Boundedness for a 3D chemotaxis-Stokes system with porous medium diffusion and tensor-valued chemotactic sensitivity

This paper deals with the following chemotaxis-Stokes system {n(t) + u center dot del(n) = Delta n(m) - del center dot (nS (x, n, c) center dot del c), x is an element of Omega, t > 0, c(t) + u center dot del(c) = Delta c - nf(c), x is an element of Omega, t > 0, u(t) = Delta u + del P + n del phi, x is an element of Omega, t > 0, del center dot u = 0, x is an element of Omega, t > 0 under no-flux boundary conditions in a bounded domain Omega subset of R-3 with smooth boundary, where m >= 1, phi is an element of W-1,W-infinity(Omega), f and S are given functions with values in [0, infinity) and R-3x3, respectively. Here S satisfies vertical bar S (x, n, c)vertical bar < S-0(c)(1 + n)(-alpha) with alpha >= 0 and some nonnegative nondecreasing function S (0). With the tensor-valued sensitivity S, this system does not possess energy-type functionals which seem to be available only when S is a scalar function. We can establish a priori estimation to overcome this difficulty and explore a relationship between m and alpha, i.e., m + alpha > 7/6, which insures the global existence of bounded weak solution. Our result covers completely and improves the recent result by Wang and Cao (Discrete Contin Dyn Syst Ser B 20:3235-3254, 2015) which asserts, just in the case m = 1, the global existence of solutions, but without boundedness, and that by Winkler (Calc Var Partial Differ Equ 54:3789-3828, 2015) which only involves the case of alpha = 0 and requires the convexity of the domain.