Time-dependent Flows and Their Applications in Parabolic-parabolic Patlak-Keller-Segel Systems Part I: Alternating Flows

TL;DR

This paper demonstrates that specific time-dependent alternating shear flows can prevent finite-time blow-up in three-dimensional parabolic-parabolic Patlak-Keller-Segel systems, which are otherwise super-critical and prone to blow-up.

Contribution

It introduces a novel class of time-dependent flows inspired by Elgindi's ideas that suppress chemotactic blow-up in super-critical PKS systems.

Findings

Alternating shear flows prevent blow-up in 3D PKS systems.

Flow design inspired by Tarek Elgindi's methods.

Suppression of finite-time blow-up with specific flows.

Abstract

We consider the three-dimensional parabolic-parabolic Patlak-Keller-Segel equations (PKS) subject to ambient flows. Without the ambient fluid flow, the equation is super-critical in three-dimension and has finite-time blow-up solutions with arbitrarily small $L^1$-mass. In this study, we show that a family of time-dependent alternating shear flows, inspired by the clever ideas of Tarek Elgindi, can suppress the chemotactic blow-up in these systems.