,ut+∇P=Δu+n∇ϕ,∇⋅u=0 in a bounded domain Ω⊂R3Ω⊂R3 with smooth boundary, where ϕ∈W1,∞(Ω)ϕ∈W1,∞(Ω), D   is a given function satisfying D(n)≥CDnm−1D(n)≥CDnm−1 for all n>0n>0 with certain CD>0CD>0, and S   is a given function with values in R3×3R3×3 which fulfills |S(x,n,c)|≤CS(1+n)−α|S(x,n,c)|≤CS(1+n)−α with some CS>0CS>0 and α>0α>0. It is proved that under the conditions m≥13 andα>65−m, and proper regularity hypotheses on the initial data, the corresponding initial–boundary problem possesses at least one global bounded weak solution. In addition, it is shown that if ξ=0ξ=0 then all solution components satisfyn(⋅,t)⇀⁎0, c(⋅,t)→0 and u(⋅,t)→0 in L∞(Ω) as t→∞t→∞.