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{n_t+u.∇n=∇.(D(n)∇n)-∇.(nS(x,n,c)∇c)+ξn-μn²,c_t+u.∇c=δc-c+n,u_t+∇P=δu+n∇φ,∇.u=0 in a bounded domain Ω⊂R³ with smooth boundary, where φ∈W^1,∞(Ω), D is a given function satisfying D(n)≥C_Dn^m-1 for all n>0 with certain C_D>0, and S is a given function with values in R^3×3 which fulfills |S(x, n, c)|≤C_S(1+n)^-α with some C_S>0 and α>0. It is proved that under the conditions m≥1/3 and α>6/5-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=0 then all solution components satisfy n(.,t)⇀*0, c(.,t)→0 and u(.,t)→0 in L∞(Ω) as t→∞.