Global boundedness of weak solutions for a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and rotation

This paper deals with the chemotaxis-Stokes system with nonlinear diffusion and rotation: n(t) + u . del n = Delta n(m) - del . (nS(x,n, c) . del c), c(t) + u . del c = Delta c - nc, u(t) + del P = Delta u + n del phi + f(x, t) and del . u = 0, in a bounded domain Omega subset of R-3, where m > 0, and phi : (Omega) over bar -> R, f : (Omega) over bar x [0, infinity) -> R-3 and S : (Omega) over bar x [0, infinity)(2) -> R-3x3 are given sufficiently smooth functions such that f is bounded in Omega x (0, infinity) and S satisfies vertical bar S(x, n, c)vertical bar <= S-0(c)(1 + n)(-alpha) for all (x, n, c) is an element of (Omega) over bar x [0, infinity)(2) with alpha > 0 and some nondecreasing function S-0: [0, infinity) -> [0, infinity). It is shown that if m + alpha > 10/9 and m + 5/4 alpha > 9/8, then for any reasonably smooth initial data, the corresponding Neumann-Neumann-Dirichlet initial-boundary problem possesses a globally bounded weak solution. This extends the previous global boundedness result for m > 9/8 and alpha = 0 [43] , and improves that for m >= 1 and m + alpha > 7/6 [34], or for m + alpha > 7/6 in the associated fluid-free system [31]. Our proof consists at its core in using, inter cilia, the maximal Sobolev regularity theory to elaborately derive some spatio-temporal estimates for the signal and the fluid equations so as to decouple the system. (C) 2019 Elsevier Inc. All rights reserved.