Suppression of blow-up in Patlak–Keller–Segel–Navier–Stokes system via the Couette flow in whole space

Abstract This paper studies the two-dimensional Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in R 2 ${\mathbb{R}}^{2}$ near the Couette flow (Ay, 0). Using the Green’s function method, we first derive enhanced dissipation estimates for the linearized system around this background flow. As a key application, we prove that solutions remain globally regular provided the shear rate A is sufficiently large.