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

In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in $\mathbb{T}\times\mathbb{R}$ with advection by a large strictly monotone shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8\pi$: solutions with mass less than $8\pi$ are global in time and there exist solutions with mass larger than $8 \pi$ which blow up in finite time \cite{Schweyer14}. 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 \cite{BedrossianHe16}, the strong shear flow has destabilizing effect in addition to the enhanced dissipation effect, which make the problem more difficult.