Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space

TL;DR

This paper demonstrates that Couette flow can suppress blow-up in 3-D Keller-Segel models by enhancing dissipation, leading to global solutions in the whole space, a significant departure from previous periodic cases.

Contribution

It introduces a new Green's function method to precisely describe enhanced dissipation and proves global existence for 3-D Keller-Segel models with Couette flow in the whole space.

Findings

Enhanced dissipation exists for all frequencies in whole space

Couette flow suppresses blow-up in 3-D Keller-Segel models

New methodology captures dissipation enhancement

Abstract

In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with Couette flow in $\mathbb{R}^3$. We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable $x$ in [2,10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism.