GLOBAL EXISTENCE OF WEAK SOLUTION IN A CHEMOTAXIS-FLUID SYSTEM WITH NONLINEAR DIFFUSION AND ROTATIONAL FLUX

In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by {n(t) + u .del n = Delta n(m) - del . (uS(x, n, c) . del c), x is an element of Omega, t > 0, c(t) + u . del c = Delta c - c + n , x is an element of Omega, t > 0, u(t) + k(u . del)u = Delta u + del p + n del phi, x is an element of Omega, t > 0 del . u = 0, x is an element of Omega, t > 0 in a bounded domain Omega subset of R-3, where k is an element of R, phi is an element of W-2,W-infinity(Omega) and the given tensor-valued function S: (Omega) over bar x [0, infinity)(2)-> R-3x3 satisfies vertical bar S(x, n, c)vertical bar <= S-0(n+1)(-alpha) for all x is an element of R-3, n >= 0, c >= 0. Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming m + alpha > 4/3 and m > 1/3.