Stabilization in a two‐dimensional fractional chemotaxis‐Navier–Stokes system with logistic source

Abstract

This paper investigates a chemotaxis‐Navier–Stokes system with logistic source and a fractional diffusion of order on the two‐dimensional periodic torus . It is proved that if and the initial data satisfy some regular conditions, the considered problem admits a global classical solution. Through constructing a new Lyapunov functional different from the previous literature, we can obtain the following asymptotic stabilization: If , the solution components and satisfy If and , where is a constant determined in (6.24), the solution components and satisfy In either case, the solution component satisfies