Global bounded solutions of the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow and logistic source

Abstract

As is well-known, the solutions to the Patlak-Keller-Segel system in 3D may blow up in finite time regardless of any initial cell mass. In this paper, we investigate the existence of global bounded solutions of the 3D Patlak-Keller-Segel-Navier-Stokes system with a large initial cell mass via Couette flow or logistic source in a finite channel. On the one hand, it is proved that as long as the Couette flow is strong enough and the initial velocity is small, the bounded solutions of the system are global in time without any limitation on the initial mass M if the logistic source term exists. On the other hand, if the logistic source term vanishes, it is proved that, as long as the Couette flow is strong enough and the initial velocity is small, the solutions are global in time under the mass condition M<8π9.