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(u0,n0,c0)∈Φ_(01)^(1)×Φ_(01)^(2)×Φ_(01)^(3),where Φ_(01)^(1)×Φ_(01)^(2)×Φ_(01)^(3) is a subspace of bmo^(−1)(R^(3))×B˙_(p,∞)^(−2+3/p) (R^(3))×(B˙_(q,∞)^(3/q) (R^(3))∩L^(∞)(R^(3))).Furthermore,we obtain that the system exists a unique global solution for any small initial data(u0,n0,c0)∈BMO^(−1)(R^(3))×B˙_(p,∞)^(−2+3/p)(R^(3))×(B_(q,∞)^(3/q)(R^(3))∩L^(∞)(R^(3))).For the difference between these spaces and known ones,our results may be regarded as a new existence theorem on the system.