A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity

We consider the three-dimensional chemotaxis-Stokes system { nt + u center dot Delta n = Delta n - del center dot(nS(x,n,c)center dot del c), x epsilon Omega, t>0 c(t) + u center dot Delta c = Delta c - nc, x epsilon Omega, t>0, u(t) + del P = Delta u +n del phi, x epsilon Omega, t>0 del center dot u = 0, x epsilon Omega, t>0 under no-flux boundary conditions in a bounded domain Omega C R-3 with smooth boundary. Here the tensor-valued sensitivity S fulfills vertical bar S(x, n, c)vertical bar <= C-s(1 + n)(-alpha) with some C-s > 0 and a > 0. It has been shown in Wang-Pang-Li (2016) that if alpha > 1/6, this system possesses a global bounded classical solution for any given initial data. We shall improve this result to the case of arbitrary alpha > 0, and thus solve the problem left in Wang-Cao (2015). (C) 2018 Elsevier Ltd. All rights reserved.