Global existence and decay rates to a self-consistent chemotaxis-fluid system

In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid: {partial derivative(t)n + u.del n = Delta n - del.(chi(c)n del c) + del.(n del phi), partial derivative(t)c + u.del c = Delta c - nf(c), partial derivative(t)u + kappa(u . del)u + del P = Delta u - n del phi + chi(c)n del c, del . u = 0 in R-d x (0, T) (d = 2, 3). One of the novelties and difficulties here is that the coupling in this model is stronger and more nonlinear than the most-studied chemotaxis-fluid model due to the additional term chi(c)n del c in the third equation. We will first establish several extensibility criteria of classical solutions, which ensure us to extend the local solutions to global ones in the three dimensional chemotaxis-Stokes case and in the two dimensional chemotaxis-Navier-Stokes version under suitable smallness assumption on parallel to c(0)parallel to(L infinity) with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential del phi at infinity. As a byproduct of the entropy functional inequality, we also establish the global-in-time existence of weak solutions to the three dimensional chemotaxis-Navier-Stokes system. To the best of our knowledge, this seems to be the first work addressing the global well-posedness and decay property of solutions to the Cauchy problem of self-consistent chemotaxis-fluid system.