Enhanced dissipation of Poiseuille flow in passive scalar equation and application to suppressing blow-up in a two-dimensional Patlak-Keller-Segel system

In this paper, we first apply the method of hypocoercivity with a time-dependent weight, introduced by Wei and Zhang (Sci. China Math. 62 (2019): 1219-1232), to establish a semigroup estimate for the passive scalar equation advected by a Poiseuille flow $ (Ay^2, 0)^\top $ in $ \mathbb{T} \times \mathbb{R} $. Our result shows that the $ x $-dependent modes of solutions to this linear problem decay at a rate proportional to $ \mathcal{O}(A^{1/2}) $, which is faster than the heat dissipation time scale $ \mathcal{O}(A) $, a phenomenon known as enhanced dissipation.This result is then applied to the classical parabolic-elliptic Patlak-Keller-Segel (PKS) system, transported by the passive Poiseuille flow $ (Ay^2, 0)^\top $, to show that the solution to the PKS system remains global in time for any large initial mass, provided the Poiseuille flow is sufficiently strong, i.e., the amplitude $ A $ is suitably large. The key decay estimates for the $ x $-dependent modes of the PKS system are derived by adapting the approach of Li, Wei, and Zhang (Comm. Pure Appl. Math. 73 (2020): 465-557).