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

This paper studies the two-dimensional Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> R² 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.