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 ) ∈ 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)×Ḃ_p,∞^-2+3/p(ℝ^3)×(Ḃ_q,∞^3/q(ℝ^3)∩ L^∞(ℝ^3)) . Furthermore, we obtain that the system exists a unique global solution for any small initial data (u_0,n_0,c_0)∈BMO^-1(ℝ^3)×Ḃ_p,∞^-2+3/p(ℝ^3)×(Ḃ_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.