Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled Keller-Segel-Stokes system {n(t) + u . del n = Delta n - del . (n S(x, n, c) . del c), (x, t) is an element of Omega x (0, T), c(t) + u . del c = Delta c - c + n, (x, t) is an element of Omega x (0, T), u(t) + del P = Delta u + n del phi, (x, t) is an element of Omega x (0, T), del. u = 0, (x, t) is an element of Omega x (0, T). Here, one of the novelties is that the chemotactic sensitivity 8 is not a scalar function but rather attains values in R-2x2, and satisfies vertical bar S (x, n, c)vertical bar <= C-S(1 + n)(-alpha) with some C-S > 0 and alpha > 0. We shall establish the existence of global bounded classical solutions for arbitrarily large initial data. In contrast to the corresponding case of scalar-valued sensitivities, this system does not possess any gradient-like structure due to the appearance of such matrix-valued S. To overcome this difficulty, we will derive a series of a priori estimates involving a new interpolation inequality. To the best of our knowledge, this is the first result on global existence and boundedness in a Keller-Segel-Stokes system with tensor-valued sensitivity, in which production of the chemical signal is involved. (C) 2015 Elsevier Inc. All rights reserved.