On the $8\pi$-subcritical mass threshold of a Patlak-Keller-Segel-Navier-Stokes system

In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $\mathbb{R}^2$, we proved that if the total mass of the cells is strictly less than $8\pi$, then classical solutions exist for any finite time and their $H^s$-Sobolev norms are almost uniformly bounded in time. On the torus $\mathbb{T}^2$, we proved that under the $8\pi$ subcritical mass constraint, the solutions are uniformly bounded in time.