Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities

The coupled chemotaxis fluid system {n(t) = Delta n - del.(nS(x, n, c) . del c) - u . del n, (x, t) is an element of Omega x (0, T), c(t) = Delta c - nc u . V c, (x, t) is an element of Omega X (0, T), u(t) = Delta u - (u . del)u + del P + n del Phi, del . u = 0, (x, t) is an element of Omega X (0, T), del c . v = (del n - nS(x,n,c) . Vc) . v = 0, u = 0, (x, t) is an element of Omega x (0, T), n(x, 0) = n(0) (x), c(x, 0) = c(0)(x), u (x, 0) = u(0)(x), x is an element of Omega, where S is an element of (C-2 ((Omega) over bar x [0, infinity)(2)))(N x N) is considered in a bounded domain Omega subset of R-N, N is an element of [2, 3), with smooth boundary. We show that it has global classical solutions if the initial data satisfy certain smallness conditions and give decay properties of these solutions.