Global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass

We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where are attracted by chemical substance transported ambient velocity, flow is forced friction induced cells. The main result paper to show global existence free-energy solutions 2D with critical subcritical mass.