Stability of a class of supercritical volume-filling chemotaxis-fluid model near Couette flow

Consider a class of chemotaxis-fluid model incorporating a volume-filling effect in the sense of Painter and Hillen [Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10(4) (2002) 501–543], which is a supercritical parabolic-elliptic Keller–Segel system. As shown by Winkler et al., for any given mass, there exists a corresponding solution of the same mass that blows up in either finite or infinite time. In this paper, we investigate the stability properties of the two dimensional Patlak–Keller–Segel-type chemotaxis-fluid model near the Couette flow [Formula: see text] in [Formula: see text] and show that the solutions are global in time as long as the initial cell mass [Formula: see text] and the shear flow is sufficiently strong ([Formula: see text] is large enough).