Suppression of blow-up in parabolic–parabolic Patlak–Keller–Segel via strictly monotone shear flows

Abstract

In this paper we consider the parabolic–parabolic Patlak–Keller–Segel models in with advection by a large strictly monotone shear flow. Without the shear flow, the model is L1 critical in two dimensions with critical mass : solutions with mass less than are global in time and there exist solutions with mass larger than which blow up in finite time (Schweyer 2014 (arXiv:1403.4975)). We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In contrast with the parabolic-elliptic case (Bedrossian and He 2016 SIAM J. Math. Anal. 49 4722–66), the strong shear flow has destabilizing effect in addition to the enhanced dissipation effect, which makes the problem more difficult.