An Application of BMO-type Space to Chemotaxis-fluid Equations

We consider a Keller-Segel model coupled to the incompressible Navier-Stokes system in 3-dimensional case. We prove that the system has a unique local solution when (u(0), n(0), c(0)) is an element of Phi(1)(01) x Phi(2)(01) x Phi(3)(01) x where Phi(1)(01) x Phi(2)(01) x Phi(3)(01) is a subspace of bmo(-1)(R-3)x(B) over dot(p,infinity)(-2+3/p) (R-3)x((B) over dot(p,infinity)(3/q) (R-3)boolean AND L-infinity (R-3)). Furthermore, we obtain that the system exists a unique global solution for any small initial data (u(0), n(0), c(0)) is an element of BMO-1(R-3) x . (B) over dot(p,infinity)(2+3/p) (R-3) x ((B) over dot(p,infinity)(3/q)(R-3) boolean AND L-infinity (R-3)). For the difference between these spaces and known ones, our results may be regarded as a new existence theorem on the system.