Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation

This paper is concerned with the following Keller-Segel-Navier-Stokes system { n(t) + u . del n = Delta n - del . (nS(x, n, c)del c), x epsilon Omega, t > 0, c(t) + u . del c = Delta c - c + n, x epsilon Omega, t > 0, u(t) + K(u . del)u = Delta u + del P + n del phi, x epsilon Omega, t > 0, del . u = 0, x epsilon Omega, t > 0 , where Omega subset of R-3 is a bounded domain with smooth boundary partial derivative Omega, K epsilon R and S denotes a given tensor-valued function fulfilling vertical bar S(x, n, c)vertical bar <= C-S / (1 + n)(alpha) with some C-S > 0 and alpha > 0. As the case K = 0 has been considered in [25], it is shown in the present paper that the corresponding initial-boundary problem with K not equal 0 admits at least one global weak solution if alpha >= 3/7. (C) 2017 Elsevier Inc. All rights reserved.