ABSTRACT The purpose of this paper is devoted to investigating a fractional parabolic‐elliptic chemotaxis system coupled with the Navier–Stokes equation in the whole space . We show the existence of a global mild solution to system (1.1) for small initial data in the scale invariant class demonstrating that and . Furthermore, by studying moments of lower order , we establish a blow‐up criterion of solution to system (1.1) with the help of the proof by contradiction.
Jiang et al. (Tue,) studied this question.