Roundedness 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 . del n = del . (D(n)del n) - del . (nS(x,n,c)del c) + xi n - mu n(2), c(t) + u . del c = Delta c -c + n, u(t) +del P = Delta u + n del phi, del . u = 0 in a bounded domain Omega subset of R-3 with smooth boundary, where phi is an element of W-1,W- infinity(Omega), D is a given function satisfying D(n) >= C(D)n(m-1) for all n > 0 with certain C-D > 0, and S is a given function with values in R-3x3 which fulfills vertical bar S(x, n, c)vertical bar <= C-S(1 + n)(-alpha) with some C-S > 0 and alpha > 0. It is proved that under the conditions m >= 1/3 and alpha > 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 xi = 0 then all solution components satisfy n(., t) ->*0, c(., t) -> 0 and u(., t) -> 0 in L-infinity(Omega) as t -> infinity. (C) 2016 Elsevier Inc. All rights reserved.