Analyticity and Existence of the Keller-Segel-Navier-Stokes Equations in Critical Besov Spaces

This paper deals with the Cauchy problem to the Keller-Segel model coupled with the incompressible 3D) Navier-Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant (C) over tilde such that the initial data (u(0), n(0) c(0)) : = (u(0)(h),u(0)(3)n(0),c(0)) satisfy (C) over tilde(parallel to(n(0), c(0))parallel to)(B)over dot(q, 1)(2 vertical bar 3/q) (R-3) x (B)over dot(q,1)(3/q) (R-3) + parallel to u(0)(h)parallel to(B) over dot (1 vertical bar 3/p)(p,1) (R-3) + parallel to u(0)(h)parallel to(B) over dot(p,1)(-1+3/)p (R-3)(alpha) parallel to u(0)(3)parallel to (B) over dot(p,1)(-1+3/)p (R-3)(1-alpha)) <= 1 for certain conditions on p, q and alpha implies the global existence of solutions with large initial verticalvelocity component.