Global well-posedness for the 2-D Patlak-Keller-Segel-Navier-Stokes system

In order to find out about the global well-posedness of the two-dimensional fluid-coupled Patlak-Keller-Segel-Navier-Stokes system in R ², some inequalities with monotonicity are constructed, and by combined use of Gronwall inequality, interpolation inequality,Hölder inequality, etc., it is worked out that the said system has a global existence of mild solution when the cell initial mass M=∫ _R² n ₀( x)d x≤8π.