Global existence and boundedness in a Keller-Segel-(Navier-)Stokes system with signal-dependent sensitivity

In this paper, we consider the following Keller-Segel(-Navier)-Stokes system {n(t) + u . del n = Delta n - del . (n chi(c)del c), x is an element of Omega, t > 0, c(t) + u . del c = Delta c - c + n, x is an element of Omega, t > 0, u(t) + k(u .del)u = Delta u + del P + n del phi, x is an element of Omega, t > 0, (star) del . u = 0, x is an element of Omega, t > 0, where Omega subset of R-N (N = 2,3) is a bounded domain with smooth boundary partial derivative Omega, kappa is an element of R and chi(c) is assumed to generalize the prototype chi(c) = chi(0)/(1 + mu c)(2), c >= 0. It is proved that i) for is kappa not equal 0 and N = 2 or kappa = 0 and N is an element of {2, 3}, the corresponding initial boundary problem admits a unique global classical solution which is bounded; ii) for is kappa not equal 0 and N = 3, the corresponding initial boundary problem possesses at least one global weak solution. (C) 2016 Elsevier Inc. All rights reserved.