Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow

Abstract

In this paper, we investigate the two-dimensional Patlak-Keller-Segel-Navier-Stokes system perturbed around the Poiseuille flow $(Ay^2,0)^{\top}$ and show that the solutions to this system are global in time if the Poiseuille flow is sufficiently strong in the sense of amplitude A large enough. This seems to be the first result showing that the Poiseuille flow can suppress the chemotactic blow-up of the solution to chemotaxis-fluid system. Our proof will be based on a weighted energy method together with the linear enhanced dissipation established by Coti Zelati-Elgindi-Widmayer (Comm. Math. Phys. 378 (2020) 987-1010).