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

TL;DR

This paper studies how shear flows influence the behavior of 3D parabolic-parabolic PKS systems, showing that strong flows can prevent blow-up when cell mass is below a threshold, despite rapid chemical gradient formation.

Contribution

It introduces analysis of shear flow effects on 3D PKS systems, revealing conditions for global regularity under strong flow and low cell mass.

Findings

Global regularity maintained with strong shear flows below mass threshold.

Strong shear flows cause rapid chemical gradient formation.

Flow strength can prevent blow-up in PKS systems.

Abstract

In this study, we investigate the behavior of three-dimensional parabolic-parabolic Patlak-Keller-Segel (PKS) systems in the presence of ambient shear flows. Our findings demonstrate that when the total mass of the cell density is below a specific threshold, the solution remains globally regular as long as the flow is sufficiently strong. The primary difficulty in our analysis stems from the fast creation of chemical gradients due to strong shear advection.