Weak solution of the two-dimensional Patlak-Keller-Segel-Navier-Stokes system with logistic source

In this paper, we consider a coupled fully parabolic Patlak-Keller-Segel-Navier-Stokes system with logistic source in $ \mathbb{R}^2 $, which is driven by friction force. The global existence and uniqueness of weak solution for this system is established by the classical energy method combined with compactness criteria and the damping effect of the logistic source.