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

TL;DR

This paper demonstrates that a sufficiently large strictly monotone shear flow can suppress blow-up in the parabolic-parabolic Patlak-Keller-Segel model on a torus cross real line, extending understanding of flow effects on chemotactic aggregation.

Contribution

It shows that strong shear flows can prevent blow-up in the parabolic-parabolic Keller-Segel model, highlighting the interplay between advection and nonlinear aggregation.

Findings

Shear flow suppresses one dimension of the dynamics.

Large shear flow prevents finite-time blow-up.

Destabilizing effects of shear are identified alongside enhanced dissipation.

Abstract

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.