Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces

In this article, we consider the Cauchy problem to Keller-Segel equations coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u(F) := e(t del)u(0); we prove that there exist 2 positive constants sigma(0) and C-0 such that if the gravitational potential phi is an element of(B) over dot(p,1)(3/p) (R-3) and the initial data (u(0), n(0), c(0)) satisfy (parallel to u(F) . del u(F)parallel to(L1(R+;(B) over dotp,1-1+3/p(R3))) + parallel to(n(0), c(0))parallel to((B) over dotq,1-2+3/q (R3)x(B) over dotq,13/q) (R-3)) x exp {C-0 (parallel to u(0)parallel to B-p,B-1-1+3/p (R3) + 1)(2)} <= sigma(0) for some p,q with 1 <= p, q < infinity, 1/p + 1/q > 1/3, 1 <= q < 6 and 1/min{p, q} - 1/max{p,q} <= 1/3, then the global solutions can be established in critical Besov spaces.