Global solutions to Keller‐Segel‐Navier‐Stokes equations with a class of large initial data in critical Besov spaces

In this article, we consider the Cauchy problem to Keller‐Segel equations coupled to the incompressible Navier‐Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u F := e t Δ u 0 ; we prove that there exist 2 positive constants σ 0 and C 0 such that if the gravitational potential ϕ ∈B ˙p , 1 3 / p ( R 3 ) and the initial data ( u 0 , n 0 , c 0 ) satisfyu F · ∇ u FL 1 ( R + ;B ˙p , 1 − 1 + 3 / p ( R 3 ) ) +n 0 , c 0B ˙q , 1 − 2 + 3 / q ( R 3 ) ×B ˙q , 1 3 / q ( R 3 )× expC 0u 0B ˙p , 1 − 1 + 3 / p ( R 3 ) + 12⩽ σ 0for some p , q with 1 ⩽ p , q < ∞ , 1 p + 1 q > 1 3 , 1 ⩽ q < 6 and 1 min { p , q } − 1 max { p , q } ⩽ 1 3 , then the global solutions can be established in critical Besov spaces.