GLOBAL SOLUTIONS TO CHEMOTAXIS-NAVIER-STOKES EQUATIONS IN CRITICAL BESOV SPACES

In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential phi and the small initial data (u(0), n(0,) c(0)) in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants sigma(0) and C-0 such that if the gravitational potential phi is an element of<(B)single over dot>(3/p)(p,1) (R-3) and the initial data (u(0), n(0), c(0)) := (u(0)(1), u(0)(2), u(0)(3), n(0), c(0)) := (u(0)(h), u(0)(3), n(0), c(0)) satisfies (parallel to u(0)(h)parallel to(<(B)single over dot>p, 1-1+3/p(R3)) + parallel to(n(0), c(0))parallel to(<(B)single over dot>q,1-2+3/q(R3) x <(B)single over dot>q, 1 3/q (R3)) ) x exp {C-0 (parallel to u(0)(3)parallel to(<(B)single over dot>p, 1-1+3/p(R3)) + 1)(2)} <= sigma(0) for some p, q with 1 < p, q < 6, 1/p + 1/q > 2/3 and 1/min{p,q} - 1/max{p, q} <= 1/3, then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field u(0)(3) in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system.