Suppression of Blow-up in Two-dimensional Keller--Segel Systems by Stochastic Couette Flows

Abstract: We study Keller--Segel systems on $\mathbb R<sup>2$ advected by a stochastic Couette flow. We prove that sufficiently strong mixing suppresses chemotactic blow-up almost surely, yielding a unique global mild solution for initial data of arbitrary mass. To prove this result, we construct maximal local mild solutions using a-priori estimates for the random solution operator of passive scalar transport equations and a fixed-point argument. Global existence then follows from a decomposition of time into blocks, a bootstrap argument and the first Borel--Cantelli lemma.