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

TL;DR

This paper demonstrates that strong shear flows can prevent blow-up in the Patlak-Keller-Segel model, making solutions globally well-behaved in both two and three dimensions by effectively reducing the model's criticality.

Contribution

It shows that sufficiently large shear flows suppress blow-up in the Patlak-Keller-Segel model, altering its criticality and ensuring global solutions in 2D and 3D.

Findings

Shear flow suppresses blow-up in 2D, ensuring global solutions.

In 3D, shear flow reduces effective criticality, preventing blow-up for subcritical mass.

Shear flow transforms the problem into a subcritical or critical regime, depending on dimension.

Abstract

In this paper we consider the parabolic-elliptic Patlak-Keller-Segel models in $\mathbb T^d$ with $d=2,3$ with the additional effect of advection by a large 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 and solutions with mass larger than $8 \pi$ with finite second moment, all blow up in finite time. In three dimensions, the model is $L^{3/2}$ critical and $L^1$ supercritical; there exists solutions with arbitrarily small mass which blow up in finite time arbitrarily fast. 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 two dimensions, the problem becomes effectively $L^1$ subcritical and so all solutions are global in time (if the shear flow is chosen large). In three dimensions, the problem is effectively $L^1$ critical, and solutions with mass less than $8\pi$ are global in time (and for all mass larger than $8\pi$, there exists solutions which blow up in finite time).