Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

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 . del(C), (x, t) is an element of Omega x (0, T), u(t) = Delta u - k(u . del)u + del P + 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), (*) is considered under the no-flux boundary conditions for n, c and the Dirichlet boundary condition for u on a bounded smooth domain Omega subset of R-N (N = 2,3), k is an element of{0,1}. We assume that S(x, n, c) is a matrix-valued sensitivity under a mild assumption such that vertical bar S(x, n, c)vertical bar < S-0(c(0)) with some non-decreasing function S-0 is an element of C-2((0, infinity)). It contrasts with the related scalar sensitivity case that (*) does not possess the natural gradient-like functional structure. Associated estimates based on the natural functional seem no longer available. In the present work, a global classical solution is constructed under a smallness assumption on parallel to C-0 parallel to(L infinity(Omega)) and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization. (C) 2016 Elsevier Inc. All rights reserved.