A Note on the Global Existence in a Fully Parabolic Patlak-Keller-Segel-Navier-Stokes System

Abstract

This paper is concerned with a fully parabolic Patlak-Keller-Segel-Navier-Stokes system

where $Ω ⊂\mathbb{R}^2$ is a smoothly bounded domain and the parameter $χ$ is positive. The main aim of this note is to show that if

then the solution of the above system is global and bounded in time.